Cube diagrams and 3-dimensional Reidemeister-like moves for knots

TL;DR

This paper introduces cube diagrams as a new way to represent knots and links, establishes their invariance properties, and constructs a knot homology equivalent to knot Floer homology, advancing knot theory methods.

Contribution

It presents cube diagrams as a novel knot representation, identifies their invariance criteria, and links them to established knot homology theories, providing new tools for knot analysis.

Findings

Cube diagrams serve as a new knot representation method.

A property of cube diagrams is a link invariant if invariant under specific moves.

Constructed a knot homology equivalent to knot Floer homology.

Abstract

In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under two types of cube diagram operations. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Floer homology.