[Paper Review] Twisted cohomology pairings of knots I; diagrammatic computation

Takefumi Nosaka|arXiv (Cornell University)|Feb 2, 2016

Geometric and Algebraic Topology16 references4 citations

TL;DR

This paper presents a diagrammatic method to compute twisted cohomology pairings for links in the 3-sphere using only link diagrams, bypassing the need for Seifert surfaces or explicit homology representatives. The key contribution is a computable formula for the bilinear form arising from cup products with local coefficients and integral homology 2-classes, which generalizes classical invariants like the Blanchfield pairing and provides a topological interpretation of quandle cocycle invariants.

ABSTRACT

We provide a diagrammatic computation for the bilinear form, which is defined as the pairing between the (relative) cup products with every local coefficients and every integral homology 2-class of every links in the 3-sphere. As a corollary, we construct bilinear forms on the twisted Alexander modules of links.

Motivation & Objective

• To develop a diagrammatic computation for twisted cohomology pairings arising from cup products with local coefficients and relative homology 2-classes in link complements.
• To overcome the difficulty of computing relative cup products in 3-manifolds with non-trivial fundamental groups, especially in the presence of boundary conditions.
• To establish a direct link between diagrammatic invariants (e.g., quandle cocycle invariants) and topological invariants via cup product pairings.
• To provide an explicit, computable formulation of the bilinear form on twisted Alexander modules for links, using only link diagrams.

Proposed method

• The method uses a diagrammatic formulation of the relative homology 2-class in the link complement, represented via chains in the fundamental groupoid of the link diagram.
• It constructs a 2-cycle in the relative chain complex $ C_2( ho_L, ho_{ ext{bdry}}; Η) $ from meridians and crossings, avoiding explicit Seifert surfaces.
• The cup product pairing is computed via a composition of maps: $ H^1(Y, ∂Y; M)^{⊗ n} \xrightarrow{\smile} H^n(Y, ∂Y; M^{⊗ n}) \xrightarrow{\langle\cdot, \mu\rangle} M^{⊗ n} \xrightarrow{\psi} A $, with $ n=2 $.
• The pairing is reduced to a sum over crossings in the diagram, using Fox derivatives and the representation $ f: \pi_1(S^3 \setminus \nu L) \to G $, with coefficients in a $ G $-module $ M $.
• The construction relies on a chain-level model of relative group homology and the use of quandle cocycles to define the pairing.
• A key technical tool is the reduction of the 2-cycle $ \hat{\mu}_{\ell} $ to a sum of standard 2-chains associated with arcs and crossings, ensuring the pairing is computable from the diagram alone.

Experimental results

Research questions

• RQ1 Can the twisted cup product pairing on the cohomology of link complements be computed directly from a link diagram without constructing Seifert surfaces?
• RQ2 How does the bilinear form on twisted Alexander modules relate to diagrammatic invariants such as quandle cocycle invariants?
• RQ3 To what extent can classical invariants like the Blanchfield pairing and Casson-Gordon local signature be recovered from this diagrammatic cup product construction?
• RQ4 Is there a canonical homomorphism from $ H_1(Y_L; M) $ to $ H^1(Y_L, \partial Y_L; M) $ that arises naturally from this construction?
• RQ5 Can the pairing be expressed purely in terms of the fundamental group and meridians using Fox calculus and group homology?

Key findings

• The paper establishes a diagrammatic formula for the twisted cohomology pairing (1) that depends only on the link diagram, meridians, and the representation $ f: \pi_1(S^3 \setminus \nu L) \to G $, eliminating the need for Seifert surfaces.
• The bilinear form on the twisted Alexander modules is explicitly computable from the diagram via a sum over crossings, using Fox derivatives and the representation data.
• For the trefoil and figure-eight knots, the method yields concrete computations that reveal non-degeneracy and symmetry properties of the pairing.
• The pairing for the $ (m,m) $-torus link $ T_{m,m} $ is computed explicitly, and the result is used in the study of 4-dimensional Lefschetz fibrations.
• The construction recovers three classical invariants: the Blanchfield pairing, twisted cup products on infinite cyclic covers, and the Casson-Gordon local signature, unifying them under a single diagrammatic framework.
• The method provides a topological interpretation of quandle cocycle invariants as special cases of the cup product pairing, linking diagrammatic invariants to cohomological structures.