The Levine-Tristram signature: a survey

Abstract

The Levine-Tristram signature associates to each oriented link L in S 3 a function σ L : S 1 → Z. This invariant can be defined in a variety of ways, and its numerous applications include the study of unlinking numbers and link concordance. In this survey, we recall the three and four dimensional definitions of σ L , list its main properties and applications, and give comprehensive references for the proofs of these statements.

Limits of the Tristram--Levine signature function

“…Surprisingly, not much is known about the topological interpretation of σ 1 . In [6, Theorem 2.1] it is proved that σ 1 ≤ r − 1; see also [4,Remark 2.2]. This statement generalizes the well-known fact that σ 1 = 0 if L is a knot.…”

Signature spectrum of positive braids

“…piecewise linear with a breakpoint at ω = ζ 3 = e 2πi 3 . Therefore, the resulting signature function for the family of braids β n = (σ 2 1 σ 2 2 ) n is also piecewise linear, with a peak (and breakpoint) of 2 3 b 1 (β n ) at ω = ζ 3 , sloping down to 1 2 b 1 (β n ) between ω = ζ 3 and ω = −1 (see also Corollary 4.4 in [6] for the value of σ ω at ω = ζ 3 ).…”

“…ω ∈ S 1 \ Q. In particular, the signature is just the difference of the numbers of positive and negative eigenvalues of the matrix M ω (see [8,12] for the original definition, and [2] for an excellent survey on the Levine-Tristram signature invariants). We use the notation B + n for the monoid of positive braids on n strands, and σ ω (β) for the Levine-Tristram signature invariant of the closure of a braid β ∈ B + n .…”

Relative genus bounds in indefinite four-manifolds

“…Levine-Tristram signatures. Given a knot K ⊂ S 3 and a value ω ∈ S 1 , the Levine-Tristram signature σ K (ω) is defined as the signature of (1 − ω)A + (1 − ω)A T , where A is a Seifert matrix for K; see [64], [37], or [9].…”