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Virtual Knot Theory and Virtual Knot Cobordism

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Book cover Knots, Low-Dimensional Topology and Applications (KNOTS16 2016)

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Abstract

This paper is an introduction to virtual knot theory and virtual knot cobordism [37, 39]. Non-trivial examples of virtual slice knots are given and determinations of the four-ball genus of positive virtual knots are explained in relation to joint work with Dye and Kaestner [12]. We study the affine index polynomial [38], prove that it is a concordance invariant, show that it is invariant also under certain forms of labeled cobordism and study a number of examples in relation to these phenomena. In particular we show how a mod-2 version of the affine index polynomial is a concordance invariant of flat virtual knots and links, and explore a number of examples in this domain.

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References

  1. V.G. Bardakov, The virtual and universal braids. Fund. Math. 184, 1–18 (2004)

    Article  MathSciNet  Google Scholar 

  2. V.G. Bardakov, Yu.A. Mikhalchishina, M.V. Neshchadim, Virtual link groups (Russian). Sibirsk. Mat. Zh. 58(5), 989–1003 (2017); Translation in Sib. Math. J. 58(5), 765–777 (2017)

    Google Scholar 

  3. V.G. Bardakov, Yu.A. Mikhalchishina, M.V. Neshchadim, Groups of virtual trefoil and Kishino knots (2018), arXiv:1804.06240v1 [math.GT]. Accessed 17 Apr 2018

  4. H. Boden, M. Chrisman, R. Gaudreau, Virtual knot cobordism and bounding the silce genus (2017), arXiv:1708.05982v2. Accessed 26 Dec 2017

  5. H. Boden, M. Nagel, Concordance group of virtual knots. Proc. Am. Math. Soc. 145(12), 5451–5461 (2017)

    Article  MathSciNet  Google Scholar 

  6. Z. Cheng, A polynomial invariant of virtual knots. Proc. Am. Math. Soc. 142(2), 713–725 (2014)

    Article  MathSciNet  Google Scholar 

  7. M. Chrisman, Band-passes and long virtual knot concordance. J. Knot Theory Ramif. 26(10), 1750057, 11 pp. (2017)

    Google Scholar 

  8. Q. Deng, X. Jin, L.H. Kauffman, Graphical virtual links and a polynomial for signed cyclic graphs. J. Knot Theory Ramif. 27(10), 1850054, 14 pp. (2018)

    Google Scholar 

  9. H.A. Dye, Vassiliev invariants from parity mappings. J. Knot Theory Ramif. 22(4), 1340008, 21 pp. (2013)

    Google Scholar 

  10. H.A. Dye, L.H. Kauffman, Minimal surface representations of virtual knots and links. Algebr. Geom. Topol. 5, 509–535 (2005)

    Article  MathSciNet  Google Scholar 

  11. H.A. Dye, L.H. Kauffman, Virtual crossing number and the arrow polynomial. JKTR 18(10), 1335–1357 (2009)

    Google Scholar 

  12. H.A. Dye, A. Kaestner, L.H. Kauffman, Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots. J. Knot Theory Ramif. 26(3), 1741001, 57 pp. (2017)

    Google Scholar 

  13. S. Eliahou, L.H. Kauffman, M. Thistlethwaite, Infinite families of links with trivial Jones polynomial. Topology 42, 155–169 (2003)

    Google Scholar 

  14. R.A. Fenn, L.H. Kauffman, V.O. Manturov, Virtual knots: unsolved problems. Fundamenta Mathematicae, in Proceedings of the Conference “Knots in Poland-2003”, vol. 188, pp. 293–323 (2005)

    Google Scholar 

  15. R. Fenn, D.P. Ilyutko, L.H. Kauffman, V.O. Manturov, Unsolved problems in virtual knot theory and combinatorial knot theory. Knots in Poland III. Part III, vol. 103 (Banach Center Publications, Institute of Mathematics, Polish Academy of Sciences, Warsaw, 2014), pp. 9–61

    Google Scholar 

  16. R.H. Fox, J.W. Milnor, Singularities of 2-spheres in 4-space and cobordism of knots. Osaka J. Math. 3, 257–267 (1966)

    MathSciNet  MATH  Google Scholar 

  17. M. Goussarov, M. Polyak, O. Viro, Finite-type invariants of classical and virtual knots. Topology 39(5), 1045–1068 (2000)

