<Emphasis Type="Italic">S</Emphasis>-graphs and braid index of links

We study the relationship between reduction operations on link diagrams and S-graphs associated with them. We are motivated by the problem of computing the braid index of a link and some well known conjectures concerning the braid index of a link and the writhe of its diagrams. Possible counterexamples are discussed in terms of both S-graphs and link diagrams. We also indicate the relation of S-graphs to singular links regarded up to an appropriate equivalence relation.     

Least squares surface approximation to scattered data using multiquadratic functions

The paper documents an investigation into some methods for fitting surfaces to scattered data. The form of the fitting function is a multiquadratic function with the criteria for the fit being the least mean squared residual for the data points. The principal problem is the selection of knot points (or base points for the multiquadratic basis functions), although the selection of the multiquadric parameter also plays a nontrivial role in the process. We first describe a greedy algorithm for knot selection, and this procedure is used as an initial step in what follows. The minimization including knot locations and the multiquadric parameter is explored, with some unexpected results in terms of “near repeated” knots. This phenomenon is explored, and leads us to consider variable parameter values for the basis functions. Examples and results are given throughout.     

Knot theory for self-indexed graphs

We introduce and study so-called self-indexed graphs. These are (oriented) finite graphs endowed with a map from the set of edges to the set of vertices. Such graphs naturally arise from classical knot and link diagrams. In fact, the graphs resulting from link diagrams have an additional structure, an integral flow. We call a self-indexed graph with integral flow a comte. The analogy with links allows us to define transformations of comtes generalizing the Reidemeister moves on link diagrams. We show that many invariants of links can be generalized to comtes, most notably the linking number, the Alexander polynomials, the link group, etc. We also discuss finite type invariants and quandle cocycle invariants of comtes.

     

Knot Placement for B-Spline Curve Approximation via $L_{∞, 1}$-Norm and Differential Evolution Algorithm

In this paper, we consider the knot placement problem in B-spline curve approximation. A novel two-stage framework is proposed for addressing this problem. In the first step, the $l_{\infty, 1}$-norm model is introduced for the sparse selection of candidate knots from an initial knot vector. By this step, the knot number is determined. In the second step, knot positions are formulated into a nonlinear optimization problem and optimized by a global optimization algorithm — the differential evolution algorithm (DE). The candidate knots selected in the first step are served for initial values of the DE algorithm. Since the candidate knots provide a good guess of knot positions, the DE algorithm can quickly converge. One advantage of the proposed algorithm is that the knot number and knot positions are determined automatically. Compared with the current existing algorithms, the proposed algorithm finds approximations with smaller fitting error when the knot number is fixed in advance. Furthermore, the proposed algorithm is robust to noisy data and can handle with few data points. We illustrate with some examples and applications.     

Boundaries of Flat Compact Surfaces in 3-Space

This paper deals with the problem `which knots or links in3-space bound flat (immersed) compact surfaces?' In aforthcoming paper by the author, it is proven that any simple closedspace curve can be deformed until it bounds a flat orientable compact(Seifert) surface. The main results of this paper are that there existknots that do not bound any flat compact surfaces. The lower bound oftotal curvature of a knot bounding an orientable nonnegatively curvedcompact surface can, for varying knot types, be arbitrarily much greaterthan the infimum of curvature needed for the knot to have its knot type.The number of 3-singular points (points of zero curvatureor if not then of zero torsion) on the boundary of a flat immersedcompact surface is greater than or equal to twice the absolute value ofthe Euler characteristic of the surface. A set of necessary and, in aweakened sense, sufficient conditions for a knot or link to be what wecall a generic boundary of a flat immersed compact surface withoutplanar regions is given.     

Topological realizations of ortho-projection graphs

We discuss the production of ortho-projection graphs from alternating knot diagrams, and introduce a more general construction of such graphs from “splittings” of closed, non-orientable surfaces. As our main result, we prove that this new topological construction generates all ortho-projection graphs. We present a minimal example of an ortho-projection graph that does not arise from a knot diagram, and provide a surface-splitting that realizes this graph.     

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 region crossing change and incidence matrix

In a recent work of Ayaka Shimizu, she studied an operation named region crossing change on link diagrams, which was proposed by Kishimoto, and showed that a region crossing change is an unknotting operation for knot diagrams. In this paper, we prove that the region crossing change on a 2-component link diagram is an unknotting operation if and only if the linking number of the diagram is even. Besides, we define an incidence matrix of a link diagram via its signed planar graph and its dual graph. By studying the relation between region crossing change and incidence matrix, we prove that a signed planar graph represents an n-component link diagram if and only if the rank of the associated incidence matrix equals c n + 1, where c denotes the size of the graph.     

