[Paper Review] On $d$-invariants and generalised Kanenobu knots

Marco Marengon|arXiv (Cornell University)|Dec 10, 2014

Geometric and Algebraic Topology20 references3 citations

TL;DR

This paper proves that for certain infinite families of L-spaces arising as branched double covers of generalized Kanenobu knots, the Ozsváth–Szabó d-invariants are unbounded both above and below. Using a relation between d-invariants and Turaev torsion in L-space branched covers, the authors establish the existence of infinitely many non-quasi-alternating, homologically thin knots with any odd determinant Δ² ≥ 25, and infinitely many hyperbolic, weight-1 manifolds not arising as surgeries on knots in S³.

ABSTRACT

We prove that for particular infinite families of $L$-spaces, arising as branched double covers, the $d$-invariants defined by Ozsváth and Szabó are arbitrarily large and small. As a consequence, we generalise a result by Greene and Watson by proving, for every odd number $Δ\geq 5$, the existence of infinitely many non-quasi-alternating homologically thin knots with determinant $Δ^2$, and a result by Hoffman and Walsh concerning the existence of hyperbolic weight $1$ manifolds that are not surgery on a knot in $S^3$.

Motivation & Objective

• To establish the unboundedness of d-invariants in specific families of L-spaces arising as branched double covers of generalized Kanenobu knots.
• To generalize Greene and Watson's result on non-quasi-alternating thin knots by extending it to all odd determinants Δ² ≥ 25.
• To extend Hoffman and Walsh's result on hyperbolic, weight-1 manifolds not arising as surgeries on knots in S³ to infinitely many such manifolds with arbitrary large odd determinant Δ².
• To provide a computational framework using Turaev torsion to determine d-invariants in L-space branched covers.
• To demonstrate that d-invariants serve as a strong obstruction to a 3-manifold being a knot surgery, particularly in the context of weight-1 manifolds.

Proposed method

• Leverages a known relation between d-invariants and Turaev torsion in L-spaces, derived from results of Mullins and Rustamov, to reduce d-invariant computation to Turaev torsion calculation.
• Constructs infinite families of generalized Kanenobu knots via 3-braids, ensuring constant determinant and homological invariants (Khovanov, odd-Khovanov, and knot Floer homologies).
• Computes the fundamental and first homology groups of the branched double covers to set up the Turaev torsion computation.
• Analyzes the coefficients of the Turaev torsion polynomial and proves they are unbounded for certain parameter families, implying unbounded d-invariants.
• Applies the duality property d(−Y, t) = −d(Y, t) to handle negative surgery slopes and extend the obstruction to knot surgeries.
• Uses the fact that d-invariants of surgeries on knots in S³ are bounded, so unbounded d-invariants imply the manifold cannot be a knot surgery.

Experimental results

Research questions

• RQ1 Can d-invariants of branched double covers of generalized Kanenobu knots be unbounded in both positive and negative directions?
• RQ2 Do there exist infinite families of non-quasi-alternating, homologically thin knots with any given odd determinant Δ² ≥ 25?
• RQ3 Are there infinitely many hyperbolic, weight-1 3-manifolds with |H₁| = Δ² that are not surgeries on knots in S³ for arbitrarily large odd Δ?
• RQ4 Can Turaev torsion computations effectively determine d-invariants in L-space branched covers?
• RQ5 To what extent do d-invariants obstruct a 3-manifold from being a knot surgery, especially in the weight-1 setting?

Key findings

• For every n ≥ 2, there exists a family of L-spaces {Σₘ}ₘ∈ℤ with |H₁(Σₘ; ℤ)| = (2n+1)² whose d-invariants are unbounded above and below.
• For every odd Δ ≥ 5, there exist infinitely many non-quasi-alternating, homologically thin knots with determinant Δ² and identical Khovanov, odd-Khovanov, and knot Floer homologies.
• The d-invariants of the branched double covers of generalized Kanenobu knots are unbounded due to unbounded coefficients in their Turaev torsion polynomials.
• For every odd Δ ≫ 0, there exist infinitely many hyperbolic, weight-1 3-manifolds M_{Δ,p} with |H₁(M_{Δ,p}; ℤ)| = Δ² that are not surgeries on any knot in S³.
• The unboundedness of d-invariants in these families provides a strong obstruction to such manifolds being knot surgeries, generalizing results by Hoffman and Walsh.
• The method using Turaev torsion to compute d-invariants in L-space branched covers is effective and yields sharp number-theoretic constraints on possible knot surgeries.