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The Levine-Tristram Signature: a Survey

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The Levine-Tristram Signature: a Survey The Levine-Tristram signature: a survey Anthony Conway Abstract The Levine-Tristram signature associates to each oriented link L in S3 a 1 function sL : S ! Z: This invariant can be defined in a variety of ways, and its nu- merous applications include the study of unlinking numbers and link concordance. In this survey, we recall the three and four dimensional definitions of sL, list its main properties and applications, and give comprehensive references for the proofs of these statements. 1 Introduction 3 1 Given an oriented link L ⊂ S , the Levine-Tristram signature is a function sL : S ! Z whose study goes back to the sixties [94, 57]. The main goal of this survey ar- ticle is to collect the various definitions of sL, while a secondary aim is to list its properties. Although some elementary arguments are outlined in the text, we pro- vide detailed external references for most of the proofs. Briefly, we will discuss the definition in terms of Seifert matrices, various 4-dimensional interpretations as well as connections to pairings on the Alexander module. The next paragraphs give the flavor of some of these constructions. Most knot theory textbooks that cover the Levine-Tristram signature introduce it using Seifert matrices [61, 49, 46, 67]. Indeed, as we review in Section2, the Levine-Tristram signature at w 2 S1 can be defined using any Seifert matrix A for L by setting T sL(w) = sign(1 − w)A + (1 − w)A : In the same section, we collect the numerous properties of sL: after listing its behav- ior under mirror images, orientation reversals and satellite operations, we review ap- Anthony Conway Department of Mathematical Sciences, Durham University, DH1 3LE, United Kingdom e-mail: [email protected] 1 2 Anthony Conway plications to unlinking numbers, link concordance and discuss various incarnations of the Murasugi-Tristram inequality [78, 94]. The signature admits several 4-dimensional interpretations: either using covers of D4 branched along surfaces cobounding L [96], or applying twisted signatures, or as invariants of the zero framed surgery along L. Before discussing these con- structions in detail in Section3, let us briefly sketch one of them. Given a locally flat 4 4 compact connected oriented surface F ⊂ D with boundary L, we set WF := D nnF ∼ and consider the coefficient system p1(WF ) ! H1(WF ) = Z ! C which maps the meridian of F to w. This gives rise to a twisted intersection form lCw (WF ) on the twisted homology C-vector space H2(WF ;Cw ) whose signature coincides with the Levine-Tristram signature: sL(w) = signlCw (WF ): Section4 is concerned with methods of extracting sK(w) from pairings on the ±1 3 Alexander module H1(XK;Z[t ]) of a knot K (here we write XK := S n nK for the exterior of K)[76, 52]. Briefly, the signature sK can be extracted by considering ±1 the primary decomposition of H1(XK;R[t ]) and by studying the Milnor pairing or the Blanchfield pairing ±1 ±1 ±1 H1(XK;Z[t ]) × H1(XK;Z[t ]) ! Q(t)=Z[t ]: In fact, as we discuss in Section5, the signature can also be understood as a signed count of SU(2) representations of p1(XK) with fixed meridional traces [64, 45], or in terms of the Meyer cocycle and the Burau representation [34]. Summariz- ing, sL admits a wealth of definitions, which never seemed to have been collected in a single article. We conclude this introduction with two remarks. Firstly, note that we mention neither the Gordon-Litherland pairing [40] nor the multivariable signature [15]. Sec- ondly, we stress that even though sL was defined 50 years ago, it continues to be actively studied nowadays. We mention some recent examples: results involving concordance properties of positive knots can be found in [3]; the behavior of sL under splicing is now understood [23]; the relation between the jumps of sL and the zeroes of DL has been clarified [38, 63]; a diagrammatic interpretation of sL (in- spired by quantum topology) is conjectured in [87]; there is a characterization of the functions that arise as knot signatures [70]; new lower bounds on unknotting num- 1 bers have been obtained via sK [71]; there is a complete description of the w 2 S at which sL is a concordance invariant [80]; and sL is invariant under topological concordance [84]. This survey is organized is as follows. In Section2, we review the Seifert matrix definition of sL and list its properties. In Section3, we outline and compare the various four dimensional interpretations of sL. In Section4, we give an overview of the definitions using the Milnor and Blanchfield pairings. In Section5, we outline additional constructions in terms of SU(2) representations and braids. The Levine-Tristram signature: a survey 3 2 Definition and properties In this section, we review the definition of the Levine-Tristram and nullity using Seifert matrices (Subsection 2.1) before listing several properties of these invariants (Subsections 2.2, 2.3 and 2.4). Knot theory textbooks which mention the Levine- Tristram signature include [61, 49, 46, 67]. 