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The number of Reidemeister moves needed for unknotting

Authors: Joel Hass and Jeffrey C. Lagarias
Journal: J. Amer. Math. Soc. 14 (2001), 399-428
MSC (1991): Primary 57M25; Secondary 11Y16, 68W40
Published electronically: January 18, 2001
MathSciNet review: 1815217
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Abstract | References | Similar Articles | Additional Information


There is a positive constant $c_1$ such that for any diagram $\mathcal{D}$ representing the unknot, there is a sequence of at most $2^{c_1 n}$ Reidemeister moves that will convert it to a trivial knot diagram, where $n$ is the number of crossings in $\mathcal{D}$. A similar result holds for elementary moves on a polygonal knot $K$ embedded in the 1-skeleton of the interior of a compact, orientable, triangulated $PL$ 3-manifold $M$. There is a positive constant $c_2$ such that for each $t \geq 1$, if $M$ consists of $t$ tetrahedra and $K$ is unknotted, then there is a sequence of at most $2^{c_2 t}$ elementary moves in $M$ which transforms $K$ to a triangle contained inside one tetrahedron of $M$. We obtain explicit values for $c_1$ and $c_2$.

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Additional Information

Joel Hass
Affiliation: Department of Mathematics, University of California, Davis, California 95616
Address at time of publication: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540

Jeffrey C. Lagarias
Affiliation: AT&T Labs – Research, Florham Park, New Jersey 07932

Keywords: Knot theory, knot diagram, Reidemeister move, normal surfaces, computational complexity
Received by editor(s): May 8, 1998
Received by editor(s) in revised form: October 10, 2000
Published electronically: January 18, 2001
Additional Notes: The first author was partially supported by NSF grant DMS-9704286.
This paper grew out of work begun while the authors were visiting the Mathematical Sciences Research Institute in Berkeley in 1996/97. Research at MSRI was supported in part by NSF grant DMS-9022140.
Article copyright: © Copyright 2001 AT&T Corp.

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