Some evaluations of Jones polynomials for certain families of weaving knots

TL;DR

This paper derives formulas for determinants of specific weaving knots, computes homology group dimensions, and provides lower bounds for their unknotting numbers, advancing understanding of their topological properties.

Contribution

It introduces new formulae for determinants and homology groups of weaving knots, offering insights into their unknotting numbers and topological invariants.

Findings

Formulas for determinants of $W(3,n)$ and $W(p,2)$

Computed homology group dimensions for double cyclic covers

Established lower bounds for unknotting numbers of certain weaving knots

Abstract

In this paper, we derive formulae for the determinant of weaving knots $W(3,n)$ and $W(p,2)$. We calculate the dimension of the first homology group with coefficients in $\mathbb{Z}_3$ of the double cyclic cover of the $3$-sphere $S^3$ branched over $W(3,n)$ and $W(p,2)$ respectively. As a consequence, we obtain a lower bound of the unknotting number of $W(3,n)$ for certain values of $n$.