Narrow Results

We study the existence of relations between the degrees of the knot polynomials and some classical knot invariants, partially confirming and extending the question of Morton on the skein polynomial and a recent question of Ferrand.

Kawakubo and Uchida showed that, if a closed oriented 4k-dimensional manifold M admits a semi-free circle action such that the dimension of the fixed point set is less than 2k, then the signature of M vanishes. In this note, by using G-signature theorem and the rigidity of the signature operator, we generalize this result to more general circle actions. Combining the same idea with the remarkable Witten–Taubes–Bott rigidity theorem, we explore more vanishing results on spin manifolds admitting such circle actions. Our results are closely related to some earlier results of Conner–Floyd, Landweber–Stong and Hirzebruch–Slodowy.

We provide linear lower bounds for the signature of positive braids in terms of the three-genus of their braid closure. This yields linear bounds for the topological slice genus of knots that arise as closures of positive braids.

Kodaira fibrations are surfaces of general type with a non-isotrivial fibration, which are differentiable fiber bundles. They are known to have positive signature divisible by $4$. Examples are known only with signature 16 and more. We review approaches to construct examples of low signature which admit two independent fibrations. Special attention is paid to ramified covers of product of curves which we analyze by studying the monodromy action for bundles of punctured curves. As a by-product, we obtain a classification of all fix-point-free automorphisms on curves of genus at most $9$.

We extend the Gordon–Litherland pairing to links in thickened surfaces, and use it to define signature, determinant and nullity invariants for links that bound (unoriented) spanning surfaces. The invariants are seen to depend only on the $S^{\ast }$-equivalence class of the spanning surface. We prove a duality result relating the invariants from one $S^{\ast }$-equivalence class of spanning surfaces to the restricted invariants of the other. Using Kuperberg’s theorem, these invariants give rise to well-defined invariants of checkerboard colorable virtual links. The determinants can be applied to determine the minimal support genus of a checkerboard colorable virtual link. The duality result leads to a simple algorithm for computing the invariants from the Tait graph associated to a checkerboard coloring. We show these invariants simultaneously generalize the combinatorial invariants defined by Im, Lee and Lee, and those defined by Boden, Chrisman and Gaudreau for almost classical links. We examine the behavior of the invariants under orientation reversal, mirror symmetry and crossing change. We give a 4-dimensional interpretation of the Gordon–Litherland pairing by relating it to the intersection form on the relative homology of certain double branched covers. This correspondence is made explicit through the use of virtual linking matrices associated to (virtual) spanning surfaces and their associated (virtual) Kirby diagrams.

In this paper we consider the generalized Smith conjecture of codimension greater than two, which says that no periodic tranformation of S^l can have the tame knotted S^h as fixed point set if l-h>2 and h>3. Using the Brieskorn spheres, this paper gives the explicit counterexamples to show that the conjecture is false in the DIFF category for the following cases: (i) l-h is even more than 2 and l is odd; (ii) 2l≤3(h+1) and h+1≡0 (mod 4).

In this paper, we relate the nullification writhe and the remaining writhe defined by C. Cerf to other link invariants. We prove that the nullification writhe of an oriented reduced alternating link diagram is equal, up to sign, to the signature of the link. Moreover, we relate the difference between the nullication writhe and the remaining writhe to invariants obtained from chessboard-coloured link diagrams such as their numbers of shaded and unshaded regions.

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 (possibly, with extra 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 crossings then the signature of the knot is negative, unless the knot is a twist knot with negative clasp. We completely determine all trivial link diagrams with two or fewer negative crossings. For a knot diagram with three negative crossings, the signature of the knot is nonpositive except for the left-handed trefoil knot. These results generalize those of Rudolph, Cochran, Gompf, Traczyk and Przytycki, solve [27, Conjecture 5], and give a partial answer to [3, Problem 2.8] about knots dominating the trefoil knot or the trivial knot. We also describe all unknotting number one positive knots.

