Quasipositivity and braid index of pretzel knots

TL;DR

This paper calculates the braid index and identifies quasipositive pretzel knots among three-stranded pretzel knots with an even number of crossings in one strand, using advanced inequalities and homomorphisms.

Contribution

It provides explicit calculations of braid index and quasipositivity for a specific class of pretzel knots, employing Morton-Franks-Williams inequalities and Khovanov-Rozansky homomorphisms.

Findings

Determined the braid index for the specified pretzel knots.

Classified which of these knots are quasipositive.

Applied advanced inequalities and homomorphisms to knot analysis.

Abstract

This short note is about three-stranded pretzel knots that have an even number of crossings in one of the strands. We calculate the braid index of such knots and determine which of them are quasipositive. The main tools are the Morton-Franks-Williams inequalities, and Khovanov-Rozansky concordance homomorphisms.