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THE POLE DIAGRAM AND THE MIYAZAWA POLYNOMIAL

ATSUSHI ISHII

Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-16, Toyonaka, Osaka 560-0043, Japan

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https://doi.org/10.1142/S0129167X08004601Cited by:12 (Source: Crossref)

Abstract

We introduce the pole diagram, which helps to retrieve information from a knot diagram when we smooth crossings. By using the notion, we define a bracket polynomial for the Miyazawa polynomial. The bracket polynomial gives a simple definition and evaluation for the Miyazawa polynomial. Then we show that the virtual crossing number of a virtualized alternating link is determined by its diagram. Furthermore, we construct infinitely many virtual link diagrams which attain the minimal real and virtual crossing numbers together.

Keywords:

AMSC: 57M27, 57M25

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