The Canonical Genus of a Classical and Virtual Knot

A diagram D of a knot defines the corresponding Gauss Diagram G _D . However, not all Gauss diagrams correspond to the ordinary knot diagrams. From a Gauss diagram G we construct closed surfaces F _G and S _G in two different ways, and we show that if the Gauss diagram corresponds to an ordinary knot diagram D, then their genus is the genus of the canonical Seifert surface associated to D. Using these constructions we introduce the virtual canonical genus invariant of a virtual knot and find estimates on the number of alternating knots of given genus and given crossing number.     

Generalizations of the Wada representations and virtual link groups

We extend the Wada representations of the classical braid group to the virtual and welded braid groups. Using the resulting representations, we construct the groups of virtual links and prove that they are link invariants. We give some examples of calculating the groups of torus (virtual) links.     

On Invariants of Virtual Links

We propose a new method of generalizing classical link invariants for the case of virtual links. In particular, we have generalized the knot quandle, the knot fundamental group, the Alexander module, and the coloring invariants. The virtual Alexander module leads to a definition of VA-polynomial that has no analogue in the classical case (i.e. vanishes on classical links).     

The polynomial invariants of twisted links

A twisted link is a generalization of a virtual link, which is related to a link diagram on a closed, possibly non-orientable surface. In this paper we generalize the Miyazawa polynomial invariant of a virtual link to an invariant of a twisted link in two formulae one of which is introduced by A. Ishii and the other by the author.     

SOME APPLICATIONS OF PLANAR GRAPH IN KNOT THEORY

The relationship between a link diagram and its corresponding planar graph is briefly reviewed.A necessary and sufficient condition is given to detect when a planar graph corresponds to a knot.The rela...     

Adaptive Surface Reconstruction Based on Tensor Product Algebraic Splines

Surface reconstruction from unorganized data points is a challenging problem in Computer Aided Design and Geometric Modeling. In this paper, we extend the mathematical model proposed by Juttler and Felis (Adv. Comput. Math., 17 (2002), pp. 135-152) based on tensor product algebraic spline surfaces from fixed meshes to adaptive meshes. We start with a tensor product algebraic B-spline surface defined on an initial mesh to fit the given data based on an optimization approach. By measuring the fitting errors over each cell of the mesh, we recursively insert new knots in cells over which the errors are larger than some given threshold, and construct a new algebraic spline surface to better fit the given data locally. The algorithm terminates when the error over each cell is less than the threshold. We provide some examples to demonstrate our algorithm and compare it with Jiittler's method. Examples suggest that our method is effective and is able to produce reconstruction surfaces of high quality.AMS subject classifications: 65D17     

The Khovanov complex for virtual links

One of the most outstanding achievements of modern knot theory is Khovanov’s categorification of Jones polynomials. In the present paper, we construct the homology theory for virtual knots. An important obstruction to this theory (unlike the case of classical knots) is the nonorientability of “atoms”; an atom is a two-dimensional combinatorial object closely related with virtual link diagrams. The problem is solved directly for the field ℤ₂ and also by using some geometrical constructions applied to atoms. We discuss a generalization proposed by Khovanov; he modifies the initial homology theory by using the Frobenius extension. We construct analogs of these theories for virtual knots, both algebraically and geometrically (by using atoms). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 127–152, 2005.     

On equivariant slice knots

We suggest a method to detect that two periodic knots are not equivariantly concordant, using surgery on factor links. We construct examples which satisfy all known necessary conditions for equivariant slice knots- Naik's and Choi-Ko-Song's improvements of classical results on Seifert forms and Casson-Gordon invariants of slice knots - but are not equivariantly slice.

     

The 3-connected matroids with circumference 6

We construct all 3-connected matroids with circumference equal to 6 having rank at least 8. A matroid belongs to this family if and only if it is a generalized parallel connection of a set of planes along a common line (which may have some virtual points).     

The Miyazawa polynomial for long virtual knots

We introduce a two-variable polynomial invariant of a long virtual knot, which dominates the Kauffman f-polynomial and the Miyazawa polynomial of the closure. Our invariant satisfies a product formula for the concatenation product of long virtual knots. It describes a formula of the Miyazawa polynomial of a ‘connected sum’ of two virtual knots. It also gives lower bounds for the real crossing number and the virtual crossing number of a long virtual knot.     

