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Concordances of sums of alternating torus knots and their mirrors to $L$-space knots

Dan Guyer, Thomas Sachen

Abstract

Continuing the work of Zemke, Livingston and Allen, we consider when linear combinations of torus knots are concordant to $L$-space knots. We begin by proving Allen's conjecture for alternating torus knots. That is, we prove that a linear combination of alternating torus knots is concordant to an $L$-space knot if and only if the connected sum is a single torus knot. Then we establish a necessary condition for when a linear combination of torus knots is concordant to an $L$-space knot.

Concordances of sums of alternating torus knots and their mirrors to $L$-space knots

Abstract

Continuing the work of Zemke, Livingston and Allen, we consider when linear combinations of torus knots are concordant to -space knots. We begin by proving Allen's conjecture for alternating torus knots. That is, we prove that a linear combination of alternating torus knots is concordant to an -space knot if and only if the connected sum is a single torus knot. Then we establish a necessary condition for when a linear combination of torus knots is concordant to an -space knot.
Paper Structure (4 sections, 9 theorems, 10 equations)

This paper contains 4 sections, 9 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Results
  4. Acknowledgements

Key Result

Proposition 2.1

Let $\{(p_i,q_i)\}_{i=1,\ldots,n}$ be a set of pairs of relatively prime positive integers with $2\leq p_i<q_i$ for all $i$ and with $n>1$. Then $\#_iT(p_i,q_i)$ is not concordant to an $L$-space knot.

Theorems & Definitions (16)

  • Conjecture : Allen Allen
  • Proposition 2.1: Livingston Livingston
  • Proposition 2.2: Allen Allen
  • Corollary 2.3
  • Remark 2.4
  • Definition 2.5: Lisca Lisca1
  • Definition 2.6: Aceto-Celoria-Park Jung
  • Proposition 2.7: Aceto-Celoria-Park Jung
  • Proposition 2.8: Aceto-Celoria-Park Jung
  • Proposition 2.9: Aceto-Celoria-Park Jung
  • ...and 6 more