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

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

2010-10-20

arxiv.org/abs/1010.4101v1

Mathematics

Geometric Topology

17 pages, many figures

Scientific paper

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.

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