It was asked by J.Birman, Williams, and L.Rudolph whether nontrivial Lorentz knots have always positive signature. Lorentz knots are examples of positive braids (in our convention they have all crossings negative so they are negative links). It was shown by L.Rudolph that positive braids have positive signature (if they represent nontrivial links). K.Murasugi has shown that nontrivial, alternating, positive links have negative signature. Here we prove in general the old folklore conjecture that nontrivial positive links have negative signature.
A conjecture of Riley about the relationship between real parabolic representations and signatures of two-bridge knots is verified for double twist knots.
The signature function of a knot is an integer-valued step function on the unit circle in the complex plane. Necessary and sufficient conditions for a function to be the signature function of a knot are presented.
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 derive a linear estimate of the signature of positive knots, in terms of their genus. As an application, we show that every knot concordance class contains at most finitely many positive knots.
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.
To each unit complex number with positive imaginary part there is defined a Tristram-Levine knot signature function. The set of all such signature functions is linearly independent as a set of functions defined on the set of all knots. The set of averaged signature functions forms a linearly independent set of homomorophisms on the knot concordance group. However, for each unit root of an Alexander polynomial, there is a slice knot with nonvanishing signature at that root and its conjugate, and nowhere else. These results hold for knots in all odd dimension.
We give necessary and sufficient conditions for an integer to be the signature of a (4q-1)-knot in the (4q+1)-sphere with a given square-free Alexander polynomial.
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 define a set of "second-order" L^(2)-signature invariants for any algebraically slice knot. These obstruct a knot's being a slice knot and generalize Casson-Gordon invariants, which we consider to be "first-order signatures". As one application we prove: If K is a genus one slice knot then, on any genus one Seifert surface, there exists a homologically essential simple closed curve of self-linking zero, which has vanishing zero-th order signature and a vanishing first-order signature. This extends theorems of Cooper and Gilmer. We introduce a geometric notion, that of a derivative of a knot…
Fox conjectured the Alexander polynomial of an alternating knot is trapezoidal, i.e. the coefficients first increase, then stabilize and finally decrease in a symmetric way. Recently, Hirasawa and Murasugi further conjectured a relation between the number of the stable coefficients in the Alexander polynomial and the signature invariant. In this paper we prove the Hirasawa-Murasugi conjecture for two-bridge knots.
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.
The signature function of a knot is a locally constant integer valued function with domain the unit circle. The jumps (i.e., the discontinuities) of the signature function can occur only at the roots of the Alexander polynomial on the unit circle. The latter are important in deforming U(1) representations of knot groups to irreducible SU(2) representations. Under the assumption that these roots are simple, we formulate a conjecture that explicitly computes the jumps of the signature function in terms of the Jones polynomial of a knot and its parallels. As evidence, we prove our conjecture for …
We introduce Tristram-Levine signatures of virtual knots and use them to investigate virtual knot concordance. The signatures are defined first for almost classical knots, which are virtual knots admitting homologically trivial representations. The signatures and $ω$-signatures are shown to give bounds on the topological slice genus of almost classical knots, and they are applied to address a recent question of Dye, Kaestner, and Kauffman on the virtual slice genus of classical knots. A conjecture on the topological slice genus is formulated and confirmed for all classical knots with up to 11 …
We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link is nonpositive with the exception of the left-handed Hopf link (with possible trivial components). We also characterize those links which have signature zero and diagrams with two negative crossings. In particular, we show that if a nontrivial knot has a diagram with two negative…
We use a link invariant defined by Cimasoni-Florens to compute ρ-invariants. This generalizes results of Cochran-Teichner and Friedl on knots to the setting of links. As an application, we prove with only twelve possible exceptions that the twist knots of algebraic order two are linearly independent in the topological concordance group.
We explain new developments in classical knot theory in 3 and 4~dimensions, i.e. we study knots in 3-space, up to isotopy as well as up to concordance. In dimension~3 we give a geometric interpretation of the Kontsevich integral (joint with Jim Conant), and in dimension 4 we introduce new concordance invariants using von Neumann signatures (joint with Tim Cochran and Kent Orr). The common geometric feature of our results is the notion of a grope cobordism.
