Every knot has characterising slopes


Lackenby, M

Publication Date: 

8 October 2018


Mathematische Annalen

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Let K be a knot in the 3-sphere. A slope p/q is said to be characterising for
K if whenever p/q surgery on K is homeomorphic, via an orientation-preserving
homeomorphism, to p/q surgery on another knot K' in the 3-sphere, then K and K'
are isotopic. It was an old conjecture of Gordon, proved by Kronheimer, Mrowka,
Ozsvath and Szabo, that every slope is characterising for the unknot. In this
paper, we show that every knot K has infinitely many characterising slopes,
confirming a conjecture of Baker and Motegi. In fact, p/q is characterising for
K provided |p| is at most |q| and |q| is sufficiently large.

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Journal Article