uu.seUppsala University Publications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Khovanov's conjecture over Z[c]
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
Article in journal (Refereed) Submitted
Identifiers
URN: urn:nbn:se:uu:diva-91044OAI: oai:DiVA.org:uu-91044DiVA: diva2:163621
Available from: 2003-11-19 Created: 2003-11-19Bibliographically approved
In thesis
1. Khovanov Homology and Link Cobordisms
Open this publication in new window or tab >>Khovanov Homology and Link Cobordisms
2003 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Khovanov homology associates to any link in three-dimensional space a bigraded chain complex of Z[c]-modules, whose graded Euler characteristic (after normalization) is the Jones polynomial of the link. The complex is invariant up to homotopy equivalence under ambient isotopy of the link. Hence, the isomorphism class of the corresponding homology module is a link invariant. If the indeterminate c is set to zero, the resulting chain complex of abelian groups is finite, and the graded Euler characteristic is still the Jones polynomial.

In this thesis, we prove that these latter homology groups are functorial under link cobordism. More precisely, every link cobordism induces a group homomorphism between the homology groups of its boundary links, invariant (up to sign) under any ambient isotopy which leaves the boundary links set-wise fixed. This proves (with a necessary modification) part of a conjecture by Khovanov (Duke Math. J. 101 (1999)). We then introduce some derived invariants of link cobordisms from a link to itself. These are graded Lefschetz numbers of the induced homomorphisms above. They can be computed directly on the chain level. The Jones polynomial appears as the graded Lefschetz number of the identity cobordism. We further prove that the statement of functoriality for homology with coefficients in Z[c] is false. This disproves the rest of the above-mentioned conjecture. Finally, we provide an elementary description of the Khovanov homology of tangles.

Place, publisher, year, edition, pages
Uppsala: Matematiska institutionen, 2003. 59 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049
Keyword
Algebra and geometry, Khovanov homology, Jones polynomial, Link cobordisms, Algebra och geometri
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-3765 (URN)91-506-1717-6 (ISBN)
Public defence
2003-12-12, Sal 111, Hus 1, Polacksbacken, Polacksbacken, Lägerhyddsvägen 2, Uppsala, 13:15
Opponent
Supervisors
Available from: 2003-11-19 Created: 2003-11-19Bibliographically approved

Open Access in DiVA

No full text

By organisation
Department of Mathematics

Search outside of DiVA

GoogleGoogle Scholar

urn-nbn

Altmetric score

urn-nbn
Total: 353 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf