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Bikei invariants and gauss diagrams for virtual knotted surfaces

Sam Nelson

Department of Mathematical Sciences, Claremont McKenna College, 850 Columbia Ave, Claremont, CA 91711, USA

E-mail Address: Sam.Nelson@cmc.edu

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and

Patricia Rivera

Department of Mathematical Sciences, Claremont McKenna College, 850 Columbia Ave, Claremont, CA 91711, USA

E-mail Address: patriciariv25@gmail.com

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https://doi.org/10.1142/S0218216516400083Cited by:3 (Source: Crossref)

Abstract

Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in $ℝ^{4}$ ; including virtual crossings allows for a theory of virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants are defined only for marked vertex diagrams representing knotted orientable surfaces; we extend these invariants to all virtual marked vertex diagrams by considering colorings by involutory biquandles, also known as bikei. We introduce a way of representing marked vertex diagrams with Gauss diagrams and use these to characterize orientability.

Keywords:

AMSC: 57M27, 57M25

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