Related papers: Studying links via plats: split and composite link…

In this paper, we study alternating links in thickened surfaces in terms of the lattices of integer flows on their Tait graphs. We use this approach to give a short proof of the first two generalised Tait conjectures. We also prove that the…

A {\em blink} is a plane graph with an arbitrary bipartition of its edges. As a consequence of a recent result of Martelli, I show that the homeomorphisms classes of closed oriented 3-manifolds are in 1-1 correspondence with specific…

In this paper, we prove intractability results about sampling from the set of partitions of a planar graph into connected components. Our proofs are motivated by a technique introduced by Jerrum, Valiant, and Vazirani. Moreover, we use…

We propose an efficient microscopic design procedure of electronic band structures having intrinsic spin and momentum dependences in spin-orbit-coupling free antiferromagnets. Our bottom-up design approach to creating desired spin-split and…

In this paper we compute a presentation for the group of ring motions of the split union of a Hopf link with Euclidean components and a Euclidean circle. A key part of this work is the study of a short exact sequence of groups of ring…

Questions that seek to determine whether a hyperplane arrangement property, be it geometric, arithmetic or topological, is of a combinatorial nature (that is determined by the intersection lattice) are abundant in the literature. To tackle…

Given any oriented link diagram, two types of new knot invariants are constructed. They satisfy some generalized skein relations. The coefficients of each invariant is from a commutative ring. Homomorphisms and representations of those…

In this paper we develop a structure called Link Algebra, in which we present a Set with two binary operations and an axiom system developed from the study of graph theory and set/antiset theory, sowing main theorems and definitions. Once…

This paper investigates the exotic phenomena exhibited by links of disconnected surfaces with boundary that are properly embedded in the 4-ball. Our main results provide two different constructions of exotic pairs of surface links that are…

Motivated by recent theoretical and experimental interest in metamaterials comprising non-local coupling terms, we present an analytic framework to realise materials with arbitrary complex dispersion relations. Building on the inverse…

Using unknotting number, we introduce a link diagram invariant of Hass and Nowik type, which changes at most by 2 under a Reidemeister move. As an application, we show that a certain infinite sequence of diagrams of the trivial…

The geometric, topological, and symplectic properties of moduli spaces (spaces of configurations modulo rotations and translations) of polygonal linkages have been studied by Kapovich, Millson, and Kamiyama, et. al. One can form a polygonal…

A 2-dimensional braid over an oriented surface-knot $F$ is presented by a graph called a chart on a surface diagram of $F$. We consider 2-dimensional braids obtained by an addition of 1-handles equipped with chart loops. We introduce moves…

This paper continues the study of decompositions of a smooth 4-manifold into two handlebodies with handles of index $\leq2$. Part I gave existence results in terms of spines and chain complexes over the fundamental group of the ambient…

The viscoplastic deformation (creep) of crystalline materials under constant stress involves the motion of a large number of interacting dislocations. Analytical methods and sophisticated `dislocation-dynamics' simulations have proved very…

This paper presents a motion analysis framework for an athlete wearing sport-specific flexible prosthesis based on the soft-rigid hybrid-link system. Such a motion analysis is a challenging problem because we need to consider the…

This paper details a series of experiments in searching for minimal energy configurations for knots and links using the computer program KnotPlot. The most interesting phenomena found in these experiments is the dependence of the…

With the idea of an eventual classification of 3-bridge links,\ we define a very nice class of 3-balls (called butterflies) with faces identified by pairs, such that the identification space is $S^{3},$ and the image of a prefered set of…

We generalize a theorem of Finkelstein and Moriah and show that if a link $L$ has a $2n$-plat projection satisfying certain conditions, then its complement contains some closed essential surfaces. In most cases these surfaces remain…

Two links are link-homotopic if they are transformed into each other by a sequence of self-crossing changes and ambient isotopies. The link-homotopy classes of 4-component links were classified by Levine with enormous algebraic…