Slipknotting in Random Diagrams

TL;DR

This paper introduces a new framework for analyzing slipknots in knot diagrams using knotoid diagrams, proving that almost all diagrams are slipknotted, and addresses related conjectures in diagram enumeration.

Contribution

It provides a novel diagram-based approach to slipknots, proving that almost all knot diagrams are slipknotted, and advances understanding of knotoid diagram enumeration.

Findings

Almost all knot diagrams are slipknotted.

Almost all unknot diagrams are slipknotted.

Addresses conjectures on knotoid diagram enumeration.

Abstract

The presence of slipknots in configurations of proteins and DNA has been shown to affect their functionality, or alter it entirely. Historically, polymers are modeled as polygonal chains in space. As an alternative to space curves, we provide a framework for working with subknots inside of knot diagrams via knotoid diagrams. We prove using a pattern theorem for knot diagrams that not only are almost all knot diagrams slipknotted, almost all unknot diagrams are slipknotted. This proves in the random diagram model a conjecture yet unproven in random space curve models. We also discuss conjectures on the enumeration of knotoid diagrams.