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Planar diagrams for local invariants of graphs in surfaces

Calvin McPhail-Snyder

Department of Mathematics, University of California, Berkeley, California 94720-3840, USA

E-mail Address: cmcs@math.berkeley.edu

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and

Kyle A. Miller

http://orcid.org/0000-0001-7400-5304

Department of Mathematics, University of California, Berkeley, California 94720-3840, USA

E-mail Address: kmill@math.berkeley.edu

Corresponding author.

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https://doi.org/10.1142/S0218216519500937Cited by:1 (Source: Crossref)

Abstract

In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These virtual graphs are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, the $S$ $S$ -polynomial, and formulate the $𝔰 𝔩 (N)$ Penrose polynomial for non-cubic graphs, giving contraction–deletion relations. The $S$ -polynomial is used to define an extension of the Yamada polynomial to virtual spatial graphs, and with it we obtain a sufficient condition for non-classicality of virtual spatial graphs. We conjecture the existence of local relations for the $S$ -polynomial at squares of integers.

Keywords:

AMSC: 05C31, 57M27, 16T30

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