Up-down colorings of virtual-link diagrams and the necessity of Reidemeister moves of type II

Kanako Oshiro; Ayaka Shimizu; Yoshiro Yaguchi

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arXiv · 1703.03573

Up-down colorings of virtual-link diagrams and the necessity of Reidemeister moves of type II

Kanako Oshiro·Ayaka Shimizu·Yoshiro Yaguchi

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Abstract

We introduce an up-down coloring of a virtual-link diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two 2-component virtual-link diagrams. By using the notion of a quandle cocycle invariant, we determine the necessity of Reidemeister moves of type II for a pair of diagrams of the trivial virtual-knot. This implies that for any virtual-knot diagram $D$, there exists a diagram $D'$ representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister move of type II.

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Kanako Oshiro, Ayaka Shimizu, Yoshiro Yaguchi. 2017-03-10. Up-down colorings of virtual-link diagrams and the necessity of Reidemeister moves of type II. https://arxiv.org/abs/1703.03573

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