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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: william.rushworth@durham.ac.uk
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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

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