Rack invariants of links in $L(p,1)$

TL;DR

This paper develops a presentation for the augmented fundamental rack of links in lens spaces and extends counting rack invariants to include the action of the lens space's fundamental group.

Contribution

It introduces a presentation for the augmented fundamental rack in $L(p,1)$ and adapts counting rack invariants to this setting, incorporating the fundamental group's action.

Findings

Extended rack invariants include $ ext{pi}_1$ action information.

Presented a new method for links in lens spaces.

Enhanced understanding of link invariants in 3-manifolds.

Abstract

We describe a presentation for the augmented fundamental rack of a link in the lens space $L(p,1)$. Using this presentation, the (enhanced) counting rack invariants that have been defined for the classical links are applied to the links in $L(p,1)$. In this case, the counting rack invariants also include the information about the action of $\pi_{1}(L(p,1))$ on the augmented fundamental rack of a link.