Every knot has characterising slopes
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 Kronhe...
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| Format: | Journal article |
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Springer Berlin Heidelberg
2018
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| _version_ | 1826260324102176768 |
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| author | Lackenby, M |
| author_facet | Lackenby, M |
| author_sort | Lackenby, M |
| collection | OXFORD |
| description | 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, Ozsváth and Szabó, 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. |
| first_indexed | 2024-03-06T19:03:49Z |
| format | Journal article |
| id | oxford-uuid:14712bef-c0b9-4c33-b46a-871ec9c466a0 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T19:03:49Z |
| publishDate | 2018 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | oxford-uuid:14712bef-c0b9-4c33-b46a-871ec9c466a02022-03-26T10:19:52ZEvery knot has characterising slopesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:14712bef-c0b9-4c33-b46a-871ec9c466a0Symplectic Elements at OxfordSpringer Berlin Heidelberg2018Lackenby, MLet 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, Ozsváth and Szabó, 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. |
| spellingShingle | Lackenby, M Every knot has characterising slopes |
| title | Every knot has characterising slopes |
| title_full | Every knot has characterising slopes |
| title_fullStr | Every knot has characterising slopes |
| title_full_unstemmed | Every knot has characterising slopes |
| title_short | Every knot has characterising slopes |
| title_sort | every knot has characterising slopes |
| work_keys_str_mv | AT lackenbym everyknothascharacterisingslopes |