Quantization of the crossing number of a knot diagram

Details Quantization of the crossing number of a knot diagram Quantization of the crossing number of a knot diagram

2011-12-08

arxiv.org/abs/1112.1809v2

Mathematics

Geometric Topology

14 pages, 14 figures

Scientific paper

We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for all the states for the plane curve. As an application, we show that every closed transversely intersected plane curve with even crossing points has two independent canonical orientations and every based closed transversely intersected plane curve with odd crossing points has two independent canonical orientations.

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