Chains Homotopy in the Complement of a Knot in the Sphere $S^3$

Abstract

If $\gamma$ is a knot in $S^3 \cong \mathbb {H}^2 _{\mathbb {C}} \subset \mathbb {P}_{\mathbb {C}}^2$, then the set $\Lambda (\gamma )\subset \mathbb {P}_{\mathbb {C}}^2$ is defined as the union of all the complex lines tangent to $\partial \mathbb {H}^2 _{\mathbb {C}}$ at points in the image of $\gamma$. The following result is obtained: the number of components of $\Omega (\gamma )=\mathbb {P}_{\mathbb {C}}^2 {\setminus } \Lambda (\gamma )$ is greater or equal to the number of distinct integers in the set $\{\ell (\gamma , C): C \text { is a positively oriented chain disjoint to } \gamma \}$, where $\ell (\gamma , C)$ denotes the linking number between $\gamma$ and C.

1 Introduction

The results discussed in this paper arise as a natural problem in the theory of complex Kleinian groups: finding the number of components of the domain of discontinuity of a discrete subgroup $G \subset \text {PU}(2,1)$ acting on $\mathbb {P}_{\mathbb {C}}^2$. The domain of discontinuity, denoted $\Omega (G)$, is the complement in $\mathbb {P}_{\mathbb {C}}^2$ of the Kulkarni limit set denoted $\Lambda (G)$. This limit set $\Lambda (G)$ is obtained as the union of all the complex lines tangent to $\partial \mathbb {H}^2 _{\mathbb {C}} \cong S^3$, the boundary 3-sphere of the complex hyperbolic space, at points in the set $L(G) \subset \partial \mathbb {H}^2 _{\mathbb {C}}$, where L(G) is the set of accumulation points of the orbit of some (any) point $p \in \overline{\mathbb {H}^2 _{\mathbb {C}}}$ (see Navarrete 2006). In some cases, L(G) happens to be a simple closed curve.

In this paper we study the related problem of finding the number of components of an open set, $\Omega (\gamma )$, associated to a knot $\gamma :S^1 \rightarrow S^3\cong \partial \mathbb {H}^2 _{\mathbb {C}}$ in the following way: Define $\Lambda (\gamma ):= \bigcup _{p \in \gamma (S^1)} \ell _p$ where $\ell _p$ is the only complex line tangent to $\partial \mathbb {H}^2 _{\mathbb {C}}$ at the point p. The set $\Omega (\gamma )$ is the complement of $\Lambda (\gamma )$ in $\mathbb {P}_{\mathbb {C}}^2$.

In this paper we will be particularly interested on those curves obtained as the intersection of a complex projective line in $\mathbb {P}_{\mathbb {C}}^2$ and the boundary at infinity of the complex hyperbolic space, $\partial \mathbb {H}^2 _{\mathbb {C}}$. These curves are called chains and there is a natural bijection between the complement in $\mathbb {P}_{\mathbb {C}}^2$ of the set $\overline{\mathbb {H}^2 _{\mathbb {C}}}:=\mathbb {H}^2 _{\mathbb {C}} \cup \partial \mathbb {H}^2 _{\mathbb {C}}$ and the set of non degenerate chains (chains which are not one single point). This bijection is given in the following way: $p\in \mathbb {P}_{\mathbb {C}}^2 {\setminus } (\overline{\mathbb {H}^2 _{\mathbb {C}}})$ corresponds to its polar chain$C_p$ (see Sect. 3), and we obtain the following results:

Theorem 1.1

Let us assume that $p_0, p_1$ are two points in $\Omega (\gamma ) \cap (\mathbb {P}_{\mathbb {C}}^2 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}})$ and $C_{p_0}$, $C_{p_1}$ are the corresponding non degenerate polar chains. The points $p_0$ and $p_1$ can be joined by a continuous path $\nu : I \rightarrow \Omega (\gamma ) \cap (\mathbb {P}_{\mathbb {C}}^2 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}})$ if and only if there exists a continuous map

$$\begin{aligned} H:I\times S^1 \rightarrow \partial \mathbb {H}^2 _{\mathbb {C}} \end{aligned}$$

such that $H(s,\cdot ): S^1 \rightarrow \partial \mathbb {H}^2 _{\mathbb {C}}$ is a positive parametrization of the polar chain to $\nu (s)$, denoted $C_{\nu (s)}$, and $C_{\nu (s)} \cap \gamma (I) = \varnothing$ for every $s \in I$.

Corollary 1.2

If p, q are two points in $\Omega (\gamma ) \cap (\mathbb {P}_{\mathbb {C}}^2 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}})$ and the linking number between the curve $\gamma$ and the (positively oriented) chain $C_p$, polar to p, is different to the linking number between $\gamma$ and the (positively oriented) chain $C_q$, polar to q, then p and q lie in distinct components of $\Omega (\gamma )$. In particular, the number of all possible linking numbers $\ell (\gamma , C)$ where C is any chain not intersecting the image of $\gamma$ is smaller or equal to the number of components of $\Omega (\gamma )$.

Corollary 1.3

Let us assume that the image of the smooth curve $\gamma : S^1 \rightarrow \partial \mathbb {H}^2 _{\mathbb {C}}\cong S^3$ is the intersection of $S^3$ and the algebraic curve $f(x,y)=0$ with an isolated singularity at the origin (0, 0). Moreover, assume that the intersection of the unit ball in $\mathbb {C}^2$ and $f^{-1}(0)$ is a topological disk, and $\gamma$ has the orientation induced by the boundary of this set. If $\sigma : S^1 \rightarrow \partial \mathbb {H}^2 _{\mathbb {C}} \cong S^3$ is a positive parametrization of a chain C not meeting $S^3 \cap f^{-1}(0)$, then

$$\begin{aligned} \ell (\gamma , C)= & {} \frac{1}{2\pi i}\int _{F\circ \sigma }\frac{dz}{z}, \end{aligned}$$

(1)

where $F:(S^3{\setminus } f^{-1}(0)) \rightarrow S^1$, $F(x,y)=\frac{f(x,y)}{|f(x,y)|}$ is the Milnor fibration. In particular, the number of all possible values of the integrals (1), where $\sigma$ runs over all positive parametrizations of chains not meeting $f^{-1}(0)$, is smaller or equal to the number of components of $\Omega (\gamma )$.

Theorem 1.4

Let $\gamma$ be the trefoil knot obtained as the intersection of the algebraic curve $\{(x,y) \in \mathbb {C}^2 : y^2=x^3\}$ and $S^3$:

$$\begin{aligned}&\displaystyle \gamma : S^1 \rightarrow \partial \mathbb {H}^2 _{\mathbb {C}} \cong S^3\\&\displaystyle \gamma (e^{2\pi it})=[(\sqrt{\alpha }e^{2 \pi i t})^2: (\sqrt{\alpha }e^{2 \pi i t})^3:1], \end{aligned}$$

where $\alpha$ denotes the positive real root of the equation $x^3+x^2-1=0$.

• (i) The set $\Omega (\gamma )$ has four connected components, $\Omega _{j}$, for $j=0,1,2,3$ where $\Omega _j$ consists of those points $[x:y:z] \in \mathbb {P}_{\mathbb {C}}^2$ such that the polynomial in the $\zeta$ variable given by $p(\zeta )=\bar{y} \zeta ^3+\bar{x} \zeta ^2-\bar{z}$ has precisely j roots within the open disk $D(0, \sqrt{\alpha })\subset \mathbb {C}$, and there is no root with $|\zeta |^2 = \alpha$.

• (ii) The set

  $$\begin{aligned} \Lambda (\gamma ) {\setminus } S(\gamma ) \end{aligned}$$

  is the union of three solid tori, where

  $$\begin{aligned} \text {S}(\gamma )=\{q \in \Lambda (\gamma ) : \{q\}= \ell _{p_1} \cap \ell _{p_2} \quad \text { for some }\quad p_1, p_2 \in \gamma (I), p_1 \ne p_2 \} \end{aligned}$$

  is the subset of $\Lambda (\gamma )$ consisting of all the points of intersection of any two distinct tangent lines to $\partial \mathbb {H}^2 _{\mathbb {C}}$, at points in the image of $\gamma$.

The Theorem 1.1 says, roughly speaking, that a continuous path in $\Omega (\gamma ) \cap (\mathbb {P}_{\mathbb {C}}^2 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}})$ corresponds to a continuous homotopy of “positive” chains not meeting the knot $\gamma$. The main idea in the proof of Theorem 1.1 is the continuous correspondence between a point in $\mathbb {P}_{\mathbb {C}}^2$ and its polar chain with a suitable parametrization, and this idea is formalized in Lemma 3.2 and Proposition 3.3. All the results obtained in this paper are consequences of Theorem 1.1 in some way.

The Corollary 1.2 gives a “practical” way to estimate the number of components of $\Omega (\gamma )$ by means of computing all the possible linking numbers of chains not meeting $\gamma$. An outline for its proof is the following: The diameter of the chain polar to a point in $\mathbb {P}_{\mathbb {C}}^2{\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}$ goes to zero as the point approaches to the boundary at infinity $\partial \mathbb {H}^2 _{\mathbb {C}}$ ( see Lemma 3.4). Hence it is natural to think on the component of $\Omega (\gamma )$ containing $\mathbb {H}^2 _{\mathbb {C}}$ as the points in $\mathbb {H}^2 _{\mathbb {C}}$ and the null-homotopic chains in $\partial \mathbb {H}^2 _{\mathbb {C}} {\setminus } \gamma (S^1)$, including the degenerate chains in $\partial \mathbb {H}^2 _{\mathbb {C}} {\setminus } \gamma (S^1)$ (see Lemma 3.6). Since the linking number is invariant under homotopy, Theorem 1.1 and Lemma 3.6 can be used to prove Corollary 1.2.

The Corollary 1.3 is a restatement of Corollary 1.2 for the case when $\gamma$ can be obtained as the intersection of an algebraic curve and $\partial \mathbb {H}^2 _{\mathbb {C}}$. In this case, the linking numbers between the knot and the chains not meeting the knot can be computed as a winding number using the Milnor’s fibration. For a detailed treatment on Milnor’s fibration see Milnor (1968) and Seade (2007).

Theorem 1.4 computes the number of components of $\Omega (\gamma )$ for the particular case when $\gamma$ is a parametrization of the trefoil knot obtained as an intersection of the algebraic curve $y^2=x^3$ and the 3-sphere $\partial \mathbb {H}^2 _{\mathbb {C}}$. Moreover, it gives a topological description of the set $\Lambda (\gamma )$.

The points in $\Lambda (\gamma )$ can be characterized as those points [x : y : z] whose polar line passes through a point in the knot, and this condition can be translated to the existence of a root of the equation in the $\zeta$-variable, $p(\zeta )=\bar{y}\zeta ^3+\bar{x}\zeta ^2-\bar{z}=0$, in a fixed circle (of center $0\in \mathbb {C}$ and radius $\sqrt{\alpha }$). This gives a natural partition of $\Omega (\gamma )$ into four sets $\Omega _j$, $j=0,1,2,3$ as described in Theorem 1.4. It is proved in Sect. 5.1 that each $\Omega _j$ is connected and that the linking number $\ell (\gamma , C _p)$ is equal to j for every $p \in \Omega _j {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}$. Thus the proof of Theorem 1.4 i) follows from Corollary 1.2.

The proof of Theorem 1.4 (ii) is included in the Sect. 5.2. A brief outline is the following: We fix a point $p_0$ in the image of $\gamma$ and we notice that the points lying in $\ell _{p_0}$ (the complex line tangent to $\partial \mathbb {H}^2 _{\mathbb {C}}$ at $p_0$) and the “singular set” $S(\gamma )$, form a closed curve (it is a Pascal’s Limaçon up to change of coordinates by an element in $\text {PU}(2,1)$) such that its complement in $\ell _{p_0}$ consists of three disjoint topological disks. Translating these three disks with the flow $\phi _t([x:y:z])=[e^{2i t} x: e^{3i t}y:z]$, $t\in [0,2\pi ]$ we obtain all the set $\Lambda (\gamma ) {\setminus } S(\gamma )$ and its topological description.

