A generating polynomial for the pretzel knot

TL;DR

This paper develops a generating polynomial that encodes the distribution of Kauffman states for specific pretzel knots with multiple tangles and half-twists, aiding in knot invariant analysis.

Contribution

It introduces a new generating polynomial for counting Kauffman states in pretzel knots with n tangles and r half-twists, providing a novel combinatorial tool.

Findings

Derived explicit formulas for the generating polynomial coefficients.

Provided statistical insights into Kauffman state distributions.

Enhanced understanding of pretzel knot invariants.

Abstract

We collect statistics which consist of the coefficients in the expansion of the generating polynomials that count the Kauffman states associated with certain classes of pretzel knots having n tangles, of r half-twists respectively.