Encoding knots by clasp diagrams

TL;DR

This paper introduces clasp diagrams as a new combinatorial method for encoding knots, simplifying the construction of knot invariants and providing explicit formulas for classical invariants like the Alexander-Conway polynomial.

Contribution

The paper presents a novel clasp diagram framework that encodes knots using full twists, offering new tools for computing invariants and understanding knot isotopy classes.

Findings

Clasp diagrams produce simple Seifert surfaces.

Explicit formula for Alexander-Conway polynomial.

Any Vassiliev invariant can be derived from clasp diagrams.

Abstract

We introduce a new combinatorial method to encode knots and links with applications to knot invariants. Clasp diagrams defined in this paper are combinatorial blueprints for building knot diagrams out of full twists on two strings rather than out of crossings. We describe an equivalence relation on clasp diagrams which produces the isotopy classes of knots as equivalence classes. This equivalence relation is generated by local moves similar to the Reidemeister moves. Clasp diagrams produce particularly simple Seifert surfaces for knots and lead to an explicit formula for the Alexander-Conway polynomial. They are also well-suited for the study of the Vassiliev invariants; we show that any such invariant can be obtained via subdiagram count in the clasp diagrams.