- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- BIRS Workshop Lecture Videos /
- 0-concordance of surface knots and Alexander ideals
Open Collections
BIRS Workshop Lecture Videos
BIRS Workshop Lecture Videos
0-concordance of surface knots and Alexander ideals Joseph, Jason
Description
Paul Melvin proved that 0-concordant 2-knots have diffeomorphic Gluck twists, but until recently there were no known proofs that there is more than one 0-concordance class. Now Sunukjian and Dai-Miller have found many examples using Heegaard Floer technology applied to the Seifert 3-manifolds which the 2-knots bound. In this talk we give another proof using Alexander ideals. The main theorem is that the Alexander ideal induces a homomorphism from the 0-concordance monoid of 2-knots to the ideal class monoid of $\mathbb{Z}[t,t^{-1}]$. A corollary is that any 2-knot with nonprincipal Alexander ideal cannot be 0-slice, and moreover has no inverse in the 0-concordance monoid. This is the first proof that the monoid is not a group, and gives another proof of the existence of infinitely many linearly independent 0-concordance classes. These techniques also apply to higher genus surfaces, where we give the first results on 0-concordance. Lastly, we show that under a mild condition on the knot group, the peripheral subgroup of a knotted surface is also a 0-concordance invariant.
Item Metadata
Title |
0-concordance of surface knots and Alexander ideals
|
Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
|
Date Issued |
2019-11-07T15:46
|
Description |
Paul Melvin proved that 0-concordant 2-knots have diffeomorphic Gluck twists, but until recently there were no known proofs that there is more than one 0-concordance class. Now Sunukjian and Dai-Miller have found many examples using Heegaard Floer technology applied to the Seifert 3-manifolds which the 2-knots bound. In this talk we give another proof using Alexander ideals. The main theorem is that the Alexander ideal induces a homomorphism from the 0-concordance monoid of 2-knots to the ideal class monoid of $\mathbb{Z}[t,t^{-1}]$. A corollary is that any 2-knot with nonprincipal Alexander ideal cannot be 0-slice, and moreover has no inverse in the 0-concordance monoid. This is the first proof that the monoid is not a group, and gives another proof of the existence of infinitely many linearly independent 0-concordance classes. These techniques also apply to higher genus surfaces, where we give the first results on 0-concordance. Lastly, we show that under a mild condition on the knot group, the peripheral subgroup of a knotted surface is also a 0-concordance invariant.
|
Extent |
49.0 minutes
|
Subject | |
Type | |
File Format |
video/mp4
|
Language |
eng
|
Notes |
Author affiliation: University of Georgia
|
Series | |
Date Available |
2020-05-06
|
Provider |
Vancouver : University of British Columbia Library
|
Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
|
DOI |
10.14288/1.0390369
|
URI | |
Affiliation | |
Peer Review Status |
Unreviewed
|
Scholarly Level |
Graduate
|
Rights URI | |
Aggregated Source Repository |
DSpace
|
Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International