Torus Knots and Dunwoody Manifolds

We obtain an explicit representation as Dunwoody manifolds of all cyclic branched coverings of torus knots of type (p,mp±1), with p > 1 and m > 0.     

Every nontrivial knot in has nontrivial A-polynomial

We show that every nontrivial knot in the -sphere has a non-trivial -polynomial.

     

CALCULATION OF HAMILTONIAN CONSTRAINT AND MANDELSTAM IDENTITIES OVER EXTENDED KNOT FAMILIES{φi}2^2,{φi}3^4 and {φi}4^6

The actions of the Hamiltonian constraint onto the members of the extended knot families {φi}2^2, {φi}3^4 and {φi}4^6
, and the check of their invariance under the Mandelstam identities are given in the extended loop representation of loop quantum gravity.     

CALCULATION OF HAMILTONIAN CONSTRAINT AND MANDELSTAM IDENTITIES OVER EXTENDED KNOT FAMILIES {ψi}22, {ψi}43 and {ψi}64

The actions of the Hamiltonian constraint onto the members of the extended knot families {ψi}22, {ψi}43 and {ψi}64, and the check of their invariance under the Mandel-stare identities are given in the extended loop representation of loop quantum gravity.     

The Compressibility of Checkerboard Surfaces of Link Diagrams

Consider the checkerboard surfaces defined by some link diagrams. When they are not orientable, one considers the boundary surfaces of small regular neighborhoods of them. This article studies the compressibility problem of these kinds of surfaces in the link complements. The problem is solved by devising a normalization theory for the compressing discs, which brings up an algorithm to read out compressibility directly from the link diagrams. As an application of the algorithm, the compressibility changes under Reidermeister moves are studied. Diagrams from the knot tables are also studied, and surprisingly, some of them are shown to define completely compressible surfaces of this kind. Infinitely many examples of non-alternating knot diagrams with incompressible surfaces of this kind are also constructed.     

有亏格为1的Heegaard分解的三维流形中的环面纽结

将存在亏格为1的Heegaard分解T’1∪F T’2的三维流形记为M=L(p,q),其中p和q是互素整数,q/p为T’2的纬线在T’1上的斜率.若环面F上的简单闭曲线γ在M中非平凡,则称γ是M中的环面纽结.本文对在M中沿环面纽结作m/n-Dehn手术所得流形进行了分类,并给出了两个实心环体沿边界上平环作融合所得流形是L(p,q)中环面纽结补的特征描述.     

精梳棉结的系统控制研究

基于新疆某纺织企业的实际生产数据，分析了精梳条棉结与细纱棉结的相关关系，探讨了精梳工序棉结产生的原因，提出了控制精梳条棉结杂质，提高精梳纱质量的一些措施和方法．     

Monohedral Idemvalent Polyhedra that are Toroids

In this article we describe polyhedra of genus one that satisfy the following two properties: They have all faces congruent (monohedral) and they have the same number of faces at each vertex (idemvalent). In particular, the construction given yields polyhedra with six congruent triangles at each vertex. The construction given is a generalization of one given half a century ago by Alaoglu and Giese, and leads to many interesting toroidal polyhedra. We end by showing how the construction can yield a monohedral idemvalent toroidal polyhedron that is knotted.     

Knots and topologically transitive flows on 3-manifolds

Suppose that ? is a nonsingular (fixed point free) flow on a smooth three-dimensional manifold M. Suppose the orbit though a point p∈M is dense in M. Let D be an imbedded disk in M containing p which is transverse to the flow. Suppose that q∈D is a point in the forward orbit of p. Under certain assumptions on M, which include the case M=S³, we prove that if q is sufficiently close to p then the orbit segment from p to q together with a compact segment in D from p to q forms a nontrivial prime knot in M.     

Knotted Paths in Percolation

We study the topology of doubly-infinite paths in the bond percolation model on the three-dimensional cubic lattice. We propose a natural definition of a knotted doubly-infinite path. We prove the existence of a critical probability p _k satisfying p _c < p _k < 1 (where p _c is the usual percolation critical probability), such that for p _c < p < p _k , all doubly-infinite open paths are knotted, while for p > p _k there are unknotted doubly-infinite paths.