Turaev genus and alternating decompositions

Abstract

We prove that the genus of the Turaev surface of a link diagram is determined by a graph whose vertices correspond to the boundary components of the maximal alternating regions of the link diagram. Furthermore, we use these graphs to classify link diagrams whose Turaev surface has genus one or two, and we prove that similar classification theorems exist for all genera.

28 pages, 24 figures. Significant changes to the proofs of Theorem 1.5 and Theorem 3.8

References in corpus (4)

• Floer homology and knot complements
• The Jones polynomial and graphs on surfaces
• Invariants for Turaev genus one links
• Link diagrams with low Turaev genus

Cited by in corpus (10)

• Invariants for Turaev genus one links
• Link diagrams with low Turaev genus
• Extremal Khovanov homology of Turaev genus one links
• Jones slopes and coarse volume of near-alternating links
• A topological characterization of toroidally alternating knots
• Near extremal Khovanov homology of Turaev genus one links
• The Jones polynomial of an almost alternating link
• On the Turaev genus of torus knots
• Geometric estimates from spanning surfaces
• On the arc index and Turaev genus of a link