Hyperbolic Knotoids

TL;DR

This paper extends the concept of hyperbolicity to knotoids, a generalization of knots, and explores their properties, including volume calculations and classifications, contributing new insights into their geometric structure.

Contribution

It introduces hyperbolicity definitions for spherical and planar knotoids, proves volume additivity under product, and provides volume tables and classifications.

Findings

Product of hyperbolic spherical knotoids is hyperbolic with additive volumes.

Determined least volume of rational spherical knotoids.

Provided comprehensive tables of hyperbolic volumes for knotoids.

Abstract

In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints. A variety of knot invariants have been extended to knotoids. Here we provide definitions of hyperbolicity for both spherical and planar knotoids. We prove that the product of hyperbolic spherical knotoids is hyperbolic and the volumes add. We also determine the least volume of a rational spherical knotoid and provide various classes of hyperbolic knotoids. We also include tables of hyperbolic volumes for both spherical and planar knotoids.