[Paper Review] Jones slopes and coarse volume of near-alternating links

Christine Ruey Shan Lee|arXiv (Cornell University)|Aug 16, 2017

Geometric and Algebraic Topology30 references3 citations

TL;DR

This paper introduces near-alternating links as a new class of links extending beyond adequate links, proving the Strong Slope Conjecture holds for them via spanning Jones surfaces. It establishes stable coefficients in their colored Jones polynomials and uses these to derive two-sided volume bounds for hyperbolic complements, demonstrating that near-alternating links satisfy the Coarse Volume Conjecture despite not being adequate.

ABSTRACT

We study near-alternating links whose diagrams satisfy conditions generalized from the notion of semi-adequate links. We extend many of the results known for adequate knots relating their colored Jones polynomials to the topology of essential surfaces and the hyperbolic volume of their complements: we show that the Strong Slope Conjecture is true for near-alternating knots with spanning Jones surfaces, their colored Jones polynomials admit stable coefficients, and the stable coefficients provide two-sided bounds on the volume of the knot complement. We also discuss extensions of these results to their Murasugi sums and a class of highly twisted links.

Motivation & Objective

• Motivate the extension of quantum invariants' topological connections beyond adequate links to a broader class of links.
• Address the question of whether the Strong Slope Conjecture and Coarse Volume Conjecture hold for non-adequate links with spanning surfaces.
• Define and characterize near-alternating links as a generalization of semi-adequate and adequate links.
• Establish that near-alternating links admit essential spanning surfaces realizing Jones slopes.
• Show that stable coefficients of the colored Jones polynomial provide coarse volume bounds for hyperbolic near-alternating links.

Proposed method

• Define near-alternating links via a 2-connected, weighted planar graph with a single negative edge of weight |r| ≥ 2, satisfying path-length and connectivity conditions.
• Use state-sum expansions of the colored Jones polynomial to isolate degree-dominating terms and analyze resolutions on crossings.
• Apply graphical skein theory to relate state surfaces of the link diagram to the reduced state graphs and their Euler characteristics.
• Prove stability of the first, second, penultimate, and last coefficients of the colored Jones polynomial by reducing to the adequate diagram Dₑ via deletion and contraction of the negative edge.
• Use Theorem 11 and Theorem 12 from [FKP08] to relate the number of twist regions to stable coefficients, enabling volume bounds.
• Establish two-sided volume bounds by combining the stable coefficient estimates with known volume inequalities for twist-reduced diagrams with ≥7 crossings per region.

Experimental results

Research questions

• RQ1 Can the Strong Slope Conjecture be realized by spanning surfaces in links that are not adequate?
• RQ2 Are stable coefficients in the colored Jones polynomial of near-alternating links preserved under diagrammatic modifications such as edge deletion and contraction?
• RQ3 Can the stable coefficients of near-alternating links provide coarse volume bounds for their hyperbolic complements?
• RQ4 Does the class of near-alternating links strictly extend the class of adequate links while preserving key quantum-topological conjectures?
• RQ5 Is there a diagrammatic condition generalizing semi-adequacy that supports both the Strong Slope Conjecture and Coarse Volume Conjecture?

Key findings

• The Strong Slope Conjecture holds for near-alternating knots with spanning Jones surfaces, as their Jones slopes are realized by essential spanning surfaces in the knot exterior.
• The first, second, penultimate, and last coefficients of the colored Jones polynomial of near-alternating links are stable, independent of the color n.
• Stable coefficients α, β, α′, β′ of the colored Jones polynomial satisfy |β| + |β′| − 1 ≤ 2(tw(D) − 1) and |β| + |β′| − 2 ≥ (tw(D) − 1)/3 for prime, twist-reduced diagrams with at least 3 crossings per twist region.
• Near-alternating links with prime, twist-reduced diagrams and ≥7 crossings per twist region are hyperbolic, enabling volume bounds via known inequalities.
• Two-sided volume bounds are established: 0.35367(|β| + |β′| − 1) < vol(S³ \ K) < 30v₃(|β| + |β′| − 2), where v₃ ≈ 1.0149.
• Near-alternating knots are not adequate, demonstrating that the Strong Slope Conjecture and Coarse Volume Conjecture hold for a strictly larger class than adequate links.