On three-fold irregular branched coverings over closed three-braid and three-bridge knots

It is well known that every closed orientable three-manifold is given as a three-fold branched covering space branched over some knot. Then it is an interesting problem that for a given knot family what kind of manifold can be got as a three-fold irregular branched covering space. K. Murasugi showed that for a closed three-braid the manifold is a lens space of type (n,1). In this paper, we will give an another proof and an algorithm to determine n for a given knot. And for a three bridge knot, we will show that its covering space is a lens space of type (p,q), and give an algorithm to determine the pair of p, q.     

NUAT T-splines of odd bi-degree and local refinement

This paper presents a new kind of spline surfaces, named non-uniform algebraic- trigonometric T-spline surfaces （NUAT T-splines for short） of odd hi-degree. The NUAT T- spline surfaces are defined by applying the T-spline framework to the non-uniform algebraic- trigonometric B-spline surfaces （NUAT B-spline surfaces）. Based on the knot insertion algorithm of the NUAT B-splines, a local refinement algorithm for the NUAT T-splines is given. This algorithm guarantees that the resulting control grid is a T-mesh as the original one. Finally, we prove that, for any NUAT T-spline of odd hi-degree, the linear independence of its blending functions can be determined by computing the rank of the NUAT T-spline-to-NUAT B-spline transformation matrix.     

Closed incompressible surfaces in the complements of positive knots

We show that any closed incompressible surface in the complement of a positive knot is algebraically non-split from the knot, positive knots cannot bound non-free incompressible Seifert surfaces and that the splittability and the primeness of positive knots and links can be seen from their positive diagrams. Received: June 28, 2000     

A Celtic Framework for Knots and Links

We describe a variant of a method used by modern graphic artists to design what are traditionally called Celtic knots, which are part of a larger family of designs called “mirror curves.” It is easily proved that every such design specifies an alternating projection of a link. We use medial graphs and graph minors to prove, conversely, that every alternating projection of a link is topologically equivalent to some Celtic link, specifiable by this method. We view Celtic representations of knots as a framework for organizing the study of knots, rather like knot mosaics or braid representations. The formalism of Celtic design suggests some new geometric invariants of links and some new recursively specifiable sequences of links. It also leads us to explore new variations of problems regarding such sequences, including calculating formulae for infinite sequences of knot polynomials. This involves a confluence of ideas from knot theory, topological graph theory, and the theory of orthogonal graph drawings.     

Virtual knots undetected by 1- and 2-strand bracket polynomials

Kishino's knot is not detected by the fundamental group or the bracket polynomial. However, we can show that Kishino's knot is not equivalent to the unknot by applying either the 3-strand bracket polynomial or the surface bracket polynomial. In this paper, we construct two non-trivial virtual knot diagrams, KD and Km, that are not detected by the 1-strand or the 2-strand bracket polynomial. From these diagrams, we construct two infinite families of non-classical virtual knot diagrams that are not detected by the bracket polynomial. Additionally, these virtual knot diagrams are trivial as flats.     

Construction of covering maps between orientable surfaces

Consider the following problem. Can a set of numbers be realized as boundary covering indices of a covering map between surfaces? How to realize them? A set of equivalent criteria for this problem are found, which can be checked by a finite algorithm. Furthermore, the algorithm will also construct a solution if such one exists. As an application, a well_known group of necessary conditions are shown to be not sufficient in infinitely many cases, while in most cases, numbers satisfying them are realizable.     

抽象经济均衡问题解的存在性及其算法

本文首先研究一类新的向量均衡问题,利用截口定理与KKM定理两种不同的工具证明此类均衡问题解的存在性,接着,把这类向量均衡问题推广到更为一般的情形,随后讨论了具有上下界的均衡问题,它是由Isac,Sehgal和Singh于1999年提出的一个公开问题,本文在一定条件下获得了一个新的解的存在性定理,并构造了一个迭代算法,讨论了算法的收敛性。     

Knots and the Bracket Calculus

In this paper, we introduce a way of encoding links (long links). This ways leads to a combinatorial representation of links by words in a given finite alphabet. We prove that the link semigroup is isomorphic to some algebraically defined semigroup with a simple system of relations. Thus, knot theory is represented as a bracket calculus: the link recognition problem is reduced to a recognition problem in this semigroup.     

Crosscap numbers of pretzel knots

The crosscap number of a knot in the 3-sphere is defined as the minimal first Betti number of non-orientable surfaces bounded by the knot. In this paper, we determine the crosscap numbers of a large class of pretzel knots. The key ingredient to obtain the result is the algorithm of enumerating all essential surfaces for Montesinos knots developed by Hatcher and Oertel.     

Excluded Minors and the Ribbon Graphs of Knots

In this article we consider minors of ribbon graphs (or, equivalently, cellularly embedded graphs). The theory of minors of ribbon graphs differs from that of graphs in that contracting loops is necessary and doing this can create additional vertices and components. Thus, the ribbon graph minor relation is incompatible with the graph minor relation. We discuss excluded minor characterizations of minor closed families of ribbon graphs. Our main result is an excluded minor characterization of the family of ribbon graphs that represent knot and link diagrams.