Counterexamples to Allen's conjectures

TL;DR

This paper provides counterexamples to Allen's conjectures by demonstrating that certain torus knots bound smooth Möbius bands with negative definite double branched covers, challenging previous assumptions in knot theory.

Contribution

The authors construct explicit counterexamples to Allen's conjectures using specific torus knots and analyze their double branched covers, revealing limitations of the conjectures.

Findings

Torus knots T(2,5) and T(2,9) bound smooth Möbius bands in the 4-ball.

Their double branched covers are negative definite.

Counterexamples disprove Allen's Conjectures 1.6 and 1.8.

Abstract

We show that the torus knots $T(2,5)$ and $T(2,9)$ bound smooth M\"{o}bius bands in the 4-ball whose double branched covers are negative definite, giving counterexamples to Conjectures 1.6 and 1.8 of Allen in [New York J. Math. 29 (2023) 1038-1059].