- Volume 55 Issue 3
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Quantization of the Crossing Number of a Knot Diagram
- KAWAUCHI, AKIO (Osaka City University Advanced Mathematical Institute) ;
- SHIMIZU, AYAKA (Department of Mathematics, Gunma National College of Technology)
- Received : 2014.01.28
- Accepted : 2014.07.14
- Published : 2015.09.23
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.
Crossing number;Oriented knot diagram;Plane curve;Warping crossing polynomial;Warping degree;Warping polynomial
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