Pointed Racks and Their Applications to Braid Theory

TL;DR

This paper introduces pointed racks, a new algebraic structure, and demonstrates their use in creating novel braid invariants that can distinguish braids more effectively than previous methods.

Contribution

The paper defines pointed racks and develops new braid invariants based on them, enhancing the tools available for braid classification and knot theory.

Findings

Integer-valued braid invariant using pointed racks

Matrix-valued invariant strengthening the integer invariant

Examples of braids distinguished by new invariants

Abstract

We define a new algebraic structure called a \emph{pointed rack} and use it to construct ambient isotopy invariants of $ n $-braids. We first introduce an integer-valued invariant of braids using pointed racks. This is then strengthened by defining a matrix-valued invariant using racks. Moreover, our invariant determines the rack coloring invariant previously defined for the closure of the braid. Finally, we include examples of braids that are distinguished by these new invariants.