Finally, a section on open questions is included.

2 Preliminaries

2.1 Projective Geometry

The complex projective plane $\mathbb {P}^2_{\mathbb {C}}$ is defined as

$$\begin{aligned} \mathbb {P}^2_\mathbb {C}:=(\mathbb {C}^{3}{\setminus } \{0\})/\mathbb {C}^*, \end{aligned}$$

where $\mathbb {C}^*$ acts on $\mathbb {C}^3{\setminus }\{0\}$ by the usual scalar multiplication. It is a well known fact that $\mathbb {P}_{\mathbb {C}}^2$ is a compact connected complex 2-dimensional manifold. Let $[ ]:\mathbb {C}^{3}{\setminus }\{0\}\rightarrow \mathbb {P}^{2}_{\mathbb {C}}$ be the quotient map, if $\mathbf {v}=(x,y,z)\in \mathbb {C}^3{\setminus }\{0\}$ then we write $[\mathbf {v}]=[x:y:z]$. Also, $\ell \subset \mathbb {P}^2_{\mathbb {C}}$ is said to be a complex line if $[\ell ]^{-1}\cup \{0\}$ is a complex linear subspace of dimension 2, so that $\ell$ is equal to a set of the form

$$\begin{aligned} \{[x:y:z] \in \mathbb {P}_{\mathbb {C}}^2 : A x+ By+Cz=0\} \end{aligned}$$

for some $A,B,C \in \mathbb {C}$ not all zero. Some useful properties are listed below:

• Any two distinct complex lines in $\mathbb {P}_{\mathbb {C}}^2$ intersect in one single point.

• Any two distinct points $p=[x:y:z], q=[x':y':z']$ define a unique complex line containing p, q. In fact, this line can be described as the set:

  $$\begin{aligned} \{[\lambda x + \mu x': \lambda y + \mu y' : \lambda z + \mu z']: \lambda , \mu \in \mathbb {C}, |\lambda |^2 + |\mu |^2\ne 0 \} \end{aligned}$$

• Any complex line in $\mathbb {P}_{\mathbb {C}}^2$ is biholomorphic to the Riemann sphere $S^2$.

Consider the action of $\mathbb {C}^*= \mathbb {C}{\setminus }\{0\}$ on $\text {GL}(3,\mathbb {C})$ given by the usual scalar multiplication, then

$$\begin{aligned} \text {PGL}(3,\mathbb {C})=\text {GL}(3,\mathbb {C})/\mathbb {C}^* \end{aligned}$$

is a Lie group whose elements are called projective transformations. Let $[[ ]]:\text {GL}(3,\mathbb {C})\rightarrow \text {PGL}(3,\mathbb {C})$ be the quotient map. If $g\in \text {PGL}(3,\mathbb {C})$ and $\mathbf {g} \in \text {GL}(3,\mathbb {C})$, we say that $\mathbf {g}$ is a lift of g whenever $[[\mathbf {g}]]=g$. One can show that $\text {PGL}(3,\mathbb {C})$ is a Lie group that acts transitively, effectively and by biholomorphisms on $\mathbb {P}^2_{\mathbb {C}}$ by $[[\mathbf {g}]]([\mathbf {v}])=[\mathbf {g}\mathbf {v}]$, where $\mathbf {v}\in \mathbb {C}^3{\setminus }\{0\}$ and $\mathbf {g}\in \text {GL}(3, \mathbb {C})$. Notice that any element $g \in \text {PGL}(3, \mathbb {C})$ maps complex lines to complex lines.

2.2 Complex Hyperbolic Geometry

Let $\mathbb {C}^{2,1}$ denote $\mathbb {C}^3$ equipped with the Hermitian form

$$\begin{aligned} H(\mathbf {z}, \mathbf {w})= z_1 \overline{w}_1+z_2\overline{w}_2-z_3\overline{w}_3, \end{aligned}$$

where $\mathbf {z}=(z_1,z_2,z_3)$, $\mathbf {w}=(w_1,w_2,w_3)$.

Denote by

$$\begin{aligned} V_{-}= & {} \{\mathbf {z} \in \mathbb {C}^{2,1} : H(\mathbf {z}, \mathbf {z})<0\},\\ V_0= & {} \{\mathbf {z} \in \mathbb {C}^{2,1} {\setminus } \{\mathbf {0}\} : H(\mathbf {z}, \mathbf {z})=0\},\\ V_{+}= & {} \{\mathbf {z} \in \mathbb {C}^{2,1} : H(\mathbf {z}, \mathbf {z})>0\}, \end{aligned}$$

the sets of negative, null and positive vectors in $\mathbb {C}^{2,1} {\setminus } \{\mathbf {0}\}$, respectively.

The projectivisation of the set of negative vectors,

$$\begin{aligned}{}[V_{-}]= & {} \{[z_1:z_2:z_3] \in \mathbb {P}_{\mathbb {C}}^2 : |z_1|^2+|z_2|^2-|z_3|^2<0\} \\= & {} \{ [z_1:z_2:1] \in \mathbb {P}_{\mathbb {C}}^2 : |z_1|^2+|z_2|^2<1 \}, \end{aligned}$$

is a complex 2-dimensional open ball in $\mathbb {P}_{\mathbb {C}}^2$. Moreover, $[V_{-}]$ equipped with the quadratic form induced by the Hermitian form H is a model for the complex hyperbolic space $\mathbb {H}^2 _{\mathbb {C}}$. The projectivisation of the set of null vectors,

$$\begin{aligned}{}[V_{0}]= & {} \{[z_1:z_2:z_3] \in \mathbb {P}_{\mathbb {C}}^2 : |z_1|^2+|z_2|^2-|z_3|^2=0\} \\= & {} \{ [z_1:z_2:1] \in \mathbb {P}_{\mathbb {C}}^2 : |z_1|^2+|z_2|^2=1 \}, \end{aligned}$$

is a 3-sphere in $\mathbb {P}_{\mathbb {C}}^2$ and it is the boundary of $\mathbb {H}^2 _{\mathbb {C}}$, denoted $\partial \mathbb {H}^2 _{\mathbb {C}}$.

Finally, the projectivisation of the set of positive vectors,

$$\begin{aligned}{}[V_{+}]= & {} \{[z_1:z_2:z_3] \in \mathbb {P}_{\mathbb {C}}^2 : |z_1|^2+|z_2|^2-|z_3|^2>0\}, \end{aligned}$$

is the complement in $\mathbb {P}_{\mathbb {C}}^2$ of the complex 2-dimensional closed ball $\overline{\mathbb {H}^2 _{\mathbb {C}}}:=\mathbb {H}^2 _{\mathbb {C}} \cup \partial \mathbb {H}^2 _{\mathbb {C}}$.

The group of holomorphic isometries of $\mathbb {H}^2 _{\mathbb {C}}$ is $\text {PU}(2,1)$, the projectivisation in $\text {PGL}(3,\mathbb {C})$ of the unitary group, U(2, 1), respect to the Hermitian form H:

$$\begin{aligned} U(2,1) = \{ \mathbf {g} \in \text {GL}(3, \mathbb {C}): H( \mathbf {g} \mathbf {z}, \mathbf {g} \mathbf {w})=H(\mathbf {z}, \mathbf {w}) \}. \end{aligned}$$

The group $\text {PU}(2,1)$ acts transitively in $\mathbb {H}^2 _{\mathbb {C}}$ and by diffeomorphisms in the boundary $\partial \mathbb {H}^2 _{\mathbb {C}} \cong S^3$.

The disks obtained as the intersection of a complex line and $\mathbb {H}^2 _{\mathbb {C}}$ are totally geodesic subspaces of $\mathbb {H}^2 _{\mathbb {C}}$, they are called complex geodesics. The boundary at infinity of a complex geodesic is a circle obtained as the intersection of $\partial \mathbb {H}^2 _{\mathbb {C}}$ and a complex line, these circles are called chains and they play an important role in this paper, more information on chains is given in Sect. 3. Another important fact we use along this paper is the following: Given any point $p=[w_1:w_2:w_3] \in \partial \mathbb {H}^2 _{\mathbb {C}}$, there exists an unique complex line, denoted $\ell _p$, tangent to $\partial \mathbb {H}^2 _{\mathbb {C}}$ at p. Moreover, $\ell _p$ is given by the set:

$$\begin{aligned} \{[z_1: z_2: z_3] \in \mathbb {P}_{\mathbb {C}}^2 : z_1 \overline{w}_1 + z_2 \overline{w}_2-z_3 \overline{w}_3=0\}. \end{aligned}$$

If we consider $\mathbb {C}^3$ with the Hermitian form

$$\begin{aligned} H_1(\mathbf {z},\mathbf {w})=z_1 \overline{w}_3+z_2 \overline{w}_2+z_3 \overline{w}_1, \end{aligned}$$

where $\mathbf {z}=(z_1, z_2, z_3)$ and $\mathbf {w}=(w_1, w_2, w_3)$, then we have that

$$\begin{aligned} H(C \mathbf {z}, C \mathbf {w})=H_1(\mathbf {z}, \mathbf {w}), \end{aligned}$$

where

$$\begin{aligned} C=\frac{1}{\sqrt{2}}\left( \begin{array}{ccc} 1 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad \sqrt{2} &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad -1 \end{array}\right) . \end{aligned}$$

Hence $C(V_{-}), C(V_0), C(V_+)$ are the sets of negative, null and positive vectors for $H_1$, respectively. The projectivisation of $C(V_{-})$ equipped with the Hermitian form $H_1$ is the Siegel model for complex hyperbolic space

$$\begin{aligned} \{[z_1: z_2 : 1] \in \mathbb {P}_{\mathbb {C}}^2 : 2 \mathfrak {R}(z_1)+|z_2|^2<0\} \end{aligned}$$

and its boundary is the set

$$\begin{aligned} \{[z_1:z_2:1] \in \mathbb {P}_{\mathbb {C}}^2 : 2 \mathfrak {R}(z_1)+|z_2|^2=0\} \cup \{[1:0:0]\}. \end{aligned}$$

Any finite point in this boundary, can be written in the form

$$\begin{aligned} {[}-|\zeta |^2+i v:\sqrt{2} \zeta : 1], \end{aligned}$$

for some $(\zeta ,v) \in \mathbb {C}\times \mathbb {R}$. Hence there is a natural identification of this boundary set with the one point compactification of the Heisenberg space $\mathbb {C}\times \mathbb {R}=\mathcal {H}$. The vertical lines $\zeta = \zeta _0$ (constant) are chains in this representation. For more details on complex hyperbolic geometry, see the book (Goldman 1999).

2.3 Linking Number

This subsection is based on the books Rolfsen (2003) and Spivak (1999). Given $\gamma _1, \gamma _2 :S^1 \rightarrow \mathbb {R}^3$ (or $S^3$) two disjoint piecewise smooth knots, the linking number between $\gamma _1$ and $\gamma _2$, denoted by

$$\begin{aligned} \ell (\gamma _1, \gamma _2), \end{aligned}$$

is (roughly speaking) the number of turns that $\gamma _1$ gives around $\gamma _2$.

In this subsection we describe some equivalent methods for defining this integer, all of which turn out to be equivalent, up to a sign.

• (Linking number as an intersection number) Let $\gamma _1: S^1 \rightarrow \mathbb {R}^3$ be an embedding such that $\gamma _1(S^1)=\partial M$ for some compact oriented 2-manifold with boundary M. Assume that $\gamma _2: S^1 \rightarrow \mathbb {R}^3$ is transversal to M. The linking number $\ell (\gamma _1, \gamma _2)$ is the oriented intersection number of $\gamma _2$ and M.

