[Paper Review] Knot invariants derived from the equivariant linking pairing

Christine Lescop|arXiv (Cornell University)|Jan 25, 2010

Geometric and Algebraic Topology22 references3 citations

TL;DR

This paper introduces a new invariant, ${\mathcal{Q}}(M,\mathbb{K})$, derived from the equivariant linking pairing in 3-manifolds with first Betti number one, using configuration space integrals and an equivariant cube construction. The invariant is valued in $\mathbb{Q}(x,y)$, satisfies a surgery formula, and is conjecturally equivalent to the two-loop part of the Kricker rational lift of the Kontsevich integral for null-homologous knots in rational homology spheres.

ABSTRACT

Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the configuration space of ordered pairs of distinct points of M. We show how to define the equivariant cube Q(M,K) of this Blanchfield pairing with respect to a framed knot K that generates H_1(M;Z)/Torsion. We present the invariant Q(M,K) and some of its properties including a surgery formula. Via surgery, the invariant Q is equivalent to an invariant Q' of null-homologous knots in rational homology spheres, that is conjecturally equivalent to the two-loop part of the Kontsevich integral. We generalize the construction of Q' to obtain a topological construction for an invariant that is conjecturally equivalent to the whole Kricker rational lift of the Kontsevich integral for null-homologous knots in rational homology spheres.

Motivation & Objective

• To define a topological invariant ${\mathcal{Q}}(M,\mathbb{K})$ for closed oriented 3-manifolds $M$ with $H_1(M;\mathbb{Z})/\text{Torsion} \cong \mathbb{Z}$, using the equivariant linking pairing.
• To construct this invariant via configuration space integrals and an equivariant cube of the Blanchfield pairing in an infinite cyclic covering.
• To establish a surgery formula relating ${\mathcal{Q}}(M,\mathbb{K})$ to the Casson-Walker invariant $\lambda(N)$ for rational homology spheres $N$.
• To generalize the construction to yield a topological invariant conjecturally equivalent to the full Kricker rational lift of the Kontsevich integral for null-homologous knots in rational homology spheres.
• To show that ${\mathcal{Q}}(M,\mathbb{K})$ induces an invariant for 3-manifolds of first Betti number one, equivalent to Ohtsuki's two-loop invariant in rank-one case.

Proposed method

• Define the equivariant linking pairing as a cohomology class in the configuration space of ordered pairs of distinct points in $M$, lifted to an infinite cyclic covering.
• Construct the equivariant cube ${\mathcal{Q}}(M,\mathbb{K})$ as a secondary invariant derived from this pairing, valued in the field of rational functions $\mathbb{Q}(x,y)$.
• Use a compactification $C_2(M)$ of the configuration space to define cycles $F_{X}, F_{Y}, F_{Z}$ dual to the linking form, and compute their triple intersection number.
• Apply a surgery formula: ${\mathcal{Q}}(M\sharp N,\mathbb{K}) = {\mathcal{Q}}(M,\mathbb{K}) + 6\lambda(N)$, where $\lambda(N)$ is the Walker-normalized Casson invariant.
• Re-express ${\mathcal{Q}}(M,\mathbb{K})$ as an invariant of null-homologous knots $\hat{K}$ in rational homology spheres via $0$-surgery on $\mathbb{K}$, leading to the invariant $\hat{{\mathcal{Q}}}$.
• Generalize $\hat{{\mathcal{Q}}}$ to a topological construction for an invariant conjecturally equivalent to the full Kricker rational lift of the Kontsevich integral.

Experimental results

Research questions

• RQ1 How can the equivariant linking pairing in a 3-manifold $M$ with $b_1(M) = 1$ be used to define a new finite-type invariant?
• RQ2 What is the relationship between the invariant ${\mathcal{Q}}(M,\mathbb{K})$ and the Casson-Walker invariant $\lambda(N)$ under connected sum with a rational homology sphere $N$?
• RQ3 Is the invariant ${\mathcal{Q}}(M,\mathbb{K})$ equivalent to the two-loop part of the Kricker rational lift of the Kontsevich integral for null-homologous knots?
• RQ4 Can the construction of ${\mathcal{Q}}(M,\mathbb{K})$ be generalized to yield a topological realization of the full Kricker rational lift?
• RQ5 What is the vector space spanned by differences ${\mathcal{Q}}(M,\mathbb{K}') - {\mathcal{Q}}(M,\mathbb{K})$ for different framed knots $\mathbb{K}'$ generating $H_1(M;\mathbb{Z})/\text{Torsion}$?

Key findings

• The invariant ${\mathcal{Q}}(M,\mathbb{K})$ is well-defined and takes values in $\mathbb{Q}(x,y)$, with ${\mathcal{Q}}(S^1 \times S^2, S^1 \times u) = 0$.
• The surgery formula states that ${\mathcal{Q}}(M\sharp N,\mathbb{K}) = {\mathcal{Q}}(M,\mathbb{K}) + 6\lambda(N)$, where $\lambda(N)$ is the Walker-normalized Casson invariant.
• The invariant ${\mathcal{Q}}(M,\mathbb{K})$ induces a well-defined invariant for 3-manifolds with first Betti number one, equivalent to Ohtsuki’s two-loop invariant in the rank-one case.
• Via $0$-surgery, ${\mathcal{Q}}(M,\mathbb{K})$ gives rise to an invariant $\hat{{\mathcal{Q}}}$ of null-homologous knots in rational homology spheres, conjecturally equivalent to the two-loop part of the Kricker rational lift.
• The construction generalizes to yield a topological invariant conjecturally equivalent to the full Kricker rational lift of the Kontsevich integral for null-homologous knots.
• The variation of ${\mathcal{Q}}$ under change of framed knot $\mathbb{K}$ is governed by the difference in their framing classes, with the change matching the universal finite type invariant variation $\frac{1}{4}(p_1(\tau) - p_1(\tau'))\xi_n$.