Classification of $k$-tangle projections using cascade representation

TL;DR

This paper introduces a cascade diagram method for classifying $k$-tangle projections, enabling effective enumeration and tabulation of projections with up to 12 crossings, advancing knot theory computational tools.

Contribution

The paper presents a novel cascade representation for $k$-tangle projections and an enumeration algorithm, expanding the catalog of known tangle projections.

Findings

Tabulated $k$-tangle projections with up to 12 crossings

Provided images of alternating $k$-tangles with 5 crossings or fewer

Developed an effective enumeration algorithm

Abstract

The paper addresses the $k$-tangle enumeration problem. We introduce a notion of cascade diagram for $k$-tangle projections. An effective enumeration algorithm for projections is proposed based on cascade representation. Tangles projections with up to 12 crossings are tabulated. We provide also pictures of alternating $k$-tangles with 5 crossing or less.