Bikei Invariants and Gauss Diagrams for Virtual Knotted Surfaces

Sam Nelson; Patricia Rivera

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arXiv · 1409.7756

Bikei Invariants and Gauss Diagrams for Virtual Knotted Surfaces

Sam Nelson·Patricia Rivera

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Abstract

Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in $\mathbb{R}^4$; 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.

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Sam Nelson, Patricia Rivera. 2014-09-27. Bikei Invariants and Gauss Diagrams for Virtual Knotted Surfaces. https://arxiv.org/abs/1409.7756

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