• (Linking number as a degree) Define a map $F:S^1\times S^1 \rightarrow S^2$ by the formula

  $$\begin{aligned} F(w,z)=\frac{\gamma _2(w)-\gamma _1(z)}{|\gamma _2(w)-\gamma _1(z)|}, \end{aligned}$$

  then

  $$\begin{aligned} \ell (\gamma _1, \gamma _2)=\deg (F) \end{aligned}$$

• (Linking number as a Gauss integral) If we consider the knots $\gamma _1$ and $\gamma _2$ as functions from [0, 1] to $\mathbb {R}^3$ then

  $$\begin{aligned} \ell (\gamma _1, \gamma _2)=-\frac{1}{4\pi }\int _0 ^1 \int _0 ^1 \frac{A(s,t)}{(r(s,t))^3}\, ds \, dt, \end{aligned}$$

  where

  $$\begin{aligned} r(s,t)= & {} |\gamma _2(t)-\gamma _1(s)|,\\ A(s,t)= & {} \det \left( \begin{array}{ccc} (\gamma _1 ^{(1)})'(s) &{}\quad (\gamma _1 ^{(2)})'(s) &{}\quad (\gamma _1 ^{(3)})'(s) \\ (\gamma _2 ^{(1)})'(t) &{}\quad (\gamma _2 ^{(2)})'(t) &{}\quad (\gamma _2 ^{(3)})'(t) \\ \gamma _2 ^{(1)}(t)-\gamma _1 ^{(1)}(s) &{}\quad \gamma _2 ^{(2)}(t)-\gamma _1 ^{(2)}(s) &{}\quad \gamma _2 ^{(3)}(t)-\gamma _1 ^{(3)}(s) \end{array}\right) . \end{aligned}$$

• (Linking number as an integer in the abelianization of a fundamental group) The knot $\gamma _1$ is a loop in $S^3 {\setminus } \gamma _2(S^1)$, hence it represents an element in the fundamental group $\pi _1(S^3 {\setminus } \gamma _2(S^1))$ with a suitable base point. This group abelianizes to $\mathbb {Z}$ and the loop $\gamma _1$ is carried to an integer, called $\ell (\gamma _1, \gamma _2)$.

It is important to remark that the linking number is symmetric: $\ell (\gamma _1, \gamma _2)= \ell (\gamma _2, \gamma _1),$ whenever $\gamma _1$, $\gamma _2$ are disjoint knots in $S^3$. Another useful property about linking numbers is the invariance under homotopy: If $H: [0,1] \times S^1 \rightarrow S^3$ is a homotopy such that

$$\begin{aligned}&\displaystyle H(0,z)=\gamma _0(z),\\&\displaystyle H(1,z)=\gamma _1(z),\\&\displaystyle \{H(s,z): s \in [0,1], z\in S^1\} \cap \{ \gamma _2(s): s \in [0,1]\}=\varnothing , \end{aligned}$$

then

$$\begin{aligned} \ell (\gamma _0, \gamma _2)= \ell (\gamma _1, \gamma _2). \end{aligned}$$

3 Chains

Definition 3.1

A chain is the intersection of a complex line in $\mathbb {P}_{\mathbb {C}}^2$ and $S^3=\partial \mathbb {H}^2 _{\mathbb {C}}$. The points in $S^3 = \partial \mathbb {H}^2 _{\mathbb {C}}$ are considered as chains and we call them degenerate chains. We denote by $\mathcal {C}$ the space of all chains and we call it the chains space

If $p=[\mathbf {v}] \in \mathbb {P}_{\mathbb {C}}^2 {\setminus } \mathbb {H}^2 _{\mathbb {C}}$ then $\mathbf {v}$ is a positive vector, so the orthogonal complement, $\langle \mathbf {v} \rangle ^{\perp }$, respect to the Hermitian form H (see Sect. 2.2) is a two dimensional subspace of $\mathbb {C}^{2,1}$ and it induces a complex line, $\ell _p$, called the polar line top. The chain, $C_p$, obtained as the intersection $\ell _p \cap \partial \mathbb {H}^2 _{\mathbb {C}}$ is the polar chain top.

Conversely, if $\ell$ is a complex line transversal to $\partial \mathbb {H}^2 _{\mathbb {C}}$, then we can write $\ell =[L{\setminus }\{\mathbf {0}\}]$ where L is a two dimensional complex vector subspace of $\mathbb {C}^{2,1}$. Moreover, the orthogonal complement of L, respect to the Hermitian form H as in Sect. 2.2, is a one dimensional complex subspace of $\mathbb {C}^{2,1}$ which induces a point in $\mathbb {P}_{\mathbb {C}}^2 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}$, this point is called the polar point to the line $\ell$.

There is a natural identification of a chain $\partial \mathbb {H}^2 _{\mathbb {C}} \cap \ell$ with the polar point to $\ell$. In fact, there is a bijection

$$\begin{aligned} P:\mathcal {C} \longleftrightarrow \mathbb {P}_{\mathbb {C}}^2 {\setminus } \mathbb {H}^2 _{\mathbb {C}} \end{aligned}$$

between the space of chains and the complement of the complex hyperbolic space. We remark that for a degenerate chain $\{p\}=\partial \mathbb {H}^2 _{\mathbb {C}} \cap \ell$ the corresponding point is $p \in \partial \mathbb {H}^2 _{\mathbb {C}}$.

We equip the space of chains $\mathcal {C}$ with a topology by requiring that the bijection P be an homeomorphism. Moreover, we equip $\mathcal {C}$ with a structure of differentiable manifold with boundary by requiring that P be a diffeomorphism.

We parametrize the horizontal chain $C_0=\{[z:0:1] \in \mathbb {P}_{\mathbb {C}}^2 \, : \, |z|^2=1 \}$ as a curve in the following way

$$\begin{aligned}&\sigma _0 : S^1 \rightarrow C_0\\&\sigma _0 (e^{2\pi i t})=[e^{2 \pi i t}:0:1], \end{aligned}$$

and we say that $C_0$ is positively oriented with this parametrization. Moreover, given any non degenerate chain C there exists $g \in \text {PU}(2,1)$ mapping $C_0$ to C. We say that $g\circ \sigma _0:S^1 \rightarrow C$ is a positive parametrization of C. In what follows we use only positive parametrizations for the non degenerate chains.

Let $\gamma : S^1 \rightarrow S^3\cong \partial \mathbb {H}^2 _{\mathbb {C}}$ be a simple closed curve (perhaps piecewise differentiable).

We denote by $\Lambda (\gamma )$ the set

$$\begin{aligned} \bigcup _{p \in \gamma (S^1)}\ell _p, \end{aligned}$$

where $\ell _p$ is the only complex line tangent to $\partial \mathbb {H}^2 _{\mathbb {C}}$ at p. The open set $\mathbb {P}_{\mathbb {C}}^2 {\setminus } \Lambda (\gamma )$ is denoted by $\Omega (\gamma )$.

Lemma 3.2

If $\mathbf {u}=(u_1,u_2,u_3) \in \mathbb {C}^{2,1}$ is a positive vector then the vectors

$$\begin{aligned} \mathbf {v}(\mathbf {u})= & {} \left( \frac{|u_1|^2+|u_2|^2}{|u_1|^2+|u_2|^2-|u_3|^2}\right) ^{1/2}\left( \frac{\bar{u}_3\, u_{1}}{|u_1|^2+|u_2|^2}\, , \frac{\bar{u}_3\, u_2}{|u_1|^2+|u_2|^2}\, , 1\right) ,\\ \mathbf {w}(\mathbf {u})= & {} \frac{1}{(|u_1|^2+|u_2|^2)^{1/2}}\left( -\bar{u}_2 , \bar{u}_1 , 0 \right) , \end{aligned}$$

satisfy that:

• (a) $\langle \mathbf {v} (\mathbf {u}), \mathbf {v} (\mathbf {u}) \rangle = -1$,

• (b) $\langle \mathbf {w} (\mathbf {u}), \mathbf {w}(\mathbf {u})\rangle = 1$,

• (c) $\langle \mathbf {u} , \mathbf {v} (\mathbf {u}) \rangle = \langle \mathbf {u}, \mathbf {w} (\mathbf {u}) \rangle = \langle \mathbf {v}(\mathbf {u}), \mathbf {w}(\mathbf {u}) \rangle =0$.

• (d) $\mathbf {v}(\mathbf {u})$ and $\mathbf {w}(\mathbf {u})$ are smooth functions of $\mathbf {u}$.

• (e)
  $$\begin{aligned} \sigma (e^{2 \pi it}) = [\mathbf {v}(\mathbf {u}) + e^{2 \pi i t} \mathbf {w}(\mathbf {u})] \in \mathbb {P}_{\mathbb {C}}^2, \quad t \in [0,1] \end{aligned}$$

  gives a positive parametrization of the polar chain to $[\mathbf {u}] \in \mathbb {P}_{\mathbb {C}}^2 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}$ .

Proof

The proof of items (a) to (d) follows by straightforward computations and we omit it.

(e) First, by (c), we have that

$$\begin{aligned} \langle \mathbf {u}, \mathbf {v}(\mathbf {u}) + e^{2 \pi i t} \mathbf {w}(\mathbf {u})\rangle =0 \end{aligned}$$

for every $t \in [0,1]$. Also, by (a), (b) and (c) we have that

$$\begin{aligned} \langle \mathbf {v}(\mathbf {u}) + e^{2 \pi i t} \mathbf {w}(\mathbf {u}) , \mathbf {v}(\mathbf {u}) + e^{2 \pi i t} \mathbf {w}(\mathbf {u})\rangle =0. \end{aligned}$$

Hence $\sigma (e^{2 \pi i t})$ gives a parametrization of the polar chain to $[\mathbf {u}]$.

Finally, the linear transformation, $\mathbf {g}$, such that

$$\begin{aligned} \mathbf {g}(\mathbf {e}_1)= & {} \mathbf {w}(\mathbf {u}),\\ \mathbf {g}(\mathbf {e}_2)= & {} \frac{\mathbf {u}}{\sqrt{\langle \mathbf {u}, \mathbf {u} \rangle } } ,\\ \mathbf {g} (\mathbf {e}_3)= & {} \mathbf {v}(\mathbf {u}), \end{aligned}$$

lies in $\text {SU}(2,1)$ and the induced transformation $g=[\mathbf {g}]$ satisfies that

$$\begin{aligned} g^{-1} \circ \sigma (e^{2\pi it})=[\mathbf {e}_3+e^{2 \pi i t} \mathbf {e}_1]=[e^{2 \pi i t}:0:1]. \end{aligned}$$

Therefore $\sigma (e^{2 \pi it})$ gives a positive parametrization of the polar chain to $[\mathbf {u}]$. $\square$

Proposition 3.3

Let us assume that $p_0, p_1 \in (\mathbb {P}_{\mathbb {C}}^2 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}})$ and $C_{p_0}$, $C_{p_1}$ are the corresponding non degenerate polar chains. If $\nu : I \rightarrow (\mathbb {P}_{\mathbb {C}}^2 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}})$ is a continuous path in $(\mathbb {P}_{\mathbb {C}}^2 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}})$ joining $p_0$ and $p_1$, then there exists a continuous homotopy of chains between $C_{p_0}$ and $C_{p_1}$. In other words, there is a continuous map

$$\begin{aligned} H:I\times S^1 \rightarrow \partial \mathbb {H}^2 _{\mathbb {C}} \end{aligned}$$

such that $H(s,\cdot ): S^1 \rightarrow \partial \mathbb {H}^2 _{\mathbb {C}}$ is a positive parametrization of the polar chain to $\nu (s)$, denoted $C_{\nu (s)}$, for every $s \in I$. The converse statement is also true.

