No Access

Graphical virtual links and a polynomial for signed cyclic graphs

Qingying Deng

School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, P. R. China

E-mail Address: qydeng.xmu@gmail.com

Search for more papers by this author

Xian’an Jin

School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, P. R. China

E-mail Address: xajin@xmu.edu.cn

Corresponding author.

Search for more papers by this author

, and

Louis H. Kauffman

Department of Mathematics, Statistics and Computer Science, 851 South Morgan Street, University of Illinois at Chicago, Chicago, Illinois 60607-7045, USA

Department of Mechanics and Mathematics, Novosibirsk State University, Novosibirsk, Russia

E-mail Address: kauffman@uic.edu

Search for more papers by this author

https://doi.org/10.1142/S0218216518500542Cited by:3 (Source: Crossref)

Abstract

For a signed cyclic graph $G$ , we can construct a unique virtual link $L$ by taking the medial construction and converting 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In this paper, we shall prove that a virtual link $L$ is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial $F [G]$ for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between $F [G]$ of a signed cyclic graph $G$ and the bracket polynomial of one of the virtual link diagrams associated with $G$ . Finally, we give a spanning subgraph expansion for $F [G]$ .

Keywords:

AMSC: 57M25, 57M27

References

1. B. Bollobás and O. Riordan, A polynomial invariant of graphs on orientable surfaces, Proc. London Math. Soc. 83(3) (2001) 513–531. Web of Science, Google Scholar
2. B. Bollobás and O. Riordan, A polynomial invariant of graphs on surfaces, Math. Ann. 323 (2002) 81–96. Web of Science, Google Scholar
3. S. Chmutov, Generalized duality for graphs on surfaces and the signed Bollobás–Riordan polynomial, J. Combin. Theory Ser. B 99 (2009) 617–638. Web of Science, Google Scholar
4. S. Chmutov, Topological tutte polynomial, preprint (2017), arXiv:1708.08132. Google Scholar
5. S. Chmutov and I. Pak, The Kauffman bracket of virtual links and the Bollobás–Riordan polynomial, Mosc. Math. J. 7(3) (2007) 409–418. Web of Science, Google Scholar
6. H. A. Dye, A. Kaestner and L. H. Kauffman, Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots, J. Knot Theory Ramifications 26(3) (2017), Article ID: 1741001, 57 pp. Link, Web of Science, Google Scholar
7. J. Ellis-Monaghan and I. Moffatt, Graphs on Surfaces: Dualities, Polynomials, and Knots (Springer, New York, 2013). Google Scholar
8. R. Fenn, D. P. Ilyutko, L. H. Kauffman and V. O. Manturov, Unsolved problems in virtual knot theory and combinatorial knot theory, in Knots in Poland III, Part III, Banach Center Publications, Vol. 103 (Polish Academy of Sciences Institute Mathematics, Warsaw, 2014), pp. 9–61. Google Scholar
9. J. Green, A table of virtual knots, http://www.math.toronto.edu/drorbn/Students/GreenJ/. Google Scholar
10. T. Imabeppu, On Sawollek polynomials of checkerboard colorable virtual links, J. Knot Theory Ramifications 25(2) (2016), Article ID: 1650010, 19 pp. Link, Web of Science, Google Scholar
11. N. Kamada, On the Jones polynomials of checkerboard colorable virtual knots, Osaka J. Math. 39(2) (2002) 325–333. Web of Science, Google Scholar
12. N. Kamada, Span of the Jones polynomial of an alternating virtual link, Algebr. Geom. Topol. 4 (2004) 1083–1101. Google Scholar
13. N. Kamada and S. Kamada, Abstract link diagrams and virtual knots, J. Knot Theory Ramifications 9(1) (2000) 93–106. Link, Web of Science, Google Scholar
14. L. H. Kauffman, New invariants in the theory of knots, Amer. Math. Monthly 95(3) (1988) 195–242. Web of Science, Google Scholar
15. L. H. Kauffman, Virtual knot theory, European J. Combin. 20(7) (1999) 663–690. Web of Science, Google Scholar
16. L. H. Kauffman, A Tutte polynomial for signed graphs, Discrete Appl. Math. 25 (1999) 105–127. Web of Science, Google Scholar
17. L. H. Kauffman, Introduction to virtual knot theory, J. Knot Theory Ramifications 21(13) (2012), Article ID: 1240007, 37 pp. Link, Web of Science, Google Scholar
18. L. H. Kauffman, Talks on Combinatorial Knot Theory, Xiamen University, June (2016). Google Scholar
19. W. T. Tutte, A contribution to the theory of chromatic polynomials, Canad. J. Math. 6 (1954) 80–91. Web of Science, Google Scholar