# On knot semigroups and Gelfand-Kirillov dimensions

## TL;DR

This paper investigates the algebraic structure of knot semigroups, proving isomorphism for double twist knots and introducing a new link invariant based on Gelfand-Kirillov dimension.

## Contribution

It confirms Vernitski's conjecture for double twist knots and introduces a novel link invariant derived from algebraic dimension.

## Key findings

- Knot semigroup of double twist knot is isomorphic to an alternating sum semigroup
- Constructed a new link invariant using Gelfand-Kirillov dimension
- Supports Vernitski's conjecture for specific knot classes

## Abstract

A knot semigroup is defined by A. Vernitski. A. Vernitski conjectured that the knot semigroup of the 2-bridge knot is isomorphic to an alternating sum semigroup. To support this conjecture, and as a first main result, we prove that the knot semigroup of the double twist knot is isomorphic to an alternating sum semigroup. Moreover, as a second main result, we construct a link invariant constructed by the Gelfand-Kiriliov dimension of algebra.