On a move reducing the genus of a knot diagram

Details On a move reducing the genus of a knot diagram On a move reducing the genus of a knot diagram

2011-11-14

arxiv.org/abs/1111.3215v2

Mathematics

Geometric Topology

12 pages, 12 figures, to appear in Indiana University Mathematics Journal

Scientific paper

For a knot diagram we introduce an operation which does not increase the genus of the diagram and does not change its representing knot type. We also describe a condition for this operation to certainly decrease the genus. The proof involves the study of a relation between the genus of a virtual knot diagram and the genus of a knotoid diagram, the former of which has been introduced by Stoimenow, Tchernov and Vdovina, and the latter by Turaev recently. Our operation has a simple interpretation in terms of Gauss codes and hence can easily be computer-implemented.

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