Seifert-Tait graphs

TL;DR

This work studies the relationship between Seifert and Tait graphs of knot diagrams and introduces Seifert-Tait knots—alternating knots whose Seifert and Tait graphs are isomorphic. It develops a framework connecting Seifert surfaces, Tait graphs, and flat diagrams via the cyclic word (Wicks form) algorithm and Gauss diagrams, proving that flat diagrams have isomorphic Seifert and Tait graphs and that, for prime knots of fixed genus $g>1$, flat diagrams dominate in number as the crossing number $n$ grows. Leveraging flype theory and generating-series arguments, the authors show that the fraction of Seifert-Tait alternating knots among prime alternating knots tends to $1$ as $n\to\infty$, establishing strong asymptotic dominance results. The findings illuminate dominant structural patterns in large-crossing alternating knots and connect graph-theoretic constructions with knot diagrams through rigorous equivalence of surfaces and a counting framework for generating series.

Abstract

We show that among alternating knots, those which have diagrams whose Seifert and Tait graphs are isomorphic are dominant.

Figures (17)

• Figure 1: Signs of oriented crossings, and the only possible splicing.
• Figure 2: Moving between a link diagram and its signed Tait graph.
• Figure 3: d-edges and c-edges before and after splicing a crossing.
• Figure 4: A diagram of the knot $8_{11}$, with an orientation.
• Figure 5: The Tait graphs $T$ and $T^{*}$. The unlabelled edges are all d-edges.

Theorems & Definitions (30)

• Definition 1.1
• Theorem
• Lemma 2.1
• Theorem 2.2: HMV
• Definition 3.1
• Definition 3.2: Alina2005
• Lemma 3.3: Cromwell2004
• Theorem 3.4
• Theorem 3.5
• Definition 4.1: Alina2005