Links and Planar Diagram Codes

TL;DR

This paper introduces Planar Diagram Codes (PD-codes), a combinatorial tool for describing link diagrams, and demonstrates their equivalence to classical link theory through algorithms and symmetry analysis.

Contribution

It formalizes PD-codes as standalone objects, provides an algorithm for reconstructing knot diagrams, and explores their symmetries and relation to Reidemeister moves.

Findings

PD-codes can be reconstructed into knot diagrams on surfaces.

PD-codes' symmetries include invertibility and chirality.

Equivalence of PD-codes modulo moves to classical link theory.

Abstract

In this paper we formalize a combinatorial object for describing link diagrams called a Planar Diagram Code. PD-codes are used by the KnotTheory Mathematica package developed by Bar-Natan, et al. We present the set of PD-codes as a stand alone object and discuss its relationship with link diagrams. We give an explicit algorithm for reconstructing a knot diagram on a surface from a PD-code. We also discuss the intrinsic symmetries of PD-codes (i.e., invertibility and chirality). The moves analogous to the Reidemeister moves are also explored, and we show that the given set of PD-codes modulo these combinatorial Reidemeister moves is equivalent to classical link theory.