Special Issue: Dedicated to 15th Anniversary of the Seminar “Knots and Representation Theory”; Guest Editors: V. O. Manturov, D. P. Ilyutko and I. M. NikonovNo Access

Rotational virtual knots and quantum link invariants

Louis H. Kauffman

Louis H. Kauffman

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

E-mail Address: kauffman@math.uic.edu

Search for more papers by this author

https://doi.org/10.1142/S0218216515410084Cited by:7 (Source: Crossref)

Abstract

This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. We give examples of non-trivial rotational virtuals that are undectable by quantum invariants.

Keywords:

AMSC: 57M27

References

1. J. S. Carter, S. Kamada and M. Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications 11(3) (2002) 311–320. Link, Web of Science, Google Scholar
2. V. G. Drinfeld, Quantum groups, in Proc. Congr. Mathematicians, Berkeley, California, USA (1987), pp. 798–820 (American Mathematical Society, Providence, RI, 1987). Google Scholar
3. H. Dye, Characterizing virtual knots, Ph.D. thesis, University of Illionois at Chicago (2002). Google Scholar
4. H. A. Dye, A. Kaestner and L. H. Kauffman, Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots, preprint (2014), arXiv:1409.5088. Google Scholar
5. H. Dye and L. H. Kauffman, Virtual knot diagrams and the Witten–Reshetikhin–Turaev invariant, J. Knot Theory Ramifications 14(8) (2005) 1045–1075. Link, Web of Science, Google Scholar
6. M. Goussarov, M. Polyak and O. Viro, Finite type invariants of classical and virtual knots, Topology 39 (2000) 1045–1068. Crossref, Web of Science, Google Scholar
7. M. A. Hennings, Hopf algebras and regular isotopy invariants for link diagrams, Math. Proc. Cambridge Philos. Soc. 109 (1991) 59–77. Crossref, Web of Science, Google Scholar
8. D. P. Ilyutko and V. O. Manturov, Cobordisms of free knots, Dokl. Akad. Nauk 429(4) (2009) 439–441 (in Russian); Dokl. Math. 80(3) (2009) 844–846. Google Scholar
9. V. F. R. Jones, A polynomial invariant for links via von Neumann algebras, Bull. Amer. Math. Soc. 129 (1985) 103–112. Crossref, Web of Science, Google Scholar
10. V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987) 335–338. Crossref, Web of Science, Google Scholar
11. V. F. R. Jones, On knot invariants related to some statistical mechanics models, Pacific J. Math. 137(2) (1989) 311–334. Crossref, Web of Science, Google Scholar
12. L. H. Kauffman, Formal Knot Theory, Lecture Notes Series, Vol. 30 (Princeton University Press, 1983). Google Scholar
13. L. H. Kauffman, State models and the Jones polynomial, Topology 26 (1987) 395–407. Crossref, Web of Science, Google Scholar
14. L. H. Kauffman, Knots and Physics (World Scientific, 1991), 2nd edn. (1993), 3rd edn. (2002), 4th edn. (2012). Link, Google Scholar
15. L. H. Kauffman, Virtual knot theory, European J. Combin. 20 (1999) 663–690. Crossref, Web of Science, Google Scholar
16. L. H. Kauffman, A survey of virtual knot theory, in Proc. Knots in Hellas ’98 (World Scientific, 2000), pp. 143–202. Google Scholar
17. L. H. Kauffman, Detecting virtual knots, Atti Sem. Mat. Fis. Univ. Modena 49 (suppl.) (2001) 241–282. Google Scholar
18. L. H. Kauffman, A self-linking invariant of virtual knots, Fund. Math. 184 (2004) 135–158. Crossref, Web of Science, Google Scholar
19. L. H. Kauffman, Virtual knot cobordism, in New Ideas in Low Dimensional Topology, eds. L. H. Kauffman and V. O. Manturov, Knots and Everything Series, Vol. 56 (World Scientific, 2014), pp. 335–377. Google Scholar
20. L. H. Kauffman and S. Lambropoulou, A categorical model for the virtual braid group, J. Knot Theory Ramifications 21(13) (2012) Article ID:1240008, 48 pp. Link, Web of Science, Google Scholar
21. L. H. Kauffman and S. J. Lomonaco, Braiding operators are universal quantum gates, New J. Phys. 6 (2004) 1–39. Crossref, Web of Science, Google Scholar
22. L. H. Kauffman and D. E. Radford, Invariants of 3-manifolds derived from finite dimensional Hopf algebras, J. Knot Theory Ramifications 4(1) (1995) 131–162. Link, Web of Science, Google Scholar
23. L. H. Kauffman and D. E. Radford, Oriented quantum algebras and invariants of knots and links, J. Algebra 246 (2001) 253–291. Crossref, Web of Science, Google Scholar
24. L. H. Kauffman and D. E. Radford, Oriented quantum algebras, categories and invariants of knots and links, J. Knot Theory Ramifications 10(7) (2001) 1047–1084. Link, Web of Science, Google Scholar
25. L. H. Kauffman and D. E. Radford, Bi-oriented quantum algebras, and a generalized Alexander polynomial for virtual links, in Diagrammatic Morphisms and Applications, Contemporary Mathematics, Vol. 318 (American Mathematical Society, Providence, RI, 2003), pp. 113–140. Crossref, Google Scholar
26. M. Khovanov and L. Rozansky, Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial, preprint (2007), arXiv:math10701333. Google Scholar
27. M. Khovanov and L. Rozansky, Matrix factorizations and link homology. II, Geom. Topol. 12(3) (2008) 1387–1425. Crossref, Web of Science, Google Scholar
28. M. Khovanov and L. Rozansky, Matrix factorizations and link homology, Fund. Math. 199(1) (2008) 191. Crossref, Web of Science, Google Scholar
29. R. Kirby, A calculus for framed links in $S^{3}$ , Invent. Math. 45 (1978) 35–56. Crossref, Web of Science, Google Scholar
30. V. O. Manturov, Parity in knot theory, Sb. Math. 201(5) (2010) 693–733 (in Russian); Math. Sb. 201(5) (2010) 65–110. Google Scholar
31. V. O. Manturov, Parity and cobordisms of free knots, Mat. Sb. 203(2) (2012) 45–76 (in Russian); Sb. Math. 203(1–2) (2012) 196–223. Google Scholar
32. V. O. Manturov and D. P. Ilyutko, Virtual Knots — The State of the Art, Series on Knots and Everything, Vol. 51 (World Scientific, 2012). Link, Google Scholar
33. N. Y. Reshetikhin, Quantized universal enveloping algebras, the Yang–Baxter equation and invariants of links, I and II, LOMI reprints E-4-87 and E-17-87, Steklov Institute, Leningrad, USSR. Google Scholar
34. N. Y. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990) 1–26. Crossref, Web of Science, Google Scholar
35. M. E. Sweedler, Hopf Algebras, Mathematics Lecture Notes Series (Benjamin, New York, 1969). Google Scholar
36. V. G. Turaev, The Yang–Baxter equations and invariants of links, Invent. Math. 92(3) (1998) 527–553. Crossref, Web of Science, Google Scholar
37. E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351–399. Crossref, Web of Science, Google Scholar
38. A. M. Kaestner and L. H. Kauffman, Parity, skein polynomials and categorification, J. Knot Theory Ramifications 21(13) (2012) 1240011, 56 pp. Link, Web of Science, Google Scholar