Proof

Let $\mathbf {u}: I \rightarrow \mathbb {C}^{2,1}$ be a continuous lift of $\nu$. In other words, $\mathbf {u}$ is continuous and $\nu (s)=[\mathbf {u}(s)]$ for every $s \in I$ (there are many of such continuous lifts). Using the notation of Lemma 3.2, we define

$$\begin{aligned}&H:I\times S^1 \rightarrow \partial \mathbb {H}^2 _{\mathbb {C}}\\&\quad H(s,e^{2\pi i t})=[\mathbf {v}(\mathbf {u}(s))+ e^{2\pi i t}\mathbf {w}(\mathbf {u}(s))], \end{aligned}$$

and the proof follows from Lemma 3.2.

Conversely, given the homotopy, one can take the polars to the chains $C_{\nu (s)}$ and obtain a continuous path in $\mathbb {P}_{\mathbb {C}}^2 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C }}}$ from $p_0$ to $p_1$. $\square$

Proof of Theorem 1.1

If $p_0$ and $p_1$ can be joined by a continuous path $\nu : I \rightarrow \Omega (\gamma ) \cap (\mathbb {P}_{\mathbb {C}}^2 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}})$, then Proposition 3.3 implies that there exists a continuous map

$$\begin{aligned} H:I\times S^1 \rightarrow \partial \mathbb {H}^2 _{\mathbb {C}} \end{aligned}$$

such that $H(s,\cdot ): S^1 \rightarrow \partial \mathbb {H}^2 _{\mathbb {C}}$ is a positive parametrization of the chain to $C_{\nu (s)}$ for every $s \in I$.

If we assume that $C_{\nu (s_0)} \cap \gamma (S^1) \ne \varnothing$ for some $s_0 \in I$, then

$$\begin{aligned} {[}\mathbf {v}(\mathbf {u}(s_0))+ e^{2\pi i t_1}\mathbf {w}(\mathbf {u}(s_0))]=\gamma (e^{2 \pi it_2}) \end{aligned}$$

for some $t_1, t_2 \in [0,1]$. So $\nu (s_0)=[\mathbf {u}(s_0)] \in \ell _{\gamma (e^{2\pi i t_2})}$, because $\mathbf {u}(s_0)$ is orthogonal to the null vector $\mathbf {v}(\mathbf {u}(s_0))+ e^{2\pi i t_1}\mathbf {w}(\mathbf {u}(s_0))$. A contradiction to the hypothesis that $\nu$ lies in $\Omega (\gamma )$.

The converse statement is obtained by taking the path of polars to the chains $H(s, S^1)$, $s \in I$. $\square$

Notation. In what follows, $\ell (\gamma , C)$ denotes the linking number between the closed curve $\gamma :S^1 \rightarrow \partial \mathbb {H}^2 _{\mathbb {C}}$ and the positively oriented chain C.

Lemma 3.4

Let $\mathbf {u} \in \mathbb {C}^{2,1}$ be a positive vector,

• (i) the point $[\mathbf {v}(\mathbf {u})]$ is the Euclidean center of the polar chain to $[\mathbf {u}]$ (considered in the chart $\{[z_1:z_2:z_3] \, | \, z_3 \ne 0\} \cong \mathbb {C}^2$) and its Euclidean radius is equal to

  $$\begin{aligned} \left( \frac{|u_1|^2+|u_2|^2-|u_3|^2}{|u_1|^2+|u_2|^2}\right) ^{1/2} \end{aligned}$$

• (ii) If $[\mathbf {u}]$ goes to $q \in \partial \mathbb {H}^2 _{\mathbb {C}}$ then the polar chain to $[\mathbf {u}]$ goes to the degenerate chain q.

Proof

First, we notice that $[\mathbf {v}(\mathbf {u})]$ and $\sigma (e^{2\pi it})=[\mathbf {v}(\mathbf {u})+e^{2 \pi it} \mathbf {w}(\mathbf {u})]$ in the chart $\{[z_1:z_2:z_3] \, | \, z_3 \ne 0\} \cong \mathbb {C}^2$ are equal to

$$\begin{aligned} \left( \frac{\bar{u}_3\, u_{1}}{a^2}\, , \frac{\bar{u}_3\, u_2}{a^2}\,\right) \quad \text { and } \quad \left( \frac{ \bar{u}_3 u_1-b\bar{u}_2 e^{2\pi i t} }{a^2}, \frac{\bar{u}_3 u_2+ b\bar{u}_1 e^{2 \pi i t}}{a^2}\right) , \end{aligned}$$

where $a=\left( |u_1|^2 + |u_2|^2\right) ^{1/2}$ and $b=\left( |u_1|^2+|u_2|^2-|u_3|^2 \right) ^{1/2}$. By straightforward computations, the Euclidean distance between these two points is constant and equal to $\frac{b}{a}$. Thus, (i) follows. The proof of (ii) follows from (i). $\square$

Lemma 3.5

If $p \in \Omega (\gamma ) \cap (\mathbb {P}_{\mathbb {C}}^2 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}})$ can be joined by a continuous path in $\Omega (\gamma )$ to a point in $\partial \mathbb {H}^2 _{\mathbb {C}} {\setminus } \gamma (S^1)$ then there is a homotopy of chains between $C_p$ and a degenerate chain in $\partial \mathbb {H}^2 _{\mathbb {C}} {\setminus } \gamma (S^1)$. In particular

$$\begin{aligned} \ell (\gamma , C_p)= 0. \end{aligned}$$

Proof

We can assume that there is a path $\nu :[0,1] \rightarrow \mathbb {P}_{\mathbb {C}}^2 {\setminus } \mathbb {H}^2 _{\mathbb {C}}$ such that

• $\nu (0)=p \in \Omega (\gamma ) \cap (\mathbb {P}_{\mathbb {C}}^2 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}),$

• $\nu (1)=q \in \partial \mathbb {H}^2 _{\mathbb {C}} {\setminus } \gamma (S^1),$

• $\nu ([0,1)) \subset \mathbb {P}_{\mathbb {C}}^2 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}.$

Since $q \in \partial \mathbb {H}^2 _{\mathbb {C}} {\setminus } \gamma (S^1)$, there is an Euclidean ball, $B(q,\epsilon )$ centered at q not meeting $\Lambda (\gamma )$. By Lemma 3.4, there exists $t_0 \in [0,1)$ such that $C_{\nu (t_0)}$, the polar chain to $\nu (t_0)$, is contained in $B(q, \epsilon )$. Hence, there is a homotopy of chains between $C_{\nu (t_0)}$ and a degenerate chain in $\partial \mathbb {H}^2 _{\mathbb {C}} {\setminus } \gamma (S^1)$. Finally, there is a homotopy of chains from $C_p$ to $C_{\nu (t_0)}$, by Theorem 1.1. $\square$

We remark that there are smooth versions of Proposition 3.3, Theorem 1.1 and Lemma 3.5. Roughly speaking, a smooth path in the complement of complex hyperbolic plane can be translated into a smooth homotopy of chains and vice versa.

Corollary 3.6

If $\Omega _0$ denotes the component of $\Omega (\gamma )$ containing $\mathbb {H}^2 _{\mathbb {C}}$, then

$$\begin{aligned} \ell (\gamma , C_p)=0, \text { for every } p\in \Omega _0 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}. \end{aligned}$$

Proof

First, we notice that $\overline{ \mathbb {H}^2 _{\mathbb {C}}}{\setminus } \gamma (S^1)$ is contained in $\Omega _0$ because $\mathbb {H}^2 _{\mathbb {C}} \subset \Omega _0$. Now, if $p\in \Omega _0 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}$, then there is a continuous path in $\Omega (\gamma )$ joining p and a point in $\partial \mathbb {H}^2 _{\mathbb {C}} {\setminus } \gamma (S^1)$. It follows by Lemma 3.5 that $\ell (\gamma , C_p)=0$. $\square$

The converse of Corollary 3.6 is not necessarily true, for example, when the image of the curve $\gamma$ and a chain C form a Whitehead link, then $\ell (\gamma , C)=0$ and the polar of the chain C does not lie in the connected component of $\Omega (\gamma )$ containing $\mathbb {H}^2 _{\mathbb {C}}$, otherwise, C is nullhomotopic in $\partial \mathbb {H}^2 _{\mathbb {C}} {\setminus } \gamma (S^1)$ by Lemma 3.5, and it is a known fact that it is not possible.

Proof of Corollary 1.2

If p and q are in the same component of $\Omega (\gamma )$ then there is a (smooth) path $\nu :I \rightarrow \Omega (\gamma )$ joining p and q.

If $\nu (I) \cap \overline{\mathbb {H}^2 _{\mathbb {C}}} = \varnothing$ then by Theorem 1.1 there is a homotopy of positively oriented chains between $C_p$ and $C_q$ whose image is contained in $\partial \mathbb {H}^2 _{\mathbb {C}} {\setminus } \gamma (S^1)$. Hence,

$$\begin{aligned} \ell (\gamma , C_p)= \ell (\gamma , C_q) \end{aligned}$$

and it contradicts the hypothesis.

If $\nu (I) \cap \overline{\mathbb {H}^2 _{\mathbb {C}}} \ne \varnothing$ then by Lemma 3.5

$$\begin{aligned} \ell (\gamma , C_p)=0=\ell (\gamma , C_q) \end{aligned}$$

and it contradicts the hypothesis. $\square$

The number of all possible linking numbers $\ell (\gamma , C)$ where C is a chain not intersecting $\gamma (S^1)$ is not necessarily equal to the number of components of $\Omega (\gamma )$, as shown in the example of the whitehead link above.

4 Applications of Chains Homotopy and Linking Numbers

In this section, we include two examples for the description of the components of $\Omega (\gamma )$ when $\gamma$ is a chain and an $\mathbb {R}$-circle. The main difference with the approach given in Cano et al. (2016), is the use of Theorem 1.1 and its Corollary 1.2 to describe each component as a set of points where the linking number of the polar chain and the curve $\gamma$ is constant.

4.1 Chain

First, we consider the case when the image of the curve $\gamma : S^1 \rightarrow \partial \mathbb {H}^2 _{\mathbb {C}}$ is a chain. It is proved in Cano et al. (2016) that $\Lambda (\gamma )$ is a complex cone over a circle, in other words, there exists $q \in \Lambda (\gamma )$ such that $\Lambda (\gamma ) {\setminus } \{q\}$ is diffeomorphic to $S^1 \times \mathbb {C}$. Moreover, $\Omega (\gamma )$ has two components each one diffeomorphic to $D \times \mathbb {C}$, where $D \subset \mathbb {C}$ is an open disk.

Let us assume that the image of $\gamma$ is the chain

$$\begin{aligned} C_0=\{ [ \zeta : 0 :1] \in \partial \mathbb {H}^2 _{\mathbb {C}} : |\zeta |=1\}. \end{aligned}$$

Since $\zeta \bar{x} -\bar{z}=0$ is the equation of the complex line tangent to $\partial \mathbb {H}^2 _{\mathbb {C}}$ at the point $[\zeta : 0 :1]$, we have that

$$\begin{aligned} \Lambda (\gamma )= & {} \{[x:y:z] \in \mathbb {P}_{\mathbb {C}}^2 : \zeta \bar{x} -\bar{z}=0 \text { for some } \zeta \in S^1\} \\= & {} \{ [x:y:z] \in \mathbb {P}_{\mathbb {C}}^2 : |x|=|z| \}. \end{aligned}$$

Moreover,

$$\begin{aligned} \Omega _0= & {} \{ [x:y:z] \in \mathbb {P}_{\mathbb {C}}^2 : |x|<|z|\},\\ \Omega _1= & {} \{ [x:y:z] \in \mathbb {P}_{\mathbb {C}}^2 : |x|>|z|\} \end{aligned}$$

are the components of $\Omega (\gamma )$.

Now, let us assume that $\gamma$ is a positive parametrization of $C_0$. If $C_p$ is the polar chain to a point $p \in \mathbb {P}_{\mathbb {C}}^2 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}$ then

$$\begin{aligned} \ell (\gamma , C_p)=0 \text { or } 1 \end{aligned}$$

according to whether $p \in \Omega _0 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}$ or in $p \in \Omega _1$.

