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A note on alternating knots in handlebodies

Lizzie Buchanan, Tanushree Shah

TL;DR

This work extends the classic Kauffman–Murasugi–Thistlethwaite paradigm for alternating knots to knots embedded in handlebodies, focusing on genus-1 (solid torus) cases. It introduces dotted-reduced diagrams and a Bataineh–Hajij type generalized Jones polynomial for links in the solid torus, proving that alternating dotted-reduced diagrams realize minimal crossing number in this setting. The core technical advance is a span analysis: for alternating dotted-reduced diagrams, the bracket span reaches $4n$ and, after normalization, the Jones span equals $n$, establishing crossing-number minimality and invariant writhe across diagrams. The results connect with Boden–Karimi–Sikora’s generalized Jones framework and extend to links in genus-$g$ handlebodies, providing tools for studying links in broader 3-manifold contexts.

Abstract

We establish a Kauffman-Murasugi-Thistlethwaite-type theorem for alternating knots in a solid torus. Specifically, we show that any dotted-reduced alternating diagram of a knot in a handlebody realizes the minimal crossing number, and that any two such diagrams of the same knot have identical writhe. The proof relies on a generalization of the Jones polynomial to the setting of handlebodies. A stronger version of this result was already proved by Boden, Karimi, and Sikora using a different generalized Jones polynomial; therefore, this text largely expands on one of the main proof tools.

A note on alternating knots in handlebodies

TL;DR

This work extends the classic Kauffman–Murasugi–Thistlethwaite paradigm for alternating knots to knots embedded in handlebodies, focusing on genus-1 (solid torus) cases. It introduces dotted-reduced diagrams and a Bataineh–Hajij type generalized Jones polynomial for links in the solid torus, proving that alternating dotted-reduced diagrams realize minimal crossing number in this setting. The core technical advance is a span analysis: for alternating dotted-reduced diagrams, the bracket span reaches and, after normalization, the Jones span equals , establishing crossing-number minimality and invariant writhe across diagrams. The results connect with Boden–Karimi–Sikora’s generalized Jones framework and extend to links in genus- handlebodies, providing tools for studying links in broader 3-manifold contexts.

Abstract

We establish a Kauffman-Murasugi-Thistlethwaite-type theorem for alternating knots in a solid torus. Specifically, we show that any dotted-reduced alternating diagram of a knot in a handlebody realizes the minimal crossing number, and that any two such diagrams of the same knot have identical writhe. The proof relies on a generalization of the Jones polynomial to the setting of handlebodies. A stronger version of this result was already proved by Boden, Karimi, and Sikora using a different generalized Jones polynomial; therefore, this text largely expands on one of the main proof tools.
Paper Structure (5 sections, 6 theorems, 20 equations, 4 figures)

This paper contains 5 sections, 6 theorems, 20 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Dotted reduced diagrams
  4. Reduced and dotted-reduced diagrams
  5. Proof of main theorem

Key Result

Proposition 2.4

For any link diagram $D$ in solid torus, and

Figures (4)

  • Figure 1: A crossing (left), its $A$-smoothing (middle), and $B$-smoothing (right)
  • Figure 2: Removing dotted-reducible crossings
  • Figure 3: Smoothing crossings
  • Figure 4: In each of the four rows, we have on the left a state $S$ in which our negative dotted-irreducible crossing is given an $A$-smoothing, and on the right the adjacent state $S'$ which is identical to $S$ except at that one crossing. The four rows show each of the four possibilities for how a state circle involving this crossing could interact with the rest of the diagram.

Theorems & Definitions (22)

  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • Corollary 3.5
  • ...and 12 more