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Twisted Blanchfield pairings and twisted signatures III: Applications

Maciej Borodzik, Anthony Conway, Wojciech Politarczyk

Abstract

This paper describes how to compute algorithmically certain twisted signature invariants of a knot $K$ using twisted Blanchfield forms. An illustration of the algorithm is implemented on $(2,q)$-torus knots. Additionally, using satellite formulas for these invariants, we also show how to obstruct the sliceness of certain iterated torus knots.

Twisted Blanchfield pairings and twisted signatures III: Applications

Abstract

This paper describes how to compute algorithmically certain twisted signature invariants of a knot using twisted Blanchfield forms. An illustration of the algorithm is implemented on -torus knots. Additionally, using satellite formulas for these invariants, we also show how to obstruct the sliceness of certain iterated torus knots.
Paper Structure (33 sections, 28 theorems, 141 equations, 3 figures, 1 algorithm)

This paper contains 33 sections, 28 theorems, 141 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Classical knot theory
  3. Twisted knot invariants
  4. Twisted Blanchfield forms and twisted signatures
  5. Twisted signatures of iterated torus knots
  6. The algorithm
  7. Discussion of the algorithm
  8. Organisation
  9. Signatures of linking forms
  10. Classification of linking forms
  11. Signatures of linking forms
  12. Twisted Blanchfield forms
  13. Twisted Blanchfield pairings
  14. A review of Powell's algorithm
  15. Twisted signatures
  16. ...and 18 more sections

Key Result

Theorem 1.3

For any $k>0$ there are $2k+1$ characters $H_1(\Sigma_2(T_{2,2k+1})) \to \mathbb{Z}_{2k+1}$ which are denoted by $\chi_\theta$ for $\theta=0,\ldots,2k$. Each of the unitary representations $\alpha(2,\chi_{\theta})$ is acyclic and, for $\theta=1,\ldots,k$, the representation $\alpha(2,\chi_{\theta})$ where the linking forms $\lambda_{\theta}^{even}$ and $\lambda_{\theta}^{odd}$ are as follows: and

Figures (3)

  • Figure 1: A diagram of $T_{2,2k+1}$ together with generators of the knot group. Arrows indicate orientation of the respective meridian when going under the knot. The blue loop is $a = x_{2k}x_{2k+1}=x_{1}x_{2}$.
  • Figure 2: Left frame: the standard genus $1$ Heegaard decomposition of $S^3$. Central frame: the knots $K_1$ and $K_2$ lying in the Heegaard torus $T$. Right frame: the neighbourhoods $\overline{\nu}(K_1)$ and $\overline{\nu}(K_2)$ of $K_1$ and $K_2$ that satisfy $T = \overline{\nu}(K_1) \cup \overline{\nu}(K_2)$ and $\overline{\nu}(K_1) \cap \overline{\nu}(K_2) = \partial \overline{\nu}(K_1) = \partial \overline{\nu}(K_2)$.
  • Figure 3: The solid torus $V_1 \cong S^1 \times I^2$ and its decomposition as a union of two $3$-balls, $B_1$ (red) and $B_2$ (the remaining part of $V_{1}$). The attaching region $\partial_+ B_1=\partial_{+,1} B_{1} \sqcup \partial_{+,2} B_{1}$ of $B_1$, thought of as a $1$-handle, is also shown.

Theorems & Definitions (66)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5
  • Remark 3.1
  • Definition 3.2
  • ...and 56 more