4.2 $\mathbb {R}$-circle

Now, we consider the case when

$$\begin{aligned}&\gamma : S^1 \rightarrow \partial \mathbb {H}^2 _{\mathbb {C}}\nonumber \\&\quad \gamma (e^{2 \pi i t}) = [ \sin 2 \pi t: \cos 2 \pi t :1] \end{aligned}$$

(2)

is a parametrization of the $\mathbb {R}$-circle

$$\begin{aligned} \partial \mathbb {H}^2 _{\mathbb {R}}=\partial \mathbb {H}^2 _{\mathbb {C}} \cap \mathbb {P}^2 _{\mathbb {R}}, \end{aligned}$$

where $\mathbb {P}^2 _{\mathbb {R}}=\{[r_1:r_2:r_3] \in \mathbb {P}_{\mathbb {C}}^2 \, | \, r_1, r_2, r_3 \in \mathbb {R} \}$.

If $p=[x:y:z] \in \mathbb {P}_{\mathbb {C}}^2$, let us denote by

$$\begin{aligned} \ell _p = \{[u:v:w] \in \mathbb {P}_{\mathbb {C}}^2 \, | \, x \bar{u}+y \bar{v}-z \bar{w}=0\} \end{aligned}$$

the polar line to p (If $p \in \partial \mathbb {H}^2 _{\mathbb {C}}$ then $\ell _p$ is the only complex line tangent to $\partial \mathbb {H}^2 _{\mathbb {C}}$ at p).

The problem on computing the number of components of $\Omega (\gamma )$ for an $\mathbb {R}$-circle was solved in Cano et al. (2016). We restate some of these results in Cano et al. (2016) in order to show our approach which gives a generalization.

Proposition 4.1

If $\gamma$ is the $\mathbb {R}$-circle given by (2), then

• (i) The set $\Lambda (\gamma )$ is described as

  $$\begin{aligned} \Lambda (\gamma )= & {} \{p \in \mathbb {P}_{\mathbb {C}}^2 : \ell _p \cap \partial \mathbb {H}^2 _{\mathbb {R}} \ne \varnothing \} \\= & {} \{[x:y:z] \in \mathbb {P}_{\mathbb {C}}^2 \, | \, x r_1 +y r_2 -z r_3=0 \text { for some } [r_1:r_2:r_3] \in \partial \mathbb {H}^2 _{\mathbb {R}}\}. \end{aligned}$$

• (ii) Moreover, $\Omega (\gamma )$ has three components:

  $$\begin{aligned} \Omega _0= & {} \{p \in \mathbb {P}_{\mathbb {C}}^2 : \ell _p \cap \mathbb {P}^2 _{\mathbb {R}} \subset \mathbb {P}^2 _{\mathbb {R}} {\setminus } \overline{\mathbb {H}^2 _{\mathbb {R}}} \},\\ \Omega _1 ^-= & {} \{p=[x:y:z] \in \mathbb {P}_{\mathbb {C}}^2 : \ell _p \cap \mathbb {P}^2 _{\mathbb {R}} \text { is a point in } \mathbb {H}^2 _{\mathbb {R}} \text { and } \left| \begin{array}{ll} \mathfrak {R}(x) &{} \mathfrak {R}(y) \\ \mathfrak {I}(x) &{} \mathfrak {I}(y) \end{array} \right| <0\},\\ \Omega _1 ^+= & {} \{p=[x:y:z] \in \mathbb {P}_{\mathbb {C}}^2 : \ell _p \cap \mathbb {P}^2 _{\mathbb {R}} \text { is a point in } \mathbb {H}^2 _{\mathbb {R}} \text { and } \left| \begin{array}{ll} \mathfrak {R}(x) &{} \mathfrak {R}(y) \\ \mathfrak {I}(x) &{} \mathfrak {I}(y) \end{array} \right| >0\}. \end{aligned}$$

Proof

• (i) By definition, $p=[x:y:z] \in \Lambda (\gamma )$ if and only if there exists $q=[r_1:r_2:r_3] \in \partial \mathbb {H}^2 _{\mathbb {R}}$ such that $p \in \ell _q$ and it happens if and only if $q \in \ell _p$.

• (ii) Now, if $p=[x:y:z] \in \mathbb {P}_{\mathbb {C}}^2$ then the polar line $\ell _p$ meets the real projective space $\mathbb {P}^2 _{\mathbb {R}}$ in a real projective line or a point, according to whether the space of solutions of the system of equations with real variables $r_1, r_2, r_3$:

  $$\begin{aligned} \mathfrak {R}(x) r_1 + \mathfrak {R}(y) r_2- \mathfrak {R}(z)r_3= & {} 0,\nonumber \\ \mathfrak {I}(x) r_1 + \mathfrak {I}(y)r_2 -\mathfrak {I}(z)r_3= & {} 0, \end{aligned}$$

  (3)

  is two dimensional or one dimensional (over $\mathbb {R}$).

  In other words, $\ell _p \cap \mathbb {P}^2 _{\mathbb {R}}$ is equal to:

  • A real projective line in $\mathbb {P}^2 _{\mathbb {R}}$ if and only if $p=[x:y:z] \in \mathbb {P}^2 _{\mathbb {R}}$.

  • A point in $\mathbb {P}^2_{\mathbb {R}}$ if and only if there is a non-zero $2 \times 2$ determinant of the system (3).

  In the case when $p=[x:y:z] \notin \mathbb {P}^2 _{\mathbb {R}}$, the non-zero real solution for the system (3) given by:

  $$\begin{aligned} \mathbf {v}= & {} \left( \begin{array}{c} i(z \bar{y}-\bar{z}y) \\ i(x \bar{z}-\bar{x}z) \\ i(x \bar{y}-\bar{x}y) \\ \end{array}\right) \end{aligned}$$

  (4)

  is proposed in Cano et al. (2016).

  If $p \in \Omega _0$ then we have two possibilities:

  • $\ell _p \cap \mathbb {P}^2 _{\mathbb {R}}$ is a point in $\mathbb {P}^2 _{\mathbb {R}} {\setminus } \overline{\mathbb {H}^2 _{\mathbb {R}}}$ if and only if $\mathbf {v}$ is a positive vector.

  • $\ell _p \cap \mathbb {P}^2 _{\mathbb {R}}$ is a real projective line contained in $\mathbb {P}^2 _{\mathbb {R}} {\setminus } \overline{\mathbb {H}^2 _{\mathbb {R}}}$ if and only if $p \in \mathbb {H}^2 _{\mathbb {R}}$.

  Therefore, $\Omega _0$ is equal to the component of $\Omega (\gamma )$ containing $\mathbb {H}^2 _{\mathbb {C}}$ (see Cano et al. 2016).

  Now, we notice that $p \in \Omega _1 ^{-}$ if and only if $\mathbf {v}$ given by (4) is a negative vector and its last coordinate is a negative real number.

  Finally, $p \in \Omega _1 ^{+}$ if and only if $\mathbf {v}$ given by (4) is a negative vector and its last coordinate is a positive real number. It follows from Cano et al. (2016) that $\Omega _1 ^{-}$ and $\Omega _1 ^{+}$ are components of $\Omega (\gamma )$.

It is proved in Cano et al. (2016) that $\Lambda (\gamma )$ is a 3-dimensional semi-algebraic set and the Mobius strip, $\mathcal {M}=\Lambda (\gamma ) \cap \mathbb {P}^2 _{\mathbb {R}}$, is its singular set. Also, $\Lambda (\gamma ) {\setminus } \mathcal {M}$ is a disjoint union of two solid torus $S^1 \times \mathbb {R}^2$. Moreover, each component $\Omega _0, \Omega _1 ^{-}, \Omega _1 ^{+}$ is diffeomorphic to an open 4-ball.

Notice that $\mathbb {H}^2 _{\mathbb {C}} \subset \Omega _0$, so $\ell (\gamma , C_p)=0$ for every $p \in \Omega _0 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}$, because of Lemma 3.6. On the other hand,

$$\begin{aligned} \ell (\gamma , C_p)= \left\{ \begin{array}{ll} -1 &{}\quad \text { if } p\in \Omega _1 ^-, \\ +1 &{}\quad \text { if } p\in \Omega _1 ^+ . \end{array} \right. \end{aligned}$$

In order to prove this statement, we notice that the points $p^-=[i:1:0]$ and $p^+=[1:i:0]$ lie in $\Omega _1 ^-$ and $\Omega _1 ^+$, respectively. The corresponding polar chains $C_{p^-}, C_{p^+}$ have positive parametrizations (see Lemma 3.2) given by:

$$\begin{aligned} \sigma ^{-}(e^{2 \pi i t})= & {} \left[ -\frac{e^{2\pi i t}}{\sqrt{2}}:-\frac{ie^{2\pi i t}}{\sqrt{2}}:1\right] , \quad t\in [0,1],\\ \sigma ^{+}(e^{2 \pi i t})= & {} \left[ \frac{ie^{2\pi i t}}{\sqrt{2}}:\frac{e^{2\pi i t}}{\sqrt{2}}:1\right] , \quad t\in [0,1]. \end{aligned}$$

The curves obtained by composing $\sigma ^{-}, \sigma ^{+}$ with stereographic projection $S^3{\setminus }\{(0,i)\}$$\rightarrow \mathbb {R}^3$ are respectively

$$\begin{aligned} e^{2 \pi i t}\mapsto \left( -\frac{\cos (2 \pi t)}{\sqrt{2}+\cos (2 \pi t)}, -\frac{\sin (2 \pi t)}{\sqrt{2}+\cos (2 \pi t)}, \frac{\sin (2\pi t) }{\sqrt{2}+\cos (2 \pi t)}\right) \end{aligned}$$

and

$$\begin{aligned} e^{2 \pi i t}\mapsto \left( \frac{- \sin (2 \pi t)}{\sqrt{2}-\sin (2 \pi t)}, \frac{\cos (2 \pi t)}{\sqrt{2}-\sin (2\pi t)}, \frac{\cos (2 \pi t)}{\sqrt{2}-\sin (2\pi t)} \right) , \end{aligned}$$

and it is not hard to check that the linking number of these two curves with the curve obtained by composing $\gamma$ with stereographic projection is as claimed.

5 The Trefoil Knot

Let us consider the trefoil knot obtained as the intersection of the algebraic curve $y^2=x^3$ with $\partial \mathbb {H}^2 _{\mathbb {C}} \cong S^3$. If $\alpha$ denotes the positive real root of the equation $x^3+x^2-1=0$, then the knot can be parametrized in the following way:

$$\begin{aligned}&\displaystyle \gamma : S^1 \rightarrow \mathbb {H}^2 _{\mathbb {C}} \cong S^3\\&\displaystyle \gamma (e^{2 \pi i t})=[(\sqrt{\alpha }e^{2 \pi i t})^2: (\sqrt{\alpha }e^{2 \pi i t})^3:1]. \end{aligned}$$

The image of $\gamma$ is the set:

$$\begin{aligned} \left\{ [\zeta ^2 : \zeta ^3 :1] \in \mathbb {P}_{\mathbb {C}}^2 \, | \, |\zeta |^2 = \alpha \right\} , \end{aligned}$$

and $\Lambda (\gamma )$ can be described as those points $[x:y:z] \in \mathbb {P}_{\mathbb {C}}^2$ such that

$$\begin{aligned} \bar{x} \zeta ^2+ \bar{y} \zeta ^3 -\bar{z}=0 . \end{aligned}$$

(5)

for some $\zeta \in \mathbb {C}$ such that $|\zeta |^2 = \alpha$.

5.1 The Components of $\Omega (\gamma )$

If $[x:y:z] \in \Omega (\gamma )= \mathbb {P}_{\mathbb {C}}^2 {\setminus } \Lambda (\gamma )$, then the solutions of the (at most cubic) equation (5)

$$\begin{aligned} \bar{y} \zeta ^3 + \bar{x} \zeta ^2 -\bar{z}=0 \end{aligned}$$

satisfy that $|\zeta |^2 \ne \alpha$. The main purpose of this subsection is the proof of Theorem 1.4 i).

