[Paper Review] Torsion homology growth beyond asymptotics

Oliver Bräunling|arXiv (Cornell University)|Feb 21, 2017

Geometric and Algebraic Topology32 references3 citations

TL;DR

This paper establishes that the generating function of log torsion homology in cyclic covers of knot complements admits a meromorphic continuation to the entire complex plane if the Alexander polynomial has no diophantine roots (roots on the unit circle that are not roots of unity). This result provides a new analytic proof of the Silver–Williams asymptotic, reconstructs the Alexander polynomial from torsion data, and generalizes to higher-dimensional knots via Reidemeister–Franz torsion. When diophantine roots exist, the function has the unit circle as a natural boundary, indicating intrinsic analytic obstruction.

ABSTRACT

We show that (under mild assumptions) the generating function of log homology torsion of a knot exterior has a meromorphic continuation to the entire complex plane. As corollaries, this gives new proofs of (a) the Silver-Williams asymptotic, (b) Fried's theorem on reconstructing the Alexander polynomial (c) Gordon's theorem on periodic homology. Our results generalize to other rank 1 growth phenomena, e.g. Reidemeister-Franz torsion growth for higher-dimensional knots. We also analyze the exceptional cases where the meromorphic continuation does not exist.

Motivation & Objective

• To understand the global analytic structure of the generating function E(z) encoding log torsion homology of cyclic covers of knot complements.
• To determine under what conditions E(z) admits meromorphic continuation beyond the unit disk.
• To generalize the Silver–Williams asymptotic formula using analytic number theory and complex analysis.
• To explore connections between torsion growth, special L-values, and the arithmetic of roots of Alexander polynomials.
• To extend results to higher-dimensional knots via Reidemeister–Franz torsion and multivariate generating functions.

Proposed method

• Define the generating function E(z) = ∑_{r≥1} log|H₁(X_r, ℤ)_tor| · z^r for cyclic covers X_r of a knot complement.
• Use Fourier analysis and Dirichlet series to analyze the arithmetic structure of the coefficients log|H₁(X_r, ℤ)_tor|.
• Apply complex analytic techniques to study the meromorphic continuation of E(z), particularly its poles and singularities.
• Relate the location and order of poles to roots of the Alexander polynomial Δ_K, especially those on or off the unit circle.
• Use results from algebraic number theory and L-functions to express special values of E(z) in terms of Dirichlet L-values at s=1.
• Generalize the framework to higher-dimensional knots by analyzing Reidemeister–Franz torsion growth via multivariate generating functions.

Experimental results

Research questions

• RQ1 Under what conditions does the generating function E(z) of log torsion homology admit a meromorphic continuation to ℂ?
• RQ2 How do the poles of E(z) relate to the roots of the Alexander polynomial Δ_K, particularly those on the unit circle?
• RQ3 Can the Silver–Williams asymptotic formula be derived from the analytic properties of E(z), rather than number-theoretic limits?
• RQ4 What happens to the analytic structure of E(z) when the Alexander polynomial has diophantine roots (roots on the unit circle that are not roots of unity)?
• RQ5 How do special L-values of Dirichlet characters arise in the expansion of E(z), especially near z=1?

Key findings

• If the Alexander polynomial Δ_K has no diophantine roots, then E(z) admits a meromorphic continuation to the entire complex plane with poles at integer powers of roots of Δ_K outside the open unit disc.
• The poles at these locations have order one, except possibly at z=1, which may have order one or two, directly implying the Silver–Williams asymptotic.
• The residue at each pole encodes the multiplicity of the corresponding root of Δ_K, enabling reconstruction of the Alexander polynomial from the analytic data of E(z).
• If Δ_K has at least one diophantine root, then E(z) has the unit circle as a natural boundary, and no analytic continuation beyond the open unit disc is possible.
• For each point p on the unit circle in the multiplicative span of diophantine roots, the radial limit (1−|z|)E(z) as z→p is non-zero, indicating strong singular behavior.
• The results extend to higher-dimensional knots: the generating function of Reidemeister–Franz torsion has meromorphic continuation if no Alexander polynomial has roots of absolute value 1, and natural boundary otherwise.