Legendrian Knots in Overtwisted Contact Structures on S3

We study the problem of classifying Legendrian knots in overtwisted contact structures on S ³. The question is whether topologically isotopic Legendrian knots have to be Legendrian isotopic if they have equal values of the well-known invariants rot and tb. We give positive answer in the case that there is an overtwisted disc intersecting none of the knots and we construct an example of a knot intersecting each overtwisted disc (this provides a counterexample to the conjecture of Eliashberg). Our proof needs some results on the structure of the group of contactomorphisms of S ³. We divide the subgroup Cont₊(S ³, ) of coorientation-preserving contactomorphisms for an overtwisted contact distribution into two classes.     

The intrinsic normal cone

Let be an algebraic stack in the sense of Deligne-Mumford. We construct a purely -dimensional algebraic stack over (in the sense of Artin), the intrinsic normal cone . The notion of (perfect) obstruction theory for is introduced, and it is shown how to construct, given a perfect obstruction theory for , a pure-dimensional virtual fundamental class in the Chow group of . We then prove some properties of such classes, both in the absolute and in the relative context. Via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory, and sometimes a perfect one. Oblatum 26-II-1996 & 27-VI-1996     

Virtual neighborhood technique for moduli spaces of holomorphic curves

We use the technique of Ruan (1999) and Li and Ruan (2001) to construct the virtual neighborhoods and show that the Gromov-Witten invariants can be defined as integrals over the top strata of the virtual neighborhoods. We prove that the invariants defined in this way satisfy all the axioms of Gromov-Witten invariants summarized by Kontsevich and Manin (1994).

     

Braids, transversal links and the Khovanov-Rozansky Theory

We establish some inequalities for the Khovanov-Rozansky cohomologies of braids. These give new upper bounds of the self-linking numbers of transversal links in standard contact which are sharper than the well-known bound given by the HOMFLY polynomial. We also introduce a sequence of transversal link invariants and discuss some of their properties.

     

The rack space

The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space and the classifying bundle is the first James bundle.

We investigate the algebraic topology of this classifying space and report on calculations given elsewhere. Apart from defining many new knot and link invariants (including generalised James-Hopf invariants), the classification theorem has some unexpected applications. We give a combinatorial interpretation for of a complex which can be used for calculations and some new interpretations of the higher homotopy groups of the 3-sphere. We also give a cobordism classification of virtual links.

     

Virtual Elements for the Reissner‐Mindlin plate problem

In this work, we present a virtual element method for the approximation of the plate bending problem in the Reissner‐Mindlin formulation. The proposed method follows the MITC approach of the FEM context. We construct a family of VEM spaces with arbitrary degree of accuracy that satisfies the conditions of the MITC philosophy. We perform some numerical tests which allow us to assess the convergence and the robustness of the method.     

Upper bound for the alternation number of a torus knot

We give an upper bound for the alternation number of a torus knot which is of either 3-, 4-, or 5-braid or of other special types. Using the inequality relating the alternation number, signature, and Rasmussen s-invariant, discovered by Abe, we determine the alternation numbers of the torus knots T(3,l), , and T(4,5). Also, for any positive integer k we construct infinitely many 3-braid knots with alternation number k.     

Seifert-fibered surgeries which do not arise from primitive/Seifert-fibered constructions

We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean's primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert-fibered surgeries arise from Dean's primitive/Seifert-fibered construction. The -pretzel knot belongs to both of the infinite families.

     

缠绕方程和DNA模型

一般化了DNA重组的缠绕模型的缠绕方程的求解方法, 同时利用有理缠绕和二桥纽结 的关系给出了某些缠绕方程的解.     

Hyperbolic knots spanning accidental Seifert surfaces of arbitrarily high genus

A method for constructing hyperbolic knots each of which bounds accidental incompressible Seifert surfaces of arbitrarily high genus is given. Mathematics Subject Classification (2000):57N10, 57M25.The author was supported in part by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.