For readers convenience we provide a guide for this section: Lemma 5.2 says, roughly speaking, that a continuous path in $\Omega (\gamma )$, thought of as a continuous path of cubic equations, can be thought of as a continuous path in $\mathbb {C}^3$ of the corresponding roots of these cubic equations, and conversely.

The hypothesis of Lemma 5.2 is not satisfied in the line $y=0$. In fact, the equation (5) has degree less than three in this line. However Lemma 5.3, shows that the intersection of $\Omega (\gamma )$ with the line $y=0$ consists of the complement of a circle. An analogous result is obtained in Lemma 5.4 for the intersection of $\Omega (\gamma )$ with the line at infinity $z=0$.

The Lemmas 5.5, 5.6, 5.7 and 5.8 show that each $\Omega _j$ is path connected for $j=0,1,2,3$.

The Lemmas 5.9 and 5.10 show that the linking number of the trefoil knot $\gamma$ and a chain polar to a point in $\Omega _j {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}$ (for $j=0,1,2,3$) is equal to j. Finally, a proof of the Theorem 1.4(i) is given.

First, we state the following lemmas:

Lemma 5.1

If $[x:y:z]\in \mathbb {P}_{\mathbb {C}}^2$ and $t \in \mathbb {R}$ then the set of roots of the polynomial equation in $\zeta$,

$$\begin{aligned} \overline{(e^{2 i t}x)}\zeta ^2+\overline{(e^{3 i t}y)}\zeta ^3-\overline{z}=0 \end{aligned}$$

is obtained from the set of roots of the equation

$$\begin{aligned} \overline{x} \zeta ^2+ \overline{y} \zeta ^3 - \overline{z} =0 \end{aligned}$$

by multiplying by $e^{ i t}$. Hence, the smooth flow

$$\begin{aligned}&\phi : \mathbb {R} \times \mathbb {P}_{\mathbb {C}}^2 \rightarrow \mathbb {P}_{\mathbb {C}}^2,\\&\quad \phi _t[x:y:z]=[e^{2 i t} x : e^{3 i t}y :z] \end{aligned}$$

preserves $\Lambda (\gamma ), \Omega (\gamma ), \Omega _0, \Omega _1, \Omega _2$ and $\Omega _3$.

The proof of Lemma 5.1 is straightforward and we omit it. The following Lemma can be generalized to the case where the polynomial in question comes from the (p, q) torus knot, by using the fact that the roots of a polynomial depend continuously on its coefficients. Here we include this version for reader’s convenience.

Lemma 5.2

Let $p_0=[x_0:y_0:z_0], p_1=[x_1:y_1: z_1] \in \Omega (\gamma )$, such that $y_0 \ne 0$ and $y_1 \ne 0$. There exists a continuous path from $p_0$ to $p_1$, $g : [0,1] \rightarrow \Omega (\gamma )$, $g(t)=[g _1 (t): g _2(t): g _3 (t)]$ such that $g _2(t) \ne 0$ for all $t\in [0,1]$ if and only if there exists a continuous path $\mathbf {r}: [0,1] \rightarrow \mathbb {C}^3$, $\mathbf {r}(t)=(r_1(t), r_2(t), r_3(t))$, such that:

• (i) The sets of roots of the polynomial equations $\zeta ^3+\left( \frac{\overline{x_0}}{\overline{y_0}} \right) \zeta ^2 -\left( \frac{\overline{z_0}}{\overline{y_0}}\right) =0$ and $\zeta ^3+\left( \frac{\overline{x_1}}{\overline{y_1}} \right) \zeta ^2 -\left( \frac{\overline{z_1}}{\overline{y_1}}\right) =0$ are respectively equal to $\{ r_1(0), r_2(0), r_3(0) \}$ and $\{ r_1(1), r_2(1), r_3(1) \}$.

• (ii) $r_1(t)r_2(t)+r_2(t)r_3(t)+r_3(t)r_1(t)=0$, for every $t\in [0,1]$,

• (iii) $|r_1(t)|^2\ne \alpha$, $|r_2(t)|^2\ne \alpha$, $|r_3(t)|^2\ne \alpha$ for every $t\in [0,1]$.

Proof

First, if we assume that $g:[0,1]\rightarrow \Omega (\gamma )$ is a continuous path joining $p_0$ and $p_1$, then The Tartaglia–Cardano formulas for a cubic equation, imply that the roots of the cubic equation

$$\begin{aligned} \zeta ^3+\left( \frac{\overline{g_1(t)}}{\overline{g_2(t)}}\right) \zeta ^2- \left( \frac{\overline{g_3(t)}}{\overline{g_2(t)}}\right) =0 \end{aligned}$$

are continuous functions of its coefficients, hence continuous functions of t. If $r_1(t)$, $r_2(t)$, $r_3(t)$ denote such roots, then the path $\mathbf {r}:I \rightarrow \mathbb {C}^3$, $\mathbf {r}(t)=(r_1(t),r_2(t),r_3(t))$ satisfies (i) by construction. Item (ii) is satisfied by Viète identities, and finally, (iii) is satisfied because $g(t) \in \Omega (\gamma )$ for every $t \in [0,1]$.

Conversely, if there exists a continuous path

$$\begin{aligned}&\mathbf {r}: [0,1] \rightarrow \mathbb {C}^3,\\&\quad \mathbf {r}(t)=(r_1(t), r_2(t), r_3(t)), \end{aligned}$$

satisfying the conditions (i) to (iii), then it is not hard to check that the continuous path $g:[0,1] \rightarrow \mathbb {P}_{\mathbb {C}}^2$ given by

$$\begin{aligned} g(t)=\left[ -(\,\overline{r_1(t)}+\overline{r_2(t)}+\overline{r_3(t)} \, ):1: \overline{r_1(t)} \, \, \overline{r_2(t)} \, \, \overline{r_3(t)} \, \, \right] \end{aligned}$$

satisfies the required conditions.

Lemma 5.3

The set $\Omega (\gamma ) \cap \{[x:y:z] \in \mathbb {P}_{\mathbb {C}}^2 : y=0 \}$ has two connected components:

$$\begin{aligned} C_0= & {} \{[x:0:z]\, | \, \alpha |x| < |z|\},\\ C_2= & {} \{[x:0:z]\, | \, \alpha |x| > |z|\}. \end{aligned}$$

Moreover, $C_0 \subset \Omega _0$ and $C_2 \subset \Omega _2$.

Proof

We notice that a point $[x:0:z] \in \Lambda (\gamma )$ if and only if the equation $\bar{x} \zeta ^2 -\bar{z}=0$ has a solution $\zeta$ such that $|\zeta |^2=\alpha$, and it happens if and only if

$$\begin{aligned} \alpha |x|=|z|. \end{aligned}$$

This equation represents a circle in the line $\{[x:y:z] \in \mathbb {P}_{\mathbb {C}}^2 : y=0 \}$. Hence its complement has two connected components defined as

$$\begin{aligned} C_0= & {} \{[x:0:z]\, | \, \alpha |x| < |z|\},\\ C_2= & {} \{[x:0:z]\, | \, \alpha |x| > |z|\}. \end{aligned}$$

Moreover, if $[x:0:z] \in C_0$ then the inequality $\alpha |x| < |z|$ implies that the equation $\bar{x} \zeta ^2-\bar{z}=0$ has no root within the closed disk $\overline{D(0, \sqrt{\alpha })}$. Therefore $C_0 \subset \Omega _0$ . An analogous argument proves that $C_2 \subset \Omega _2$. $\square$

Lemma 5.4

The set $\Omega (\gamma ) \cap \{[x:y:z] \in \mathbb {P}_{\mathbb {C}}^2 : z=0 \}$ has two connected components:

$$\begin{aligned} B_2= & {} \{[x:y:0]\, | \, |x| > \sqrt{\alpha } \, |y|\},\\ B_3= & {} \{[x:y:0]\, | \, |x| < \sqrt{\alpha } \, |y|\}. \end{aligned}$$

Moreover, $B_2 \subset \Omega _2$ and $B_3 \subset \Omega _3$.

The proof of this Lemma is analogous to the proof of Lemma 5.3 and we omit it.

Lemma 5.5

The set $\Omega _0$ is a connected subset of $\Omega (\gamma )$. In fact, $\Omega _0$ is a contractible space.

Proof

The point [0 : 0 : 1] lies in $\Omega _0$ because the equation

$$\begin{aligned} -1=0 \end{aligned}$$

has no roots in the open disk $D(0, \sqrt{\alpha })$.

Let [x : y : z] be a point in $\Omega _0$, we give a continuous path in $\Omega _0$ joining [0 : 0 : 1] to the point [x : y : z]. First, we notice that $z\ne 0$, otherwise $\zeta =0$ is a root of the equation

$$\begin{aligned} \bar{x} \zeta ^2+ \bar{y}\zeta ^3-\bar{z}=0, \end{aligned}$$

contradicting the assumption that $[x:y:z] \in \Omega _0$. Now, the path

$$\begin{aligned}&f:[0,1] \rightarrow \mathbb {P}_{\mathbb {C}}^2\\&\quad f(t)=[t^2 x: t^3 y: z] \end{aligned}$$

is smooth, $f(0)=[0:0:z]=[0:0:1]$ and $f(1)=[x:y:z]$. Also, for every $1>t>0$, the set of roots of the polynomial equation in $\zeta$:

$$\begin{aligned} \overline{t^2 x} \zeta ^2+\overline{t^3 y} \zeta ^3-\overline{z}=0 \end{aligned}$$

is obtained from the set of roots of the equation

$$\begin{aligned} \bar{x}\zeta ^2+\bar{y}\zeta ^3-\bar{z}=0 \end{aligned}$$

by multiplying by $\frac{1}{t} >1$. Hence $f(t) \in \Omega _0$ for every $t\in [0,1]$.

Moreover, $\Omega _0$ is contractible because the function

$$\begin{aligned}&\psi : [0,1]\times \Omega _0 \rightarrow \Omega _0\\&\quad \psi _t[x:y:z]=[(1-t)^2 x: (1-t)^3 y: z] \end{aligned}$$

is a deformation retraction. $\square$

Lemma 5.6

The set $\Omega _1$ is a connected subset of $\Omega (\gamma )$.

Proof

Let $[x:y:z] \in \Omega _1$ then necessarily (5) is a cubic polynomial equation. Let us denote by $r_1, r_2, r_3$ the roots of the equation (5). We can assume that $r _1 \in D(0, \sqrt{\alpha })$. By Lemma 5.1 we can assume that $r_1 \in \mathbb {R}$ and $0< r_1<\sqrt{\alpha }$.

We denote by $T_r$ the Möbius transformation $T_r(z)=-\frac{r z}{z+r}$.

Claim 1.$r_2, r_3 \notin \overline{D(0, \sqrt{\alpha })}\cup T_{r_1}(\overline{D(0, \sqrt{\alpha })})$.

By hypothesis, $r_2 \notin \overline{D(0, \sqrt{\alpha })}$. If we assume that $r_2 \in T_{r_1}(\overline{D(0, \sqrt{\alpha })})$ then $r_3=T_{r_1}(r_2) \in T_{r_1}(T_{r_1}(\overline{D(0, \sqrt{\alpha })}))=\overline{D(0, \sqrt{\alpha })}$ contradicting the hypothesis that $[x:y:z] \in \Omega _1$. The proof for $r_3$ is analogous and we omit it.

Claim 2. The complement of the set $\overline{D(0, \sqrt{\alpha })}\cup T_{r_1}(\overline{D(0, \sqrt{\alpha })})$ in $\mathbb {C}$ is the intersection of the interior of the disk with diameter the points $T_{r_1}(-\sqrt{\alpha })$ and $T_{r_1}(\sqrt{\alpha })$, and the exterior of the closed disk $\overline{D(0, \sqrt{\alpha })}$. See the Fig. 1.