    Article  MathSciNet  Google Scholar 

  18. N. Gügümcü, L.H. Kauffman, New invariants of knotoids. Eur. J. Combin. 65, 186–229 (2017)

    Article  MathSciNet  Google Scholar 

  19. N. Gügümcü, L.H. Kauffman, On the height of knotoids, Algebraic Modeling of Topological and Computational Structures and Applications. Springer Proceedings in Mathematics & Statistics, vol. 219 (Springer, Cham, 2017), pp. 259–281

    Google Scholar 

  20. A. Henrich, A sequence of degree one Vassiliev invariants for virtual knots. J. Knot Theory Ramif. 19(4), 461–487 (2010)

    Article  MathSciNet  Google Scholar 

  21. D.P. Ilyutko, V.O. Manturov, Cobordisms of free knots (Russian). Dokl. Akad. Nauk 429(4), 439–441 (2009); Translation in Dokl. Math. 80(3), 844–846 (2009)

    Google Scholar 

  22. V.F.R. Jones, A polynomial invariant for links via von Neumann algebras. Bull. Am. Math. Soc. 129, 103–112 (1985)

    Article  Google Scholar 

  23. V.F.R. Jones, Hecke algebra representations of braid groups and link polynomials. Ann. Math. 126, 335–338 (1987)

    Article  MathSciNet  Google Scholar 

  24. V.F.R. Jones, On knot invariants related to some statistical mechanics models. Pac. J. Math. 137(2), 311–334 (1989)

    Article  MathSciNet  Google Scholar 

  25. D.A. Joyce, A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebr. 23(1), 37–65 (1982)

    Article  MathSciNet  Google Scholar 

  26. N. Kamada, Y. Miyazawa, A 2-variable polynomial invariant for a virtual link derived from magnetic graphs. Hiroshima Math. J. 35(2), 309–326 (2005)

    Article  MathSciNet  Google Scholar 

  27. L.H. Kauffman, Formal Knot Theory. Lecture Notes Series, vol. 30 (Princeton University Press, Princeton, 1983). Dover Publications (2011)

    Google Scholar 

  28. L.H. Kauffman, State models and the Jones polynomial. Topology 26, 395–407 (1987)

    Article  MathSciNet  Google Scholar 

  29. L.H. Kauffman, On Knots (Princeton University Press, Princeton, 1987)

    Google Scholar 

  30. L.H. Kauffman, Knots and Physics (World Scientific Publishers, Singapore, 1991). Second Edition (1993), Third Edition (2002), Fourth Edition (2012)

    Google Scholar 

  31. L.H. Kauffman, Virtual knot theory. Eur. J. Comb. 20, 663–690 (1999)

    Article  MathSciNet  Google Scholar 

  32. L.H. Kauffman, A survey of virtual knot theory, in Proceedings of Knots in Hellas ’98 (World Scientific Publishers, Singapore, 2000), pp. 143–202

    Google Scholar 

  33. L.H. Kauffman, Detecting virtual knots, Atti. Sem. Mat. Fis. Univ. Modena Supplemento al, vol. IL (2001), pp. 241–282

    Google Scholar 

  34. L.H. Kauffman, A self-linking invariant of virtual knots. Fund. Math. 184, 135–158 (2004), arXiv:math.GT/0405049

  35. L.H. Kauffman, Knot diagrammatics, in Handbook of Knot Theory ed. by W. Menasco, M. Thistlethwaite (Elsevier, Amsterdam, 2005), pp. 233–318

    Google Scholar 

  36. L.H. Kauffman, H.A. Dye, Minimal surface representations of virtual knots and links. Algebr. Geom. Topol. 5(2005), 509–535 (2005)

    MathSciNet  MATH  Google Scholar 

  37. L.H. Kauffman, Introduction to virtual knot theory. J. Knot Theory Ramif. 21(13), 1240007, 37 pp. (2012)

    Google Scholar 

  38. L.H. Kauffman, An affine index polynomial invariant of virtual knots. J. Knot Theory Ramif. 22(4), 1340007, 30 pp. (2013)

    Google Scholar 

  39. L.H. Kauffman, Virtual knot cobordism, in New Ideas in Low Dimensional Topology, ed. by L.H. Kauffman, V.O. Manturov. Series on Knots and Everything, vol. 56 (World Scientific Publishing, Hackensack, 2015), pp. 335–377

    Google Scholar 

  40. L.H. Kauffman, Virtual knot cobordism and the affine index polynomial. J. Knot Theory Ramif. 1843017, 29 pp. (2018)

    Google Scholar 

  41. L.H. Kauffman, S. Lambropoulou, Virtual braids. Fund. Math. 184, 159–186 (2004)

    Article  MathSciNet  Google Scholar 

  42. L.H. Kauffman, S. Lambropoulou, Virtual braids and the L-move. J. Knot Theory Ramif. 15(6), 773–811 (2006)

    Article  MathSciNet  Google Scholar 

  43. L.H. Kauffman, S. Lambropoulou, The L-move and virtual braids, Intelligence of Low Dimensional Topology 2006. Series on Knots and Everything, vol. 40 (World Scientific Publishing, Hackensack, 2007), pp. 133–142

