Rational-Slice Knots via Strongly Negative-Amphicheiral Knots

We show that certain satellite knots of every strongly negative-amphicheiral rational knot are rational-slice knots. This proof also shows that the 0-surgery manifold of a certain strongly negative amphicheiral knot such as the figure-eight knot bounds a compact oriented smooth 4-manifold homotopy equivalent to the 2-sphere such that a second homology class of the 4-manifold is represented by a smoothly embedded 2-sphere if and only if the modulo two reduction of it is zero.     

Rational-slice Knots via Strongly Negative-amphicheiral Knots

We show that certain satellite knots of every strongly negative-amphicheiral rational knot are rational-slice knots. This proof also shows that the O-surgery manifold of a certain strongly negative amphicheiral knot such as the figure-eight knot bounds a compact oriented smooth 4-manifold homotopy equivalent to the 2-sphere such that a second homology class of the 4-manifold is represented by a smoothly embedded 2-sphere if and only if the modulo two reduction of it is zero.     

On Knots

Book Reviews

On KnotsLouis H. Kauffman: (Annals of Mathematics Studies, Vol. 115), Princeton University Press, Princeton, New Jersey, 1987, xv + 480pp     

纽结与多项式

In this paper, we deal with some corresponding relations between knots and polynomials by using the basic properties of knot polynomials （such as, some special values of knot polynomials, the Arf invariant and derivative of knot polynomials）. We give necessary and sufficient conditions that a Laurent polynomial with integer coefficients, whose breadth is less than five, is the Jones polynomial of a certain knot.     

Strongly n-Trivial Knots

The authors prove in this paper that, given any knot k of genusg,k fails to be strongly n-trivial for all n with n 6g –3. 2000 Mathematics Subject Classification 57M99.     

Knots in Riemannian manifolds

In this paper we study submanifolds with nonpositive extrinsic curvature in a positively curved manifold. Among other things we prove that, if ${K\subset (S^n, g)}$ is a totally geodesic submanifold of codimension 2 in a Riemannian sphere with positive sectional curvature where n ≥ 5, then K is homeomorphic to S ^n–2 and the fundamental group of the knot complement ${\pi _1(S^n-K)\cong \mathbb{Z}}$ .     

Theory of Thickened Knots

Mathematical Notes -     

纽结的n-Gordian复形

定义了纽结的n-Gordian复形,这是纽结Gordian复形的一种推广,并证明了当n为正偶数时,对任意纽结K和任意正整数r,存在一个包含K且共含r个纽结的集合,使得此集合中任意两个纽结的n-Gordian距离为1.     

Trefoil Knots without Tritangent Planes

A quick proof is given that trefoil knots constructed explicitlyas (3,2) curves on a torus by stereographic projection fromS³ have no tritangent planes.     

Torus Knots and Mirror Symmetry

We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the full ${{\rm Sl}(2, \mathbb {Z})}$ symmetry of the spectral curve of the resolved conifold, and should be regarded as the mirror of the topological D-brane associated with torus knots in the large N Gopakumar–Vafa duality. Moreover, we derive the curve as the large N limit of the matrix model computing torus knot invariants.     

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.     

Three-Page Embeddings of Singular Knots

A semigroup with 15 generators and 84 relations is constructed. The center of the semigroup is in a one-to-one correspondence with the set of all isotopy classes of nonoriented singular knots (links with finitely many double intersections in general position) in ³.     

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.     

Virtual Knots and Infinite-Dimensional Lie Algebras

In the present work, we construct an invariant of virtual knots valued in (infinite-dimensional) Lie Algebras and establish some properties of it. This leads to some heuristic ideas how to construct quandles and extract (virtual) link invariants.     

Inscribing Smooth Knots with Regular Polygons

A regular n-gon inscribing a knot is a sequence of n pointson a knot, such that the distances between adjacent points areall the same. It is shown that any smooth knot is inscribedby a regular n-gon for any n. 2000 Mathematics Subject Classification57M25 (primary).