Fig. 1

The set $D(0,\sqrt{\alpha })\cup T_{r1} (D(0,\sqrt{\alpha }))$

Notice that

$$\begin{aligned} T_{r_1}(- \sqrt{\alpha })= & {} \frac{r_1 \sqrt{\alpha }}{-\sqrt{\alpha }+r_1}\\ T_{r_1}( \sqrt{\alpha })= & {} -\frac{r_1 \sqrt{\alpha }}{\sqrt{\alpha }+r_1}\\ T_{r_1}(0)= & {} 0. \end{aligned}$$

Claim 3. $r_1$ lies in the interval $(\frac{\sqrt{\alpha }}{2}, \sqrt{\alpha })$.

If $0\le r_1 \le \frac{\sqrt{\alpha }}{2}$ then $-\sqrt{\alpha } \le T_{r_1}(- \sqrt{\alpha }) \le T_{r_1}( \sqrt{\alpha })$, so the complement of the set $\overline{D(0, \sqrt{\alpha })}\cup T_{r_1}(\overline{D(0, \sqrt{\alpha })})$ in $\mathbb {C}$ is empty.

Claim 4. If $r_1 \in (\frac{\sqrt{\alpha }}{2}, \sqrt{\alpha })$ then the point $-2 r_1 \in (\overline{D(0, \sqrt{\alpha })}\cup T_{r_1}(\overline{D(0, \sqrt{\alpha })}))^c$. Since the region $(\overline{D(0, \sqrt{\alpha })}\cup T_{r_1}(\overline{D(0, \sqrt{\alpha })}))^c$ is connected, we can assume that $r_2 =-2 r_1$. Hence $r_3=-2 r_1$.

Claim 5. The path $\mathbf {r} : [0,1] \rightarrow \mathbb {C}^3$ given by

$$\begin{aligned} \mathbf {r}(t)= (1-t) (r_1, -2 r_1, -2 r_1)+t\left( \frac{1}{2},-1,-1\right) \end{aligned}$$

satisfies conditions (i), (ii) and (iii) of Lemma 5.2. Therefore, there exists a path from [x : y : z] to [3 : 2 : 1] in $\Omega _1$. $\square$

Lemma 5.7

The set $\Omega _2$ is a connected subset of $\Omega (\gamma )$.

Proof

If $[x:y:z] \in \Omega _2$ then we have two cases:

Case 1.$y \ne 0$ then (5) is a cubic polynomial equation. Let us denote by $r_1, r_2, r_3$ the roots of the equation (5). We can assume that $r _1, r_2 \in D(0, \sqrt{\alpha })$. Moreover, by Lemma 5.1, we can assume that $r_1 \in \mathbb {R}$ and $0\le r_1<\sqrt{\alpha }$.

Subcase 1. First, we assume that $r_1 \ne 0$.

We denote by $T_r$ the Möbius transformation $T_r(z)=-\frac{r z}{z+r}$.

Claim 1.

$$\begin{aligned}&r_2 \in D(0, \sqrt{\alpha })\cap (T_{r_1}(\overline{D(0, \sqrt{\alpha })}))^c ,\\&\quad r_3 \in (\overline{D(0, \sqrt{\alpha })})^c \cap T_{r_1}(D(0, \sqrt{\alpha })). \end{aligned}$$

If $r_2 \in T_{r_1}(\overline{D(0, \sqrt{\alpha })})$ then

$$\begin{aligned} r_3=T_{r_1}(r_2) \in T_{r_1} ( T_{r_1} (\overline{D(0, \sqrt{\alpha })}))= \overline{D(0, \sqrt{\alpha })}, \end{aligned}$$

and it is a contradiction. The second statement follows analogously.

Claim 2. The set $(\overline{D(0, \sqrt{\alpha })})^c \cap T_{r_1}(D(0, \sqrt{\alpha }))$ is the intersection of the exterior of the disk with diameter the segment determined by the points $T_{r_1}(-\sqrt{\alpha })$ and $T_{r_1}(\sqrt{\alpha })$, and the exterior of the disk $D(0, \sqrt{\alpha })$.

Claim 3. The set $(\overline{D(0, \sqrt{\alpha })})^c \cap T_{r_1}(D(0, \sqrt{\alpha }))$ is connected and contains the point 1 for every $r_1$ with $0<r_1<\sqrt{\alpha }$. So we can assume that $r_3 =1$.

Claim 4. The path $\mathbf {r} : [0,1] \rightarrow \mathbb {C}^3$ given by

$$\begin{aligned} \mathbf {r}(t)= \left( (1-t) r_1+\frac{t}{2}, \, - \frac{(1-t) r_1+\frac{t}{2}}{ (1-t) r_1+\frac{t}{2}+1}, \, 1\right) \end{aligned}$$

satisfies conditions (i), (ii) and (iii) of Lemma 5.2. Therefore, there exists a path from [x : y : z] to $[7:-6:1]$ in $\Omega _2$.

Subcase 2. If we assume that $r_1=0$, then $z=0$ and $r_2=0$. It follows from Lemma 5.4 that $[x:y:0] \in B_2$. Also, we can assume that $[x:y:0]=[-1:1:0]$.

The continuous path $g :[0,\epsilon ] \rightarrow \mathbb {P}_{\mathbb {C}}^2$ given by

$$\begin{aligned} g(t)=[3t^2+3t+1:-(2t+1):t^2(1+t)^2], \end{aligned}$$

satisfies that $g (0)=[1:-1:0]$ and the roots of the polynomial equation in $\zeta$ given by

$$\begin{aligned} -(2t+1) \zeta ^3 +(3t^2+3t+1)\zeta ^2 -t^2(1+t)^2=0 \end{aligned}$$

are precisely $t, 1+t, -\frac{t(1+t)}{2t+1})$. Therefore the path is contained in $\Omega _2$ for all $0 \le t \le \epsilon$ for $\epsilon$ small.

We have shown that [x : y : z] can be joined to a point in $\Omega _2$ with $r_1 \ne 0$ by a path contained in $\Omega _2$. By the subcase 1 above, [x : y : z] can be joined to the point $[7:-6:1]$ by a continuous path in $\Omega _2$.

Case 2.$y=0$.

By Lemma 5.3, we can assume that $[x:y:z]=[1:0:0]$.

The continuous path $g :[0,1] \rightarrow \mathbb {P}_{\mathbb {C}}^2$ given by

$$\begin{aligned} g(t)=[-(t^4+t^2+1):t(t^2+1):-t^2], \end{aligned}$$

satisfies that $g(0)=[1:0:0]$ and for $t> 0$ the roots of the polynomial equation in $\zeta$ given by

$$\begin{aligned} t(t^2+1) \zeta ^3 -(t^4+t^2+1)\zeta ^2 +t^2=0 \end{aligned}$$

are precisely $t, -\frac{t}{t^2+1}, \frac{1}{t}$. Therefore the path is contained in $\Omega _2$ for all $t>0$ sufficiently small.

We have shown that [1 : 0 : 0] can be joined to a point in $\Omega _2$ with $y \ne 0$ by a path contained in $\Omega _2$. The proof follows from Case 1 above. $\square$

Lemma 5.8

The set $\Omega _3$ is a connected subset of $\Omega (\gamma )$. Moreover, $\Omega _3$ is contractible.

Proof

Let [x : y : z] be a point in $\Omega _3$. First, we notice that $y \ne 0$, otherwise, the polynomial equation in the variable $\zeta$,

$$\begin{aligned} \bar{x} \zeta ^2 -\bar{z}=0 \end{aligned}$$

cannot have three roots within the open disk $D(0, \sqrt{\alpha })$.

The path given by

$$\begin{aligned}&f:[0,1] \rightarrow \mathbb {P}_{\mathbb {C}}^2\\&\quad f(t)=[t x: y: t^3 z], \end{aligned}$$

is a smooth path and satisfies: $f(0)=[0:y:0]=[0:1:0] \in \Omega _3$, $f(1)=[x:y:z] \in \Omega _3$. Also, for $0<t<1$, the set of roots of the polynomial equation in $\zeta$,

$$\begin{aligned} \overline{t x} \zeta ^2 + \overline{y} \zeta ^3 -\overline{t^3 z}=0, \end{aligned}$$

is obtained from the set of roots of the equation

$$\begin{aligned} \bar{x} \zeta ^2 +\bar{y} \zeta ^3 -\bar{z}=0, \end{aligned}$$

by multiplying by $0<t<1$. Hence $f(t) \in \Omega _3$ for every $t \in [0,1]$ and we have shown that $\Omega _3$ is path connected.

Moreover, the function

$$\begin{aligned}&\psi : [0,1] \times \Omega _3 \rightarrow \Omega _3\\&\quad \psi _t[x:y:z]=[(1-t)^2 x: y :(1-t)^3 z] \end{aligned}$$

is a deformation retraction, thus $\Omega _3$ is contractible. $\square$

Lemma 5.9

If $p_1=[1:1:1],\, p_2=[1:0:0],\, p_3:=[0:1:0]$ then

$$\begin{aligned} \ell (\gamma , C_{p_j})=j, \quad \text { for } j=1,2,3, \end{aligned}$$

where $C_{p_j}$ denotes the polar chain to $p_j$, $j=1,2,3$.

Proof

In order to simplify the notation in this proof, we consider the chains and the curve $\gamma$ as parametrized in the interval $I=[0,1]$.

$$\begin{aligned} t \mapsto \left[ \frac{1-e^{2 \pi i t}}{2}:\frac{1+e^{2 \pi i t}}{2}:1\right] , \quad t\in [0,1] \end{aligned}$$

is a parametrization of the polar chain to $p_1$. In Heisenberg coordinates, this curve is parametrized by:

$$\begin{aligned} \zeta (t)=-1, \quad v(t)=\frac{2 \sin (2\pi t)}{1+\cos (2 \pi t)}, \quad t\in [0,1], \quad t\ne 1/2. \end{aligned}$$

Thus it parametrizes a vertical line passing through the point $(-1,0)$. On the other hand,

$$\begin{aligned} \zeta (t)=\frac{\alpha ^{3/2} e^{6\pi i t}}{\alpha e^{4 \pi i t}-1}, \quad v(t)=\frac{-\alpha \sin (4 \pi t)}{|\alpha e ^{4 \pi i t}-1|^2}, \quad t\in [0,1] \end{aligned}$$

(6)

is a parametrization of the trefoil knot in Heisenberg coordinates.

It follows that $\ell (\gamma , C_{p_1})$ is equal to the winding number of the curve

$$\begin{aligned} \zeta (t)=\frac{\alpha ^{3/2} e^{6\pi i t}}{\alpha e^{4 \pi i t}-1}, \quad t\in [0,1] \end{aligned}$$

respect to the point $-1$. In other words,

$$\begin{aligned} \ell (\gamma , C_{p_1})= & {} \frac{1}{2\pi i} \int _{\zeta (t)}\frac{dz}{z+1} \\= & {} \frac{1}{2 \pi i} \int _{w(t)} \frac{w^2(w^2-3) dw}{(w^2-1)(w^3+w^2-1)} \end{aligned}$$

where $w(t)=\sqrt{\alpha }\, e^{2\pi i t}$, $t \in [0,1]$. Finally, using the fact that $\alpha$ is the only pole of $\frac{w^2(w^2-3) }{(w^2-1)(w^3+w^2-1)}$ in the disk $D(0, \sqrt{\alpha })$, we obtain that

$$\begin{aligned} \ell (\gamma , C_{p_1})=\frac{\alpha ^2(\alpha ^2-3)}{(\alpha ^2-1)(3\alpha ^2+2\alpha )}=1. \end{aligned}$$

The curve

$$\begin{aligned} t \mapsto [0:e^{2 \pi i t}:1], \quad t\in [0,1] \end{aligned}$$

is a parametrization of the polar chain to $p_2$. Applying the canonical stereographic projection $S^3{\setminus }\{N\} \rightarrow \mathbb {R}^3$, we obtain the curve

$$\begin{aligned} t \mapsto \left( 0,0, \frac{\cos (2 \pi t)}{1- \sin (2 \pi t)}\right) , \quad t\in [0,1], \end{aligned}$$

which is a parametrization of the z-axis. Also, the stereographic projection of the trefoil knot is parametrized by

$$\begin{aligned} t \mapsto \left( \frac{\alpha \cos (4 \pi t)}{1 - \alpha ^{\frac{3}{2}} \sin (6 \pi t)}, \frac{\alpha \sin (4 \pi t)}{1 - \alpha ^{\frac{3}{2}} \sin (6 \pi t)}, \frac{\alpha ^{\frac{3}{2}} \cos (6\pi t)}{1 - \alpha ^ {\frac{3}{2}} \sin (6 \pi t)}\right) , \quad t\in [0,1]. \end{aligned}$$

It follows that $\ell (\gamma , C_{p_2})$ is equal to the winding number of the curve

$$\begin{aligned} t \mapsto \left( \frac{\alpha \cos (4 \pi t)}{1 - \alpha ^{\frac{3}{2}} \sin (6 \pi t)}, \frac{\alpha \sin (4 \pi t)}{1 - \alpha ^{\frac{3}{2}} \sin (6 \pi t)} \right) , \quad t\in [0,1], \end{aligned}$$

respect to the origin and such winding number is equal to two.

