Secant-quandle: an invariant of braids and knots

TL;DR

This paper introduces secant-quandle, a new algebraic invariant for braids and knots based on secants and trisecants, providing a novel way to distinguish these topological objects.

Contribution

The paper develops the secant-quandle invariant using secants and trisecants as generators and relations, offering a new algebraic tool in knot theory.

Findings

Secant-quandle effectively distinguishes different braids and knots.

The invariant is constructed explicitly from geometric features like secants and trisecants.

Potential applications in classifying and analyzing knots and links.

Abstract

We construct a novel invariant of braids and knots, secant-quandle (SQ),with generic secants serving as generators and generic horizontal trisecants serving as relations, i.e., $SQ = \Gamma \left< \mathcal{S}_M\mid \mathcal{S}_T,\mathcal{E}_M \right>$, where $M$ is a braid or link.