Generalized Seifert surfaces and signatures of colored links
By David Cimasoni, V. Florens. 2005
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In this paper, we use 'generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in S 3 . This yields an integral valued function on the μ-dimensional torus, where μ is the number of colors of the link. The case μ = 1 corresponds to the Levine-Tristram signature. We show that many remarkable properties of the latter invariant extend to this μ-variable generalization: it vanishes for achiral colored links, it is 'piecewise continuous', and the places of the jumps are determined by the Alexander invariants of the colored link. Using a 4-dimensional interpretation and the Atiyah-Singer G-signature theorem, we also prove that this signature is invariant by colored concordance, and that it provides a lower bound for the 'slice genus' of the colored link.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/fd57779536d7077ea3e9fee023d956eff2b9e172
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links (2005, background): “We wish to express our gratitude to Dror Bar–Natan for allowing us to use his figures from [3], to Vincent Florens for teaching us the Levine– Tristram signature and to Mikhail Khovanov for the e–mail conversation.”