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Alexander invariants for virtual knots

Hans U. Boden

Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S-4K1, Canada

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Emily Dies

Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S-4K1, Canada

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Anne Isabel Gaudreau

Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S-4K1, Canada

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Adam Gerlings

Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S-4K1, Canada

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Eric Harper

Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S-4K1, Canada

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, and

Andrew J. Nicas

Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S-4K1, Canada

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https://doi.org/10.1142/S0218216515500091Cited by:21 (Source: Crossref)

Abstract

Given a virtual knot K, we introduce a new group-valued invariant VG_K called the virtual knot group, and we use the elementary ideals of VG_K to define invariants of K called the virtual Alexander invariants. For instance, associated to the zeroth ideal is a polynomial H_K(s, t, q) in three variables which we call the virtual Alexander polynomial, and we show that it is closely related to the generalized Alexander polynomial G_K(s, t) introduced by Sawollek; Kauffman and Radford; and Silver and Williams. We define a natural normalization of the virtual Alexander polynomial and show it satisfies a skein formula. We also introduce the twisted virtual Alexander polynomial associated to a virtual knot K and a representation ϱ : VG_K → GL_n(R), and we define a normalization of the twisted virtual Alexander polynomial. As applications we derive bounds on the virtual crossing numbers of virtual knots from the virtual Alexander polynomial and twisted virtual Alexander polynomial.

Keywords:

AMSC: 57M25, 57M27, 20C15

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