Exceptional surgery curves in triangulated 3-manifolds
For the purposes of this paper, Dehn surgery along a curve K in a 3-manifold M with slope r is `exceptional' if the resulting 3-manifold M_K(r) is reducible or a solid torus, or the core of the surgery solid torus has finite order in the fundamental group of M_K(r). We show that, providing the...
| Main Author: | |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
1999
|
| _version_ | 1826295957688418304 |
|---|---|
| author | Lackenby, M |
| author_facet | Lackenby, M |
| author_sort | Lackenby, M |
| collection | OXFORD |
| description | For the purposes of this paper, Dehn surgery along a curve K in a 3-manifold M with slope r is `exceptional' if the resulting 3-manifold M_K(r) is reducible or a solid torus, or the core of the surgery solid torus has finite order in the fundamental group of M_K(r). We show that, providing the exterior of K is irreducible and atoroidal, and the distance between r and the meridian slope is more than one, and a homology condition is satisfied, then there is (up to ambient isotopy) only a finite number of such exceptional surgery curves in a given compact orientable 3-manifold M, with the boundary of M a (possibly empty) union of tori. Moreover, there is a simple algorithm to find all these surgery curves, which involves inserting tangles into the 3-simplices of any given triangulation of M. As a consequence, we deduce some results about the finiteness of certain unknotting operations on knots in the 3-sphere. |
| first_indexed | 2024-03-07T04:08:57Z |
| format | Journal article |
| id | oxford-uuid:c72c60e6-57ff-4807-bb2b-94579767566e |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T04:08:57Z |
| publishDate | 1999 |
| record_format | dspace |
| spelling | oxford-uuid:c72c60e6-57ff-4807-bb2b-94579767566e2022-03-27T06:43:08ZExceptional surgery curves in triangulated 3-manifoldsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:c72c60e6-57ff-4807-bb2b-94579767566eEnglishSymplectic Elements at Oxford1999Lackenby, MFor the purposes of this paper, Dehn surgery along a curve K in a 3-manifold M with slope r is `exceptional' if the resulting 3-manifold M_K(r) is reducible or a solid torus, or the core of the surgery solid torus has finite order in the fundamental group of M_K(r). We show that, providing the exterior of K is irreducible and atoroidal, and the distance between r and the meridian slope is more than one, and a homology condition is satisfied, then there is (up to ambient isotopy) only a finite number of such exceptional surgery curves in a given compact orientable 3-manifold M, with the boundary of M a (possibly empty) union of tori. Moreover, there is a simple algorithm to find all these surgery curves, which involves inserting tangles into the 3-simplices of any given triangulation of M. As a consequence, we deduce some results about the finiteness of certain unknotting operations on knots in the 3-sphere. |
| spellingShingle | Lackenby, M Exceptional surgery curves in triangulated 3-manifolds |
| title | Exceptional surgery curves in triangulated 3-manifolds |
| title_full | Exceptional surgery curves in triangulated 3-manifolds |
| title_fullStr | Exceptional surgery curves in triangulated 3-manifolds |
| title_full_unstemmed | Exceptional surgery curves in triangulated 3-manifolds |
| title_short | Exceptional surgery curves in triangulated 3-manifolds |
| title_sort | exceptional surgery curves in triangulated 3 manifolds |
| work_keys_str_mv | AT lackenbym exceptionalsurgerycurvesintriangulated3manifolds |