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Knots with unknotting number 1 and essential Conway spheres

Cameron McA Gordon and John Luecke

Algebraic & Geometric Topology 6 (2006) 2051–2116

arXiv: math.GT/0601265


For a knot K in S3, let T(K) be the characteristic toric sub-orbifold of the orbifold (S3,K) as defined by Bonahon–Siebenmann. If K has unknotting number one, we show that an unknotting arc for K can always be found which is disjoint from T(K), unless either K is an EM–knot (of Eudave-Muñoz) or (S3,K) contains an EM–tangle after cutting along T(K). As a consequence, we describe exactly which large algebraic knots (ie, algebraic in the sense of Conway and containing an essential Conway sphere) have unknotting number one and give a practical procedure for deciding this (as well as determining an unknotting crossing). Among the knots up to 11 crossings in Conway’s table which are obviously large algebraic by virtue of their description in the Conway notation, we determine which have unknotting number one. Combined with the work of Ozsváth–Szabó, this determines the knots with 10 or fewer crossings that have unknotting number one. We show that an alternating, large algebraic knot with unknotting number one can always be unknotted in an alternating diagram.

As part of the above work, we determine the hyperbolic knots in a solid torus which admit a non-integral, toroidal Dehn surgery. Finally, we show that having unknotting number one is invariant under mutation.

unknotting number 1, Conway spheres, algebraic knots, mutation
Mathematical Subject Classification 2000
Primary: 57N10
Secondary: 57M25
Forward citations
Received: 9 January 2006
Accepted: 29 September 2006
Published: 19 November 2006
Cameron McA Gordon
Department of Mathematics
The University of Texas at Austin
1 University Station C1200
Austin, TX 78712-0257
John Luecke
Department of Mathematics
The University of Texas at Austin
1 University Station C1200
Austin, TX 78712-0257