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One-sided Levine–Tristram signature limits exist

Statement

Assume the hypotheses and conventions of the Levine–Tristram form definition. At every root $ω_{0} \in S^{1} ∖ {1}$ of $Δ_{K}$ , each one-sided limit of the nonroot signature as the parameter moves along the unit circle to $ω_{0}$ exists and is an integer.

Statement references

The matrix and one-sided limit convention

Facts & Assumptions

Given: Full AC, an oriented knot, its Seifert matrix $V$ , and an Alexander root $ω_{0} \neq 1$ .

[A1]

Full AC and the chosen surface/basis supply a finite real Seifert matrix. Seifert pairing and matrix of an oriented knot, The Axiom of Choice

[F1]

$H_{e^{i θ}}$ is a continuous Hermitian matrix family, and its singular parameters away from $1$ are exactly the zeros of a nonzero Alexander polynomial. Levine–Tristram Hermitian form and signature, The singular parameters of the Levine–Tristram form, The square Seifert presentation generates the knot Alexander polynomial

[F2]

Ordered Hermitian eigenvalues vary continuously, indeed Lipschitzly in operator norm. Ordered eigenvalues of Hermitian matrices are operator-norm Lipschitz

[F3]

A continuous real function on an interval cannot take both positive and negative values without vanishing between them. Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$

Proof

1.1A1F1F2givenchoose

The entries of $H_{e^{i θ}}$ are continuous functions of $θ$ . By [F2], each ordered eigenvalue $λ_{j} (θ)$ is continuous. The determinant presentation in [F1] is a nonzero Laurent polynomial because its value at $t = 1$ is $\pm 1$ ; hence it has only finitely many roots on the unit circle. Choose $ϵ > 0$ so that each punctured one-sided arc $(θ_{0} - ϵ, θ_{0})$ and $(θ_{0}, θ_{0} + ϵ)$ contains no other root. On each such arc, [F1] makes every eigenvalue nonzero.

2.1F1F3step 1.1algebra∎

On either arc, a continuous real function $λ_{j}$ that never vanishes has constant sign: if it took both signs, [F3] would give a zero. Thus every positive/negative eigenvalue count is constant on each one-sided arc, and the signature there is a fixed integer. Each one-sided limit is that integer.

Depends on

Used by

Cited to discharge well-definedness by Levine–Tristram Hermitian form and signature.

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Sources

Anthony Conway, The Levine–Tristram signature: a survey, 2019–20 MATRIX Annals (2021), pp. 31–56 (arXiv:1903.04477) (standard reference, not scraped)