According to a formula by Gordon and Litherland, the signature of a knot K can be computed as the signature of a Goeritz matrix of K minus a suitable correction term, read off from the diagram of K. In this article, we consider the family of two bridge knots K(p/q) and compute the signature of their Goeritz matrices in terms of the coefficients of the continued fraction expansion of p/q. In many cases we also compute the value of the correction term. We show that for every two bridge knot K(p/q), there are "even continued fraction expansions" of p/q, for which the correction term vanishes, the…
New lower bounds on the unknotting number of a knot are constructed from the classical knot signature function. These bounds can be twice as strong as previously known signature bounds. They can also be stronger than known bounds arising from Heegaard Floer and Khovanov homology. Results include new bounds on the Gordian distance between knots and information about four-dimensional knot invariants. By considering a related non-balanced signature function, bounds on the unknotting number of slice knots are constructed; these are related to the property of double-sliceness.
This paper concerns twisted signature invariants of knots and 3-manifolds. In the fibered case, we reduce the computation of these invariants to the study of the intersection form and monodromy on the twisted homology of the fiber surface. Along the way, we use rings of power series to obtain new interpretations of the twisted Milnor pairing introduced by Kirk and Livingston. This allows us to relate these pairings to twisted Blanchfield pairings. Finally, we study the resulting signature invariants, all of which are twisted generalisations of the Levine-Tristram signature.
We construct many examples of non-slice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional topological knot concordance group. The bottom part of the filtration exhibits all classical concordance invariants, including the Casson-Gordon invariants. As a first step, we construct an infinite sequence of new obstructions that vanish on slice knots. These take values in the L-theory of skew fields associated to certain {\em universal} groups. Finally, we…
A knot in the 3-sphere is said to have zero negative unknotting number if it can be transformed into the unknot by performing only positive crossing changes. In this paper, we provide an obstruction for a knot to having zero negative unknotting number, and discuss its application to two classes of knots.
We show that a finite numerical boundary slope of an essential surface in the exterior of a Montesinos knot is bounded above and below in terms of the numbers of positive/negative crossings of a specific minimal diagram of the knot.
Ascending numbers are determined for 64 knots with at most n=10 crossings. After proving the theorem about the signature of alternating knot families, we distinguished all families of knots obtained from generating alternating knots with at most 10 crossings, for which the unknotting number can be confirmed by using the general formulae for signatures. For 11 families of knots general formulae are obtained for their ascending numbers.
We use twisted Alexander polynomials to show that certain algebraically slice 2-bridge knots are not topologically slice, even though all prime power Casson-Gordon signatures vanish. We also provide some computations indicating the efficacy of Casson-Gordon signatures in obstructing the smooth sliceness of 2-bridge knots.
Band surgery is an operation relating pairs of knots or links in the three-sphere. We prove that if two quasi-alternating knots $K$ and $K'$ of the same square-free determinant are related by a band surgery, then the absolute value of the difference in their signatures is either 0 or 8. This obstruction follows from a more general theorem about the difference in the Heegaard Floer $d$-invariants for pairs of L-spaces that are related by distance one Dehn fillings and satisfy a certain condition in first homology. These results imply that $T(2, 5)$ is the only torus knot $T(2, m)$ with $m$ squa…
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;\…
In this paper, we consider generalizations of the Alexander polynomial and signature of 2-bridge knots by considering the Gordon-Litherland bilinear forms associated to essential state surfaces of the 2-bridge knots. We show that the resulting invariants are well-defined and explore properties of these invariants. Finally we realize the boundary slopes of the essential surfaces as a difference of signatures of the knot.
In this note we prove that for any finite quandle $X$ and any 2-cocycle $ϕ\in Z^2_{Q\pm}(X; \mathds{Z})$, the cocycle invariant $Φ^{\pm}_ϕ(K)$ is trivial for all knots $K$.
The doubly slice genus of a knot in the 3-sphere is the minimal genus among unknotted orientable surfaces in the 4-sphere for which the knot arises as a cross-section. We use the classical signature function of the knot to give a new lower bound for the doubly slice genus. We combine this with an upper bound due to C. McDonald to prove that for every nonnegative integer $N$ there is a knot where the difference between the slice and doubly slice genus is exactly $N$, refining a result of W. Chen which says this difference can be arbitrarily large.
