Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The homomorphism defect of an extended Levine-Tristram signature via twisted homology

Alice Merz

Abstract

Taking the Levine-Tristram signature of the closure of a braid defines a map from the braid group to the integers. A formula of Gambaudo and Ghys provides an evaluation of the homomorphism defect of this map in terms of the Burau representation and the Meyer cocycle. In 2017 Cimasoni and Conway generalized this formula to the multivariable signature of the closure of coloured tangles. In the present paper, we extend even further their result by using a different 4-dimensional interpretation of the signature. We obtain an evaluation of the additivity defect in terms of the Maslov index and the isotropic functor $\mathscr{F}_ω$. We also show that in the case of coloured braids this defect can be rewritten in terms of the Meyer cocycle and the coloured Gassner representation, making it a direct generalization of the formula of Gambaudo and Ghys.

The homomorphism defect of an extended Levine-Tristram signature via twisted homology

Abstract

Taking the Levine-Tristram signature of the closure of a braid defines a map from the braid group to the integers. A formula of Gambaudo and Ghys provides an evaluation of the homomorphism defect of this map in terms of the Burau representation and the Meyer cocycle. In 2017 Cimasoni and Conway generalized this formula to the multivariable signature of the closure of coloured tangles. In the present paper, we extend even further their result by using a different 4-dimensional interpretation of the signature. We obtain an evaluation of the additivity defect in terms of the Maslov index and the isotropic functor . We also show that in the case of coloured braids this defect can be rewritten in terms of the Meyer cocycle and the coloured Gassner representation, making it a direct generalization of the formula of Gambaudo and Ghys.

Paper Structure

This paper contains 18 sections, 24 theorems, 149 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Some basic definitions
  3. Levine-Tristram signature
  4. Multivariate Levine-Tristram signature
  5. Twisted homology
  6. Novikov-Wall non-additivity
  7. The isotropic functor
  8. The category of coloured tangles
  9. The isotropic category
  10. The isotropic functor
  11. Non-additivity of multivariate signature
  12. Proof of the main theorem
  13. A reduction
  14. The manifold $P(\tau_1, \tau_2)$.
  15. The manifold $C(\tau)$.
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\tau_1$ and $\tau_2$ be $(c,c)$-tangles. Then we have for all $\omega=(\omega_1,\ldots,\omega_\mu)\in (S^1\setminus\{1\})^\mu$ such that $\prod_{j=1}^\mu \omega_j^{i_j} \neq 1.$ Here $\overline{\tau}$ denotes the reflection of the tangle $\tau$ across a horizontal plane with opposite orientation and $\mathop{\mathrm{id}}\nolimits_c$ denotes the trivial

Figures (12)

  • Figure 1: A clasp intersection.
  • Figure 2: The manifold $Y$ in the setting of the Novikov-Wall non-additivity theorem.
  • Figure 3: The composition of two coloured tangles. Here $\tau_1$ is a $(c,c')$-tangle with $c=(-2)$ and $c'=(-1,+1,-2)$, while $\tau_2$ is a $(c',c")$-tangle with $c"=(-1,+1,-2)$.
  • Figure 4: The closure of a tangle.
  • Figure 5: An example of $D_n$ for $\mu=1$.
  • ...and 7 more figures

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Example 2.5
  • Remark 2.6
  • Theorem 2.7
  • Definition 2.8
  • ...and 43 more