Constrained knots in lens spaces

TL;DR

This paper introduces constrained knots in lens spaces, classifies them using knot Floer homology, and explores their properties, including hyperbolic structures and relations to surgeries on links in $S^3$.

Contribution

It provides a complete classification of constrained knots via $ ext{HFK}$ calculations and presentations of knot groups, expanding understanding of their structure and examples.

Findings

Constrained knots include 2-bridge knots and simple knots in lens spaces.

The $ ext{HFK}$ of constrained knots is thin, enabling classification.

Many constrained knots have hyperbolic complements with simple ideal triangulations.

Abstract

In this paper, we study a special family of $(1,1)$ knots called constrained knots, which includes 2-bridge knots in the 3-sphere $S^3$ and simple knots in lens spaces. Constrained knots are parameterized by five integers and characterized by the distribution of spin$^c$ structures in the corresponding $(1,1)$ diagrams. The knot Floer homology $\widehat{HFK}$ of a constrained knot is thin. We obtain a complete classification of constrained knots based on the calculation of $\widehat{HFK}$ and presentations of knot groups. We provide many examples of constrained knots constructed from surgeries on links in $S^3$, which are related to 2-bridge knots and 1-bridge braids. We also show many examples of constrained knots whose knot complements are orientable hyperbolic 1-cusped manifolds with simple ideal triangulations.