Boundary-twisted normal form and the number of elementary moves to unknot

TL;DR

This paper introduces a new normal form for surfaces in triangulated 3-manifolds, providing a simpler proof and significantly improved bounds on the number of moves needed to unknot, with implications for Reidemeister move complexity.

Contribution

It presents a boundary-twisted normal form for surfaces and improves upper bounds on elementary moves to unknot and Reidemeister moves needed for unknotting.

Findings

Simpler proof for bounds on elementary moves to unknot.

New upper bound of 2^{120t+14} for elementary moves.

Improved upper bound of 2^{10^5 n} for Reidemeister moves.

Abstract

Suppose $K$ is an unknot lying in the 1-skeleton of a triangulated 3-manifold with $t$ tetrahedra. Hass and Lagarias showed there is an upper bound, depending only on $t$, for the minimal number of elementary moves to untangle $K$. We give a simpler proof, utilizing a normal form for surfaces whose boundary is contained in the 1-skeleton of a triangulated 3-manifold. We also obtain a significantly better upper bound of $2^{120t+14}$ and improve the Hass--Lagarias upper bound on the number of Reidemeister moves needed to unknot to $2^{10^5 n}$, where $n$ is the crossing number.