Tight contact structures on Seifert surface complements

2017/09/29 by Tamás Kálmán, Daniel V. Mathews · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Connection (principal bundle) #Euler characteristic #Euler's formula #Geometric and Algebraic Topology #Knot (papermaking) #Mathematical Dynamics and Fractals #Seifert surface #Surface (topology) #math.GT #math.SG #msc:57M15 #msc:57M27 #msc:57R17 #msc:57R58

paper · pdf · doi:10.1112/topo.12144

published in Journal of Topology 13(2), 730-776 (Wiley) · 45 pages, 19 figures

arxiv created 2017/09/29 · openalex created_date 2017/10/06 · openalex publication_date 2020/03/18 · arxiv updated 2020/03/25 · openalex updated_date 2026/08/05

Abstract

We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda's method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of the contact structures are identified with hypertrees in a certain hypergraph. Using earlier work, this establishes a connection between contact topology and the Homfly polynomial. We also show that the contact invariants of our tight contact structures form a basis for sutured Floer homology. Finally, we relate our methods and results to Kauffman's formal knot theory.

Citations

• Minimal genus Seifert surfaces for alternating links
• Giroux torsion and twisted coefficients
• Contact structures, sutured Floer homology and TQFT
• Holomorphic disks and three-manifolds invariants: Properties and applications
• Classification of overtwisted contact structures on 3-manifolds
• Holomorphic disks and topological invariants for closed three-manifolds
• Chip-Firing and Rotor-Routing on Directed Graphs
• Root polytopes, Tutte polynomials, and a duality theorem for bipartite graphs
• Characteristic Classes. (AM-76)

Cited by

• Contact structures, excisions and sutured monopole Floer homology