A lower bound on tunnel number degeneration

Trent Schirmer

arXiv:1407.3554·math.GT·July 20, 2016

A lower bound on tunnel number degeneration

Trent Schirmer

PDF

TL;DR

This paper establishes a lower bound on the tunnel number degeneration of knots under connected sum operations, using bounds on Heegaard genus in specific 3-manifold amalgamations, advancing understanding of knot complexity.

Contribution

It introduces a theorem that bounds Heegaard genus in certain 3-manifold amalgamations, leading to a new inequality for the tunnel number of connected sum knots.

Findings

01

Proves $t(K_1igoplus K_2) ext{ is at least } ext{max}igrace{t(K_1), t(K_2)igrace}$.

02

Establishes a lower bound on tunnel number degeneration.

03

Provides a method to analyze tunnel number behavior under knot connected sums.

Abstract

We prove a theorem which bounds Heegaard genus from below under special kinds of toroidal amalgamations of $3$ -manifolds. As a consequence, we conclude $t (K_{1} # K_{2}) \geq max {t (K_{1}), t (K_{2})}$ for any pair of knots $K_{1}, K_{2} \subset S^{3}$ , where $t (K)$ denotes the tunnel number of $K$ .

Peer Reviews

No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.

Videos

No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.