Vassiliev invariants for pretzel knots

TL;DR

This paper computes low-order Vassiliev invariants for pretzel knots, revealing their polynomial structure and properties, which advances understanding of knot invariants in topology.

Contribution

It provides explicit calculations of Vassiliev invariants up to order six for pretzel knots, highlighting their polynomial form and topological properties.

Findings

Vassiliev invariants are symmetric polynomials in parameters

Invariants depend on g+1 parameters for pretzel knots

Invariants are topologically and integrally well-behaved

Abstract

We compute Vassiliev invariants up to order six for arbitrary pretzel knots, which depend on $g+1$ parameters $n_1,\ldots,n_{g+1}$. These invariants are symmetric polynomials in $n_1,\ldots,n_{g+1}$ whose degree coincide with their order. We also discuss their topological and integer-valued properties.