The Levine-Tristram signature associates to each oriented link $L$ in $S^3$ a function $σ_L \colon S^1 \to \mathbb{Z}.$ 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.
We prove that the maximum of the Levine-Tristram signature function of a torus knot satisfies a reduction formula analogous to a result by Gordon-Litherland-Murasugi for the classical signature.
We show that under a precise condition on the single variable Alexander polynomial, the limit at $1$ of the Tristram--Levine signature of a link is determined by the linking matrix.
We give an explicit construction of linearly independent families of knots arbitrarily deep in the (n)-solvable filtration of the knot concordance group using the ρ^1-invariant. A difference between previous constructions of infinite rank subgroups in the concordance group and ours is that the deepest infecting knots in the construction we present are allowed to have vanishing Tristram-Levine signatures.
Echeverria recently introduced an invariant for a smoothly embedded torus in a homology $S^1\times S^3$, using gauge theory for singular connections. We define a new topological invariant of such an embedded torus, analogous to the classical Levine-Tristram invariant of a knot. In the 3-dimensional situation, a count of singular connections on a knot complement reproduces the Levine-Tristram invariant. We compute the invariant for a number of examples embedded tori, and show that our topological invariant is the same as what one might expect from Echeverria's invariant. Langte Ma has subsequen…
We give a new proof that the Levine-Tristram signatures of a link give lower bounds for the minimal sum of the genera of a collection of oriented, locally flat, disjointly embedded surfaces that the link can bound in the 4-ball. We call this minimal sum the 4-genus of the link. We also extend a theorem of Cochran, Friedl and Teichner to show that the 4-genus of a link does not increase under infection by a string link, which is a generalised satellite construction, provided that certain homotopy triviality conditions hold on the axis curves, and that enough Milnor's invariants of the infection…
Taking the Levine-Tristram signature of the closure of a braid defines a map from the braid group to the integers. A formula of Gambaudo and Ghys provides an evaluation of the homomorphism defect of this map in terms of the Burau representation and the Meyer cocycle. In 2017 Cimasoni and Conway generalized this formula to the multivariable signature of the closure of coloured tangles. In the present paper, we extend even further their result by using a different 4-dimensional interpretation of the signature. We obtain an evaluation of the additivity defect in terms of the Maslov index and the …
Given a knot $K$ inside an integer homology sphere $Y$, the Casson-Lin-Herald invariant can be interpreted as a signed count of conjugacy classes of irreducible representations of the knot complement into $SU(2)$ which map the meridian of the knot to a fixed conjugacy class. It has the interesting feature that it determines the Tristram-Levine signature of the knot associated to the conjugacy class chosen. Turning things around, given a 4-manifold $X$ with the integral homology of $S^{1}\times S^{3}$, and an embedded torus $T$ inside $X$ such that $H_{1}(T;\mathbb{Z})$ surjects onto $H_{1}(X;\…
We derive a lower bound for all Levine-Tristram signatures of positive braid links, linear in terms of the first Betti number. As a consequence, we obtain upper and lower bounds on the ratio of fixed pairs of Levine-Tristram signature invariants, valid uniformly on all positive braid monoids.
We study properties of the signature function of the torus knot $T_{p,q}$. First we provide a very elementary proof of the formula for the integral of the signatures over the circle. We obtain also a closed formula for the Tristram--Levine signature of a torus knot in terms of Dedekind sums.
We present a proof of Litherland's formula for the Tristram-Levine signature of a satellite knot in terms of its constituents. Litherland's original proof used more advanced algebraic techniques, while ours uses only linear algebra and some basic results in knot theory.
We compute the average Tristram---Levine signature of any graph link with positive weights in a three sphere, generalizing the results of Kirby and Melvin. The main tools are the Neumann's algorithm for computing the equivariant signatures of graph links and the Reciprocity Law for Dedekind sums.
We use a knot invariant, namely the Tristram--Levine signature to study deformations of singular points of plane curves. We find a bound on the sum of M numbers over all singularities of a generic fiber in terms of the M number of the singularity at the central fiber and some topological data.
Let $D$ be an oriented link diagram with the set of regions $\operatorname{r}_{D}$. We define a symmetric map (or matrix) $\operatornameτ_{D}\colon\operatorname{r}_{D}\times \operatorname{r}_{D} \to \mathbb{Z}[x]$ that gives rise to an invariant of oriented links, based on a slightly modified $S$-equivalence of Trotter and Murasugi in the space of symmetric matrices. In particular, for real $x$, the negative signature of $\operatornameτ_{D}$ corrected by the writhe is conjecturally twice the Tristram--Levine signature function, where $2x=\sqrt{t}+\frac1{\sqrt{t}}$ with $t$ being the indetermin…
We study the four-genus of linear combinations of torus knots: aT(p,q) # -bT(p',q'). Fixing positive p, q, p', and q', our focus is on the behavior of the four-genus as a function of positive a and b. Three types of examples are presented: in the first, for all a and b the four-genus is completely determined by the Tristram-Levine signature function; for the second, the recently defined Upsilon function of Ozsvath-Stipsicz-Szabo determines the four-genus for all a and b; for the third, a surprising interplay between signatures and Upsilon appears.
We refine prior bounds on how the multivariable signature and the nullity of a link change under link cobordisms. The formula generalizes a series of results about the 4-genus having their origins in the Murasugi-Tristram inequality, and at the same time extends previously known results about concordance invariance of the signature to a bigger set of allowed variables. Finally, we show that the multivariable signature and nullity are also invariant under $0.5$-solvable cobordism.
We use Morse theoretical arguments to study algebraic curves in C^2. We take an algebraic curve C in C^2 and intersect it with a family of spheres with fixed origin and varying radii. We explain in detail how does the resulting link change when we cross a singular point of C. Applying link invariants as Murasugi's signature and Levine--Tristram signatures we obtain some informations about possible singularities of a curve C in terms of its topology.
Homology of the circle with non-trivial local coefficients is trivial. From this well-known fact we deduce geometric corollaries concerning links of codimension two. In particular, the Murasugi-Tristram signatures are extended to invariants of links formed of arbitrary oriented closed codimension two submanifolds of an odd-dimensional sphere. The novelty is that the submanifolds are not assumed to be disjoint, but are transversal to each other, and the signatures are parametrized by points of the whole torus. Murasugi-Tristram inequalities and their generalizations are also extended to this se…