It is known that each of the successive quotient groups of the grope and solvable filtrations of the knot concordance group has an infinite rank subgroup. The generating knots of these subgroups are constructed using iterated doubling operators. In this paper, for each of the successive quotients of the filtrations we give a new infinite rank subgroup which trivially intersects the previously known infinite rank subgroups. Instead of iterated doubling operators, the generating knots of these new subgroups are constructed using the notion of algebraic $n$-solutions, which was introduced by Coch…
In this paper we compute the signature for a family of knots $W(k,n)$, the weaving knots of type $(k,n)$. By work of E.~S.~Lee the signature calculation implies a vanishing theorem for the Khovanov homology of weaving knots. Specializing to knots $W(3,n)$, we develop recursion relations that enable us to compute the Jones polynomial of $W(3,n)$. Using additional results of Lee, we compute the ranks of the Khovanov Homology of these knots. At the end we provide evidence for our conjecture that, asymptotically, the ranks of Khovanov Homology of $W(3,n)$ are {\it normally distributed}.
We define a notion of complexity for shake-slice knots which is analogous to the definition of complexity for h-cobordisms studied by Morgan-Szabó. We prove that for each framing $n \ne 0$ and complexity $c \ge 0$, there is an $n$-shake-slice knot with complexity at least $c$. Our construction makes use of dualizable patterns, and we include a crash course in their properties. We bound complexity by studying the behavior of the classical knot signature and the Levine-Tristram signature of a knot under the operation of twisting algebraically-one strands.
We show that bilinear cup products with local coefficients of closed 3-manifolds recover some twisted pairings of infinite covers and the Casson-Gordon local signatures. As a result, we further give diagrammatic computations of the pairings and signatures.
We find a formula for the L2 signature of a (p,q) torus knot, which is the integral of the omega-signatures over the unit circle. We then apply this to a theorem of Cochran-Orr-Teichner to prove that the n-twisted doubles of the unknot, for n not 0 or 2, are not slice. This is a new proof of the result first proved by Casson and Gordon.
Let $K\subset S^3$ be a Fox $p$-colored knot and assume $K$ bounds a locally flat surface $S\subset B^4$ over which the given $p$-coloring extends. This coloring of $S$ induces a dihedral branched cover $X\to S^4$. Its branching set is a closed surface embedded in $S^4$ locally flatly away from one singularity whose link is $K$. When $S$ is homotopy ribbon and $X$ a definite four-manifold, a condition relating the signature of $X$ and the Murasugi signature of $K$ guarantees that $S$ in fact realizes the four-genus of $K$. We exhibit an infinite family of knots $K_m$ with this property, each w…
For $\ell >1$, we develop $L^{(2)}$-signature obstructions for $(4\ell-3)$-dimensional knots with metabelian knot groups to be doubly slice. For each $\ell>1$, we construct an infinite family of knots on which our obstructions are non-zero, but for which double sliceness is not obstructed by any previously known invariant.
An oriented compact 4-manifold $V$ with boundary $S^3$ is called a positon (resp. negaton) if its intersection form is positive definite (resp. negative definite) and it is simply connected. In this paper, we prove that there exist infinitely many knots which cannot bound null-homologous disks either in positons or in negatons. As a consequence, we find knots that cannot be unknotted either by only positive crossing changes or by only negative crossing changes.
We show that a positive braid knot has maximal topological 4-genus exactly if it has maximal signature invariant. As an application, we determine all positive braid knots with maximal topological 4-genus and compute the topological 4-genus for all positive braid knots with up to 12 crossings.
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.
Recently, Mullins calculated the Casson-Walker invariant of the 2-fold cyclic branched cover of an oriented link in S^3 in terms of its Jones polynomial and its signature, under the assumption that the 2-fold branched cover is a rational homology 3-sphere. Using elementary principles, we provide a similar calculation for the general case. In addition, we calculate the LMO invariant of the p-fold branched cover of twisted knots in S^3 in terms of the Kontsevich integral of the knot.
Given a knot K in the three-sphere, we address the question: which Dehn surgeries on K bound negative-definite four-manifolds? We show that the answer depends on a number m(K), which is a smooth concordance invariant. We study the properties of this invariant, and compute it for torus knots.
Using the Gordon-Litherland pairing, one can define invariants (signature, nullity, determinant) for ${\mathbb Z}/2$ null-homologous links in thickened surfaces. In this paper, we study the concordance properties of these invariants. For example, if $K \subset Σ\times I$ is ${\mathbb Z}/2$ null-homologous and slice, we show that its signatures vanish and its determinants are perfect squares. These statements are derived from a cobordism result for closed unoriented surfaces in certain 4-manifolds. The Brown invariants are defined for ${\mathbb Z}/2$ null-homologous links in thickened surfaces.…
