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Alexander and writhe polynomials for virtual knots

Blake Mellor

Mathematics Department, Loyola Marymount University, Los Angeles, CA 90045-2659, USA

https://doi.org/10.1142/S0218216516500504Cited by:10 (Source: Crossref)

Abstract

We give a new interpretation of the Alexander polynomial $Δ_{0}$ for virtual knots due to Sawollek [On Alexander–Conway polynomials for virtual knots and Links, preprint (2001), arXiv:math/9912173] and Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications12 (2003) 987–1000], and use it to show that, for any virtual knot, $Δ_{0}$ determines the writhe polynomial of Cheng and Gao [A polynomial invariant of virtual links, J. Knot Theory Ramifications22(12) (2013), Article ID: 1341002, 33pp.] (equivalently, Kauffman’s affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramifications22(4) (2013), Article ID: 1340007, 30pp.]). We also use it to define a second-order writhe polynomial, and give some applications.

Keywords:

AMSC: 57M25

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