# Regional knot invariants

## TL;DR

This paper introduces a new regional knot invariant called a tridle, constructed from planar regions and crossings, which generalizes existing invariants like the Alexander polynomial through algebraic structures.

## Contribution

It develops the concept of a tridle as a regional knot invariant and extends it to linear tridles, enabling polynomial invariants similar to the Alexander polynomial.

## Key findings

- Defined a new regional knot invariant called a tridle
- Established a method to derive polynomial invariants from linear tridles
- Connected the invariant to existing knot invariants like the Alexander polynomial

## Abstract

In this paper, a regional knot invariant is constructed. Like the Wirtinger presentation of a knot group, each planar region contributes a generator, and each crossing contributes a relation. The invariant is call a tridle of the link. As in the quandle theory, one can define Alexander quandle and get Alexander polynomial from it. For link diagram, one can also define a linear tridle and its presentation matrix. A polynomial invariant can be derive from the matrix just like the Alexander polynomial case.