Tunnel number degeneration under the connected sum of prime knots

TL;DR

This paper investigates how the tunnel number of knots behaves under connected sum operations, providing counterexamples to a conjecture that it cannot decrease by more than one.

Contribution

It constructs infinitely many counterexamples showing the tunnel number can degenerate by more than one when taking connected sums of prime knots.

Findings

Counterexamples to the conjecture on tunnel number degeneration

Demonstration of tunnel number decrease greater than one

Analysis of 2-string free tangle decompositions

Abstract

We study 2-string free tangle decompositions of knots with tunnel number two. As an application, we construct infinitely many counter-examples to a conjecture in the literature stating that the tunnel number of the connected sum of prime knots doesn't degenerate by more than one.