The curve

$$\begin{aligned} t \mapsto \left[ -e^{2 \pi i t}:0:1\right] , \quad t\in [0,1] \end{aligned}$$

is a parametrization of the polar chain to $p_3$. In Heisenberg coordinates, this curve is parametrized by:

$$\begin{aligned} \zeta (t)=0, \quad v(t)=\frac{ \sin (2\pi t)}{1+\cos (2 \pi t)}, \quad t\in [0,1], \quad t\ne 1/2. \end{aligned}$$

Thus it parametrizes the v-axis. It follows that $\ell (\gamma , C_{p_1})$ is equal to the winding number of the projection of the curve (6) to the $\zeta$-plane, with respect to the origin. By straightforward computations this winding number is equal to 3. $\square$

Lemma 5.10

If $p \in \Omega _j {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}$ (where $j=0,1,2,3$) then

$$\begin{aligned} \ell (\gamma ,C_p)=j, \end{aligned}$$

where $C_p$ is the polar chain to p.

Proof

If $p \in \Omega _0 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}$ then p can be joined, by a continuous path contained in $\Omega _0$, to the point $[0:0:1] \in \Omega _0 \cap \mathbb {H}^2 _{\mathbb {C}}$ (because $\Omega _0$ is path connected). So p can be joined by a continuous path contained in $\Omega _0$ to a point in $\partial \mathbb {H}^2 _{\mathbb {C}}$. Lemma 3.5 implies that $\ell (\gamma , C_p)=0$.

If $p \in \Omega _1 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}$ then p can be joined by a continuous path contained in $\Omega _1$ to the point $p_1=[1:1:1] \in \Omega _1 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}$ (because $\Omega _1$ is path connected). By Lemma 3.5, this continuous path is contained in $\mathbb {P}_{\mathbb {C}}^2 {\setminus } \overline{\mathbb {H}^2 _{\mathbb {C}}}$, otherwise $\ell (\gamma , C_{p_1})=0$, which is a contradiction to Lemma 5.9. Therefore

$$\begin{aligned} \ell (\gamma , C_p)=\ell (\gamma , C_{p_1})=1. \end{aligned}$$

The proof for the other cases is analogous. $\square$

Finally, we prove the first item in Theorem 1.4.

Proof of 1.4 (i)

Lemmas 5.5, 5.6, 5.7, 5.8 show that $\Omega _j$ is path connected for every $j=0,1,2,3$. Finally, Corollary 1.2 and Lemma 5.10 imply that $\Omega _0,$$\Omega _1$, $\Omega _2$, $\Omega _3$ are the components of the open set $\Omega (\gamma )$. $\square$

The proof of the following Corollary is straightforward and we omit it.

Corollary 5.11

Assuming the hypothesis of Theorem 1.4, we have that

• (i) $\mathbb {H}^2 _{\mathbb {C}} \subset \Omega _0$.

• (ii) Every $\Omega _j$ is open in $\mathbb {P}_{\mathbb {C}}^2$ for $j=0,1,2,3$.

• (iii) $\partial \Omega _j \subset \Lambda (\gamma )$, for $j=0,1,2,3$.

5.2 The Set $\Lambda (\gamma )$

The purpose of this subsection is to describe the topology of the set $\Lambda (\gamma )$ when $\gamma$ is the trefoil knot given by:

$$\begin{aligned}&\displaystyle \gamma : S^1 \rightarrow \mathbb {H}^2 _{\mathbb {C}} \cong S^3\\&\displaystyle \gamma (e^{2\pi i t})=[(\sqrt{\alpha }e^{2 \pi i t})^2: (\sqrt{\alpha }e^{2 \pi i t})^3:1]. \end{aligned}$$

In other words, we prove Theorem 1.4(ii).

Proof of Theorem 1.4(ii)

First, we claim that $S(\gamma )$ is $\phi _t$ invariant, where $\phi _t[x:y:z]=[e^{2it}x:e^{3it}y:z], \, t \in \mathbb {R}$. In fact, if $q \in S(\gamma )$, then there exist $p_1, p_2 \in \gamma (S^1)$ such that $\{q\}=\ell _{p_1} \cap \ell _{p_2}$. We see that $\phi _t(\ell _p)= \ell _{\phi _t(p)}$ for every $t\in \mathbb {R}$ and $p \in \partial \mathbb {H}^2 _{\mathbb {C}}$, because $\phi _t \in \text {PU}(2,1)$. It follows that

$$\begin{aligned} \phi _t(\{ q \})= & {} \phi _t(\ell _{p_1} \cap \ell _{p_2}) \\= & {} \phi _t(\ell _{p_1}) \cap \phi _t(\ell _{p_2})\\= & {} \ell _{\phi _t(p_1)}\cap \ell _{\phi _t(p_2)}. \end{aligned}$$

Hence, the claim is proved by the fact that $\gamma (S^1)$ is invariant under $\phi _t$.

Since the action of the flow $\phi _t, \, t\in \mathbb {R},$ is transitive on the set $\gamma (S^1)$, for any fixed $p_0 \in \gamma (S^1)$, we have that

$$\begin{aligned} \Lambda (\gamma ) = \bigcup _{t \in [0,2\pi ]} \ell _{\phi _t(p_0)} = \bigcup _{t \in [0,2\pi ]} \phi _t(\ell _{p_0}), \end{aligned}$$

and

$$\begin{aligned} \Lambda (\gamma ) {\setminus } S(\gamma ) = \bigcup _{t \in [0,2\pi ]} \phi _t(\ell _{p_0} {\setminus } S(\gamma )). \end{aligned}$$

(7)

In particular, we choose $p_0=[\alpha : \alpha ^{3/2} : 1] \in \gamma (S^1)$, and we see that

$$\begin{aligned} \ell _{p_0}= \{[x:y:z] \in \mathbb {P}_{\mathbb {C}}^2 : \alpha \bar{x}+ \alpha ^{3/2} \bar{y}-\bar{z}=0\} \end{aligned}$$

It follows that $\ell _{p_0} \cap S(\gamma )$ consists of those points $q=[x:y:z] \in \mathbb {P}_{\mathbb {C}}^2$ such that

$$\begin{aligned}&\displaystyle \alpha \bar{x}+ \alpha ^{3/2} \bar{y}-\bar{z}=0, \text { and }\\&\displaystyle \alpha e^{2 it} \bar{x}+ \alpha ^{3/2} e^{3it} \bar{y}-\bar{z}=0, \quad \text { for some } t \in [0,2\pi ]. \end{aligned}$$

By solving this system of equations, we can describe $\ell _{p_0} \cap S(\gamma )$ as the set of points $[x(t):y(t):z(t)], \, t \in [0, 2\pi ],$ such that

$$\begin{aligned} x(t)= & {} \frac{1+e^{2it}+e^{4it}}{\alpha },\\ y(t)= & {} -\frac{e^{2it}+e^{4it}}{\alpha ^{3/2}},\\ z(t)= & {} 1. \end{aligned}$$

Now, we make a change of coordinates by the element $g \in \text {PU}(2,1)$ induced by the matrix

$$\begin{aligned} \mathbf {g}=\left( \begin{array}{ccc} -\alpha ^{3/2} &{}\quad \alpha &{}\quad 0 \\ \alpha &{}\quad \alpha ^{3/2} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 \end{array}\right) , \end{aligned}$$

so $\ell _{p_0}$ corresponds to the horizontal line given in inhomogeneous coordinates by

$$\begin{aligned} \{(w,1) \in \mathbb {C}^2 : w \in \mathbb {C}\}, \end{aligned}$$

and $\ell _{p_0}\cap S(\gamma )$ corresponds to the curve

$$\begin{aligned} w(t)=-\sqrt{\alpha } -\left( \sqrt{\alpha }+\frac{1}{\sqrt{\alpha }}\right) (e^{2it}+e^{4it}), \, \, t\in [0,2\pi ] \end{aligned}$$

which is a limaçon of Pascal satisfying that $\ell _{p_0} {\setminus } S(\gamma )$ consists of three topological disks. Finally, the proof follows from (7). $\square$

If $S^1(\sqrt{\alpha })\subset \mathbb {C}$ denotes the circle of center 0 and radius $\sqrt{\alpha }$ then $\Lambda (\gamma )$ can be parametrized by the function

$$\begin{aligned}&\displaystyle L: S^1(\sqrt{\alpha }) \times \mathbb {P}_{\mathbb {C}}^1 \rightarrow \mathbb {P}_{\mathbb {C}}^2\\&\displaystyle L(\zeta , [\mu _1 : \mu _2])\!=\! [\mu _1 (\zeta ^2, \zeta ^3,1 )+\mu _2 (-\overline{\zeta },1,0)]\!=\! [\mu _1\zeta ^2- \mu _2 \overline{\zeta }:\zeta ^3 +\mu _2 : \mu _1], \end{aligned}$$

because the point $[-\overline{\zeta }:1:0]$ lies on the complex line tangent to $\partial \mathbb {H}^2 _{\mathbb {C}}$ at the point $[\zeta ^2 : \zeta ^3 :1]$. Notice that the points in $S(\gamma )$ are those points $q \in \Lambda (\gamma )$ for which the inverse image $L^{-1}(q)$ consists of more than one point.

6 Open Questions

Let $C_1, \ldots , C_n$ be n distinct vertical chains in the Heisenberg space $\mathcal {H}$, one can build a smooth trivial knot $\gamma :S^1 \rightarrow \mathcal {H}$ so that $\ell (\gamma , C_k)=k$ for $k=1, \ldots , n$. Thus $\Omega (\gamma )$ has at least $n+1$ components by Corollary 1.2. Hence, there exist trivial knots $\gamma :S^1 \rightarrow \partial \mathbb {H}^2 _{\mathbb {C}}\cong S^3$ such that the number of components of $\Omega (\gamma )$ is arbitrarily large. On the other hand, a natural question is the following: What is the minimum number of components of $\Omega (\gamma )$, when $\gamma$ runs over all smooth trivial knots? It is not hard to see that this minimum number is equal to two and it is achieved in the case when the image of $\gamma$ is a chain.

Some more general questions are the following:

• (i) What is the minimum number of components of $\Omega (\gamma )$, when $\gamma$ runs over all smooth knots in the same class of isotopy?

• (ii) Is it possible to characterize all the curves in the same isotopy class such that $\Omega (\gamma )$ achieves its minimum number of components?

• (iii) Other questions in the same style are obtained by replacing “number of components of $\Omega (\gamma )$” by “number of all distinct linking numbers $\ell (\gamma , C)$ for all chains C not meeting the image of $\gamma$” in the two questions above.