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Crossing numbers of cable knots

Efstratia Kalfagianni, Rob Mcconkey

Abstract

We use the degree of the colored Jones knot polynomials to show that the crossing number of a $(p,q)$-cable of an adequate knot with crossing number $c$ is larger than $q^2\, c$. As an application we determine the crossing number of $2$-cables of adequate knots. We also determine the crossing number of the connected sum of any adequate knot with a $2$-cable of an adequate knot.

Crossing numbers of cable knots

Abstract

We use the degree of the colored Jones knot polynomials to show that the crossing number of a -cable of an adequate knot with crossing number is larger than . As an application we determine the crossing number of -cables of adequate knots. We also determine the crossing number of the connected sum of any adequate knot with a -cable of an adequate knot.
Paper Structure (6 sections, 11 theorems, 39 equations, 4 figures)

This paper contains 6 sections, 11 theorems, 39 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Crossing numbers of cables of adequate knots
  3. Preliminaries
  4. Lower bounds and admissible knots
  5. Non-adequacy results
  6. Composite non-adequate knots

Key Result

Theorem 1.1

For any adequate knot $K$ with crossing number $c(K)$, and any coprime integers $p,q$, we have $c(K_{p,q})\geq q^2 \, c(K)+1$.

Figures (4)

  • Figure 1: The $A$- and $B$-resolution and the corresponding edges of ${\mathbb G}_A(D)$ and ${\mathbb G}_B(D)$.
  • Figure 2: A positive crossing and a negative crossing.
  • Figure 3: Three positive (left) and three negative (right) twists on four strands.
  • Figure 4: A diagram of the (-1,2)-cable of the figure eight knot and its all-$B$ state graph.

Theorems & Definitions (22)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 12 more