In this paper, we present the Goeritz matrix for checkerboard colorable virtual links or, equivalently, checkerboard colorable links in thickened surfaces S_g × [0, 1], which is an extension of the Goeritz matrix for classical knots and links in ℝ³. Using this, we show that the signature, nullity and determinant of classical oriented knots and links extend to those of checkerboard colorable oriented virtual links.

An H(2)-move is an unknotting operation of a knot, which is performed by adding a half-twisted band. We define the H(2)-Gordian distance of two knots to be the minimum number of H(2)-moves needed to transform one into the other. We give several methods to estimate the H(2)-Gordian distance of knots. Then we give a table of H(2)-Gordian distances of knots with up to 7 crossings.

We show for an alternating knot the minimal boundary slope of an essential spanning surface is given by the signature plus twice the minimum degree of the Jones polynomial and the maximal boundary slope of an essential spanning surface is given by the signature plus twice the maximum degree of the Jones polynomial. For alternating Montesinos knots, these are the minimal and maximal boundary slopes.

An oriented 2-component link is called band-trivializable, if it can be unknotted by a single band surgery. We consider whether a given 2-component link is band-trivializable or not. Then we can completely determine the band-trivializability for the prime links with up to 9 crossings. We use the signature, the Jones and Q polynomials, and the Arf invariant. Since a band-trivializable link has 4-ball genus zero, we also give a table for the 4-ball genus of the prime links with up to 9 crossings. Furthermore, we give an additional answer to the problem of whether a (2n + 1)-crossing 2-bridge knot is related to a (2, 2n) torus link or not by a band surgery for n = 3, 4, which comes from the study of a DNA site-specific recombination.

A conjecture of Riley about the relationship between real parabolic representations and signatures of two-bridge knots is verified for double twist knots.

In this paper, we consider generalizations of the Alexander polynomial and signature of 2-bridge knots by considering the Gordon–Litherland bilinear forms associated with 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 essential surfaces as differences of signatures of the knot.

For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an ascending diagram. We study the ascending number of a knot by analyzing the Conway polynomial. In this paper, we give a sharper lower bound of the ascending number of a knot and newly determine the ascending number for 26 prime knots up to 10 crossings.

In this paper, we introduce the Goeritz matrix for a pseudo-link whose entries lie in the Laurent polynomial ring $\mathbb{Z}[u^{-1},u]$, which generalizes the Goeritz matrix for a classical link. We show that the determinant of a modified Goeritz matrix gives a Laurent polynomial invariant for pseudo-links in one variable u with integer coefficients. We also introduce the notions of the signature, determinant, and nullity of pseudo-links. Further, we discuss some properties of the invariants and compute the polynomials and those numerical invariants for several pseudo-knot families.

We compute the smooth 4-genera of the prime knots with 12 crossings whose values, as reported on the KnotInfo website, were unknown. This completes the calculation of the smooth 4-genus for all prime knots with 12 or fewer crossings.

In this paper, we examine the properties of the Jones polynomial using dimensionality reduction learning techniques combined with ideas from topological data analysis. Our data set consists of more than 10 million knots up to 17 crossings and two other special families up to 2001 crossings. We introduce and describe a method for using filtrations to analyze infinite data sets where representative sampling is impossible or impractical, an essential requirement for working with knots and the data from knot invariants. In particular, this method provides a new approach for analyzing knot invariants using Principal Component Analysis. Using this approach on the Jones polynomial data, we find that it can be viewed as an approximately three-dimensional subspace, that this description is surprisingly stable with respect to the filtration by the crossing number, and that the results suggest further structures to be examined and understood.

In this paper, we study conditions for $2$-component pretzel links to be concordant to the trivial link.

We define the 2-signatures, 2-nullities and Arf invariants (when possible) for links which are null-homologous modulo two in a rational homology three-sphere. We define these invariants using the Goeritz form on non-oriented spanning surfaces. We develop their cobordism properties from this point of view. We give a good way to index these invariants. We also define d-signatures and d-nullities for links which are null-homologous modulo d in a rational homology sphere from the point of view of branched covers. We index d-signatures and d-nullities and develop their cobordism properties. Finally we define Arf invariants (when possible) in a general closed 3-manifold using spin structures.