    Google Scholar 

  44. L.H. Kauffman, S. Lambropoulou, A categorical structure for the virtual braid group. Commun. Algebr. 39(12), 4679–4704 (2011)

    Article  MathSciNet  Google Scholar 

  45. L.H. Kauffman, S. Lambropoulou, A categorical model for the virtual braid group. J. Knot Theory Ramif. 21(13), 1240008, 48 pp. (2012)

    Google Scholar 

  46. G. Kuperberg, What is a virtual link? Algebr. Geom. Topol. 3, 587–591 (2003)

    Article  MathSciNet  Google Scholar 

  47. V.O. Manturov, On invariants of virtual links. Acta Appl. Math. 72, 295–309 (2002)

    Article  MathSciNet  Google Scholar 

  48. V.O. Manturov, Parity in knot theory. Math. sb. 201(5), 693–733 (2010). Original Russian Text in Mathematical Sbornik 201(5), 65–110 (2010)

    Google Scholar 

  49. V.O. Manturov, Virtual Knot Theory - The State of the Art (World Scientific Publishing, Singapore, 2012)

    Google Scholar 

  50. V.O. Manturov, Parity and cobordisms of free knots (Russian). Mat. Sb. 203(2), 45–76 (2012); Translation in Sb. Math. 203(1–2), 196–223 (2012)

    Google Scholar 

  51. S.V. Matveev, Distributive groupoids in knot theory (Russian). Mat. Sb. (N.S.) 119(161), 78–88 (1982). 160(1)

    Google Scholar 

  52. J.A. Rasmussen, Khovanov homology and the slice genus. Invent. Math. 182(2), 419–447 (2010)

    Article  MathSciNet  Google Scholar 

  53. S. Satoh, Virtual knot presentation of ribbon torus-knots. JKTR 9(4), 531–542 (2000)

    MathSciNet  MATH  Google Scholar 

  54. J. Sawollek, An orientation-sensitive Vassiliev invariant for virtual knots. J. Knot Theory Ramif. 12(6), 767–779 (2003)

    Article  MathSciNet  Google Scholar 

  55. J. Scott Carter, S. Kamada, M. Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms. Knots 2000 Korea, Vol. 1 (Yongpyong). J. Knot Theory Ramif. 11(3), 311–322 (2002)

    Google Scholar 

  56. Y. Takeda, Introduction to virtual surface-knot theory. JKTR 21(14), 1250131, 6 pp. (2012)

    Google Scholar 

  57. V. Turaev, Cobordisms of words. Commun. Contemp. Math. 10(suppl. 1), 927–972 (2008)

    Article  MathSciNet  Google Scholar 

  58. E. Witten, Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (contract no. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation).

Author information

Authors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science (m/c 249), University of Illinois, 851 South Morgan Street, Chicago, IL, 60607-7045, USA

    Louis H. Kauffman

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

    Louis H. Kauffman

Authors
  1. Louis H. Kauffman
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Correspondence to Louis H. Kauffman .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Williams College, Williamstown, MA, USA

    Colin C. Adams

  2. Department of Mathematics, University of Texas at Austin, Austin, TX, USA

    Cameron McA. Gordon

  3. Department of Mathematics, Vanderbilt University, Nashville, TN, USA

    Vaughan F.R. Jones

  4. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, USA

    Louis H. Kauffman

  5. School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Athens, Greece

    Sofia Lambropoulou

  6. Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA, USA

    Kenneth C. Millett

  7. Department of Mathematics, Columbian College of Arts & Sciences, George Washington University, Washington, DC, USA

    Jozef H. Przytycki

  8. Department of Mathematics and Applications, University of Milano-Bicocca, Milano, Italy

    Renzo Ricca

  9. Department of Mathematics, North Carolina State University, Raleigh, NC, USA

    Radmila Sazdanovic

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Kauffman, L.H. (2019). Virtual Knot Theory and Virtual Knot Cobordism. In: Adams, C., et al. Knots, Low-Dimensional Topology and Applications. KNOTS16 2016. Springer Proceedings in Mathematics & Statistics, vol 284. Springer, Cham. https://doi.org/10.1007/978-3-030-16031-9_4

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