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Doubled Khovanov Homology

Published online by Cambridge University Press:  20 November 2018

William Rushworth
William Rushworth*
Affiliation:
Department of Mathematical Sciences, Durham University, United Kingdom, e-mail: [email protected]
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Abstract

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We define a homology theory of virtual links built out of the direct sum of the standard Khovanov complex with itself, motivating the name doubled Khovanov homology. We demonstrate that it can be used to show that some virtual links are non-classical, and that it yields a condition on a virtual knot being the connect sum of two unknots. Further, we show that doubled Khovanov homology possesses a perturbation analogous to that defined by Lee in the classical case, and we define a doubled Rasmussen invariant. This invariant is used to obtain various cobordism obstructions; in particular, it is an obstruction to sliceness. Finally, we show that the doubled Rasmussen invariant contains the odd writhe of a virtual knot and use this to show that knots with non-zero odd writhe are not slice.

Keywords

57M2557M2757N70Khovanov homologyvirtual knot concordancevirtual knot theory
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 5 , 01 October 2018 , pp. 1130 - 1172
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Bar-Natan, D., On Khovanov's categorification of the Jones polynomial. Algebr. Geom. Topol. 2 (2002), 337–370. http://dx.doi.Org/10.2140/agt.2002.2.337Google Scholar
[2] Bar-Natan, D. and Burgos-Soto, H., Khovanov homology for alternating tangles. J. Knot Theory Ramifications 23 (2014) no. 2, 1450013. http://dx.doi.org/10.1142/S0218216514500138Google Scholar
[3] Bar-Natan, D. and Morrison, S., The Karoubi envelope and Lee's degeneration of Khovanov homology. Algebr. Geom. Topol. 6 (2006), 1459–1469. http://dx.doi.Org/10.2140/agt.2006.6.1459Google Scholar
[4] Dye, H. A., Virtual knots undetected by 1- and 2-strand bracket polynomials. Topology Appl. 153 (2005), no. 1, 141–160. http://dx.doi.Org/10.1016/j.topol.2005.01.028Google Scholar
[5] Dye, H. A., Kaestner, A., and Kauffman, L. H., Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots. J. Knot Theory Ramifications 26 (2017) no. 3, 1741001. http://dx.doi.Org/10.1142/S021 821 651 7410012Google Scholar
[6] Green, J., A table of virtual knots. https://vwvw.math.toronto.edu/drorbn/Students/GreenJ/index.htmlGoogle Scholar
[7] Kauffman, L. H., Virtual knot theory. European J. Combin. 20 (1999), no. 7, 663–690. http://dx.doi.Org/10.1006/eujc.1999.0314Google Scholar
[8] Kauffman, L. H., A self-linking invariant of virtual knots. Fund. Math. 184 (2004), 135–158. http://dx.doi.org/10.4064/fm184-0-10Google Scholar
[9] Kauffman, L. H., Virtual knot cobordism. In: New ideas in low dimensional topology, Ser. Knots Everything, 56, World Sci. Publ., Hackensack, NJ, 2015. http://dx.doi.Org/10.1142/9789814630627J3009Google Scholar
[10] Kauffman, L. H. and Radford, D., Bi-oriented quantum algebras, and a generalized alexander polynomial for virtual links. In: Diagrammatic morphisms and applications (San Francisco, CA, 2000), Contemp. Math., 318, American Mathematics Society, Providence, RI, 2003. http://dx.doi.org/10.1090/conm/318/05548Google Scholar
[11] Khovanov, M., A categorification of the Jones polynomial. Duke Math. J. 101 (2000), no. 3, 359–426. http://dx.doi.Org/10.1215/S0012-7094-00-10131-7Google Scholar
[12] Kishino, T. and Satoh, S., A note on non-classical virtual knots. J. Knot Theory Ramifications 13 (2004), no. 7, 845–856. http://dx.doi.Org/10.1142/S0218216504003548Google Scholar
[13] Kuperberg, G., What is a virtual link? Algebr. Geom. Topol. 3 (2003), 587–591. http://dx.doi.org/10.2140/agt.20033.587Google Scholar
[14] Lee, E. S., An endomorphism of the Khovanov invariant. Adv. Math. 197 (2005), no. 2, 554–586. http://dx.doi.Org/10.101 6/j.aim.2 004.10.01 5Google Scholar
[15] Manturov, V. O., Khovanov homology for virtual links with arbitrary coefficients. J. Knot Theory Ramifications 16 (2007), no. 3, 345–377. http://dx.doi.org/10.1142/S0218216507005336Google Scholar
[16] Manturov, V. O., Compact and long virtual knots. Trans. Moscow Math. Soc. 69 (2008), 1–26. http://dx.doi.org/10.1090/S0077-1554-08-00168-4Google Scholar
[17] Manturov, V. O. and Ilyutko, D. P., Virtual knots: the state of the art. Series on Knots and Everything, 51, World Scientific, Hackensack, NJ, 2013.Google Scholar
[18] Nelson, S., Unknotting virtual knots with Gauss diagram forbidden moves. J. Knot Theory Ramifications 10 (2001) no. 6,931-935. http://dx.doi.org/10.1142/S0218216501001244Google Scholar
[19] Rasmussen, J., Khovanov homology and the slice genus. Invent. Math. 182 (2010), no. 2, 419–447. http://dx.doi.org/10.1007/s00222-010-0275-6Google Scholar
[20] Rushworth, W., Computations of the slice genus of virtual knots. arxiv:1706.08279Google Scholar
[21] Shumakovitch, A., Patterns in odd Khovanov homology. J. Knot Theory Ramifications 20 (2011), no. 1, 203–222. http://dx.doi.org/10.1142/S0218216511008802Google Scholar
[22] Silver, D. S. and Williams, S. G., On a class of virtual knots with unit j ones polynomial. J. Knot Theory Ramifications 13 (2004), no. 3, 367–371. http://dx.doi.Org/10.1142/S0218216504003196Google Scholar
[23] Tubbenhauer, D., Virtual Khovanov homology using cobordisms. J. Knot Theory Ramifications 23 (2014) no. 9, 1450046. http://dx.doi.org/10.1142/S0218216514500461Google Scholar
[24] Turaev, V. and Turner, P., Unoriented topological quantum field theory and link homology. Algebr. Geom. Topol. 6 (2006), 1069–1093. http://dx.doi.Org/10.2140/agt.2006.6.1069Google Scholar