A family of transversely nonsimple knots

TL;DR

This paper uses knot Floer homology to identify an infinite family of prime knots that are transversely nonsimple, and explores the connections between grid diagrams, braids, and Legendrian and transverse knots in contact geometry.

Contribution

It introduces an infinite family of transversely nonsimple prime knots and analyzes their properties using knot Floer homology, linking combinatorial and geometric knot invariants.

Findings

Existence of an infinite family of transversely nonsimple prime knots.

Establishment of a relationship between grid diagrams, braids, and contact geometric knots.

Abstract

We apply knot Floer homology to exhibit an infinite family of transversely nonsimple prime knots starting with $10_{132}$. We also discuss the combinatorial relationship between grid diagrams, braids, and Legendrian and transverse knots in standard contact $\mathbb{R}^3$.