2.1 The definition via Seifert surfaces A Seifert surface for an oriented link L is a compact oriented surface F whose ori- ented boundary is L. While a Seifert surface may be disconnected, we require that it has no closed components. Since F is oriented, it admits a regular neighborhood homeomorphic to F ×[−1;1] in which F is identified with F ×f0g. For e = ±1, the e 3 push off maps i : H1(F;Z) ! H1(S n F;Z) are defined by sending a (homology class of a) curve x to ie (x) := x × feg. The Seifert pairing of F is the bilinear form H1(F;Z) × H1(F;Z) ! Z (a;b) 7! `k(i−(a);b): A Seifert matrix for an oriented link L is any matrix representing the Seifert pair- ing. Although Seifert matrices do not provide link invariants, their so-called S- equivalence class does [61, Chapter 8]. Given a Seifert matrix A, observe that the matrix (1 − w)A + (1 − w)AT is Hermitian for all w lying in S1. Definition 1. Let L be an oriented link, let F be a Seifert surface for L with b0(F) components and let A be a matrix representing the Seifert pairing of F. Given w 2 S1, the Levine-Tristram signature and nullity of L at w are defined as T sL(w) := sign((1 − w)A + (1 − w)A ); T hL(w) := null((1 − w)A + (1 − w)A ) + b0(F) − 1: These signatures and nullities are well defined (i.e. they are independent of the choice of the Seifert surface) [61, Theorem 8.9] and, varying w along S1, pro- 1 duce functions sL;hL : S ! Z. The Levine-Tristram signature is sometimes called the w-signature (or the equivariant signature or the Tristram-Levine signature), while sL(−1) is referred to as the signature of L or as the Murasugi signature of L [78]. The definition of sL(w) goes back to Tristram [94] and Levine [57]. Remark 1. We note that sL and hL are piecewise constant: both observations follow from the fact that the Alexander polynomial DL(t) can be computed (up to its in- T determinacy) by the formula DL(t) = det(tA − A ). This latter fact also shows that, 1 given w 2 S n f1g, the nullity hL(w) vanishes if and only if DL(w) 6= 0. Moreover, tor the discontinuities of sL only occur at zeros of (t − 1)DL (t) [38, Theorem 2.1]. 4 Anthony Conway Several authors assume Seifert surfaces to be connected, and the nullity is then simply defined as the nullity of the matrix H(w) = (1 − w)A + (1 − w)AT . The extra flexibility afforded by disconnected Seifert surfaces can for instance be taken advantage of when studying the behavior of the signature and nullity of boundary links. Remark 2. Since the matrix (1 − w)A + (1 − w)AT vanishes at w = 1, we shall fre- 1 1 quently think of sL and hL as functions on S∗ := S n f1g. Note nevertheless that 1 for a knot K, the function sK vanishes in a neighborhood of 1 2 S [59, page 255], while for a m-component link, one can only conclude that the limits of jsL(w)j are at most m − 1 as w approaches 1. 2.2 Properties of the signature and nullity This subsection discusses the behaviour of the signature and nullity under operations such as orientation reversal, mirror image, connected sums and satellite operations. The following proposition collects several properties of the Levine-Tristram sig- nature. Proposition 1. Let L be a m-component oriented link and let w 2 S1. 1. The Levine-Tristram signature is symmetric: sL(w) = sL(w). 2. The integer sL(w) + hL(w) − m + 1 is even. m p 1 3. If DL(w) 6= 0, then sL(w) = m − sgn(i ∇L( w)) mod 4. 4. If mL denotes the mirror image of L, then smL(w) = −sL(w). 5. If rL is obtained by reversing the orientation of each component of L, then srL(w) = sL(w). 6. Let L0 and L00 be two oriented links. If L is obtained by performing a single con- 0 00 nected sum between a component of L and a component of L , then sL(w) = sL0 (w) + sL00 (w). 7. The signature is additive under the disjoint sum operation: if L is the link ob- 0 00 tained by taking the disjoint union of two oriented links L and L , then sL(w) = sL0 (w) + sL00 (w).
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