Abstract
The present paper is devoted to graph-links with many components and consists of two parts. In the first part of the paper we classify vertices of a labeled graph according to the component they belong to. Using this classification, we construct an invariant of graph-links. This invariant shows that the labeled second Bouchet graph generates a nonrealizable graph-link.
In the second part of the work we introduce the notion of an oriented graph-link. We define a writhe number for the oriented graph-link and we get an invariant of oriented graph-links, the Jones polynomial, by normalizing the Kauffman bracket with the writhe number.
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References
D. Bar-Natan and S. Garoufalidis, “On the Melvin–Morton–Rozansky conjecture,” Invent. Math., 125, 103–133 (1996).
A. Bouchet, “Circle graph obstructions,” J. Combin. Theory Ser. B, 60, 107–144 (1994).
S. V. Chmutov, S. V. Duzhin, and S.K. Lando, “Vassiliev knot invariants. I, II, III,” Adv. Sov. Math., 21, 117–147 (1994).
S. Chmutov, S. Duzhin, and J. Mostovoy, Introduction to Vassiliev Knot Invariants, Cambridge University Press, Cambridge (2012).
M. Cohn and A. Lempel, “Cycle decomposition by disjoint transpositions,” J. Combin. Theory Ser. A, 13, 83–89 (1972).
M. Goussarov, M. Polyak, and O. Viro, “Finite type invariants of classical and virtual knots,” Topology, 39, 1045–1068 (2000).
D.P. Ilyutko, “Framed 4-valent graphs: Euler tours, Gauss circuits and rotating circuits,” Sb. Math., 202, No. 9, 5303–1326 (2011).
D.P. Ilyutko, “An equivalence between the set of graph-knots and the set of homotopy classes of looped graphs,” J. Knot Theory Ramif., 21, No. 1, DOI: 10.1142/S0218216512500010 (2012).
D.P. Ilyutko and V. O. Manturov, “Introduction to graph-link theory,” J. Knot Theory Ramif., 18, No. 6, 791–823 (2009).
D.P. Ilyutko and V. O. Manturov, “Graph-links,” Dokl. Math., 80, No. 2, 739–742 (2009).
D.P. Ilyutko and V.O. Manturov, “Graph-links,” in: Proceedings of the Advanced Summer School on Knot Theory, Trieste, Italy, May 11–29, 2009; World Scientific, Hackensack; ICTP, Trieste; 135–161 (2012).
D.P. Ilyutko, V.O. Manturov, and I. M. Nikonov, “Parity in knot theory and graph-links,” J. Math. Sci., 193, No. 6, 809–965 (2013).
N. Kamada and S. Kamada, “Abstract link diagrams and virtual knots,” J. Knot Theory Ramif., 9, No. 1, 93–109 (2000).
L. H. Kauffman, “Virtual Knots,” talks at MSRI Meeting, January 1997 and AMS meeting at University of Maryland, College Park, March 1997.
L. H. Kauffman, “Virtual knot theory,” Eur. J. Combin., 20, No. 7, 663–690 (1999).
V.O. Manturov, Knot Theory, CRC-Press, Boca Raton (2004).
V.O. Manturov, Knot Theory [in Russian], RCD, Moscow–Izhevsk (2005).
V.O. Manturov, “On free knots,” arXiv: math.GT/0901.2214.
V.O. Manturov, “On free knots and links,” arXiv: math.GT/0902.0127.
V.O. Manturov, “Parity in knot theory,” Sb. Math., 201, No. 5, 693–733 (2010).
V.O. Manturov and D. P. Ilyutko, Virtual Knots: The State of the Art, World Scientific, Singapore (2013).
G. Moran, “Chords in a circle and linear algebra over GF(2),” J. Combin. Theory Ser. A, 37, 239–247 (1984).
V. V. Prasolov and A.B. Sossinsky, Knots, Links, Braids and 3-Manifolds. An Introduction to the New Invariants in Low-Dimensional Topology, AMS, Providence (1996).
R.C. Read and P. Rosenstiehl, “On the Gauss crossing problem,” Colloq. Math. Soc. Janos Bolyai, 18, 843–876 (1978).
K. Reidemeister, Knotentheorie, Springer, Berlin (1932).
E. Soboleva, “Vassiliev knot invariants coming from Lie algebras and 4-invariants,” J. Knot Theory Ramif., 10, No. 1, 161–169 (2001).
L. Traldi, “A bracket polynomial for graphs. II. Links, Euler circuits and marked graphs,” J. Knot Theory Ramif., 19, 547–586 (2010).
L. Traldi, “A bracket polynomial for graphs. III. Vertex weights,” arXiv: math.GT, math.CO/ 0905.4879.
L. Traldi, “Binary nullity, Euler circuits and interlace polynomials,” arXiv: math.CO/0903.4405.
L. Traldi and L. Zulli, “A bracket polynomial for graphs,” J. Knot Theory Ramif., 18, 1681–1709 (2009).
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 51, Topology, 2013.
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Ilyutko, D.P., Safina, V.S. Graph-Links: Nonrealizability, Orientation, and Jones Polynomial. J Math Sci 214, 632–664 (2016). https://doi.org/10.1007/s10958-016-2803-4
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DOI: https://doi.org/10.1007/s10958-016-2803-4