相关论文: A Study of Gram Determinants in Knot Theory

The study of homological invariants such as Tor, Ext and local cohomology modules constitutes an important direction in commutative algebra. Explicit descriptions of these invariants are notoriously difficult to find and often involve…

We determine p-colorability of the paradromic rings. These rings arise by generalizing the well-known experiment of bisecting a Mobius strip. Instead of joining the ends with a single half twist, use $m$ twists, and, rather than bisecting…

We consider a natural model of random knotting- choose a knot diagram at random from the finite set of diagrams with n crossings. We tabulate diagrams with 10 and fewer crossings and classify the diagrams by knot type, allowing us to…

A geometric argument is given to prove that the Seifert genus of a positive knot equals its slice genus. A combinatorial invariant, giving a lower bound for the slice genus, is formulated for arbitrary knots. Properties and applications of…

We associate with a matrix over an arbitrary field an infinite family of matrices whose sizes vary from one to infinity; their entries are traces of powers of the original matrix. We explicitly evaluate the determinants of matrices in our…

We define a broad class of graphs that generalize the Gordian graph of knots. These knot graphs take into account unknotting operations, the concordance relation, and equivalence relations generated by knot invariants. We prove that…

Pseudodiagrams are diagrams of knots where some information about which strand goes over/under at certain crossings may be missing. Pseudoknots are equivalence classes of pseudodiagrams, with equivalence defined by a class of…

In this paper we investigate the descriptional complexity of knot theoretic problems and show upper bounds for planarity problem of signed and unsigned knot diagrams represented by Gauss words. Since a topological equivalence of knots can…

The leading coefficient of the Alexander polynomial of a knot is the most informative element in this invariant, and the growth of orders of the first homology of cyclic branched covering spaces is also a familiar subject. Accordingly,…

In an earlier paper, we introduced ``bordered knot algebras'', which are graded algebras indexed by a pair of integers (m,k). In a subsequent paper, we introduced a two-parameter family of differential graded algebra, the ``pong algebras'',…

In this note we give concise formulas, which lead to a simple and fast computer program that computes a powerful knot invariant. This invariant $\rho_1$ is not new, yet our formulas are by far the simplest and fastest: given a knot we write…

Graded type theories are an emerging paradigm for augmenting the reasoning power of types with parameterizable, fine-grained analyses of program properties. There have been many such theories in recent years which equip a type theory with…

We present a novel approach to the concept of gluing in mathematics by introducing the notions of a gluing data category and a gluing data functor. Our work provides a formal categorical characterization of the notion of gluing in algebraic…

The notion of the genus of a quadratic form is generalized to vertex operator algebras. We define it as the modular braided tensor category associated to a suitable vertex operator algebra together with the central charge. Statements…

We define a knot to be $\gamma_0$-sharp if its Seifert genus is detected by the concordance invariant $\gamma_0$, which arises from the immersed curve formalism in bordered Heegaard Floer homology. We show that a connected sum of…

We explain new developments in classical knot theory in 3 and 4~dimensions, i.e. we study knots in 3-space, up to isotopy as well as up to concordance. In dimension~3 we give a geometric interpretation of the Kontsevich integral (joint with…

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by…

We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing…

To a special type of grope embedded in 4-space, that we call an admissible grope, we associate a length function for each real number q at least 1. This gives rise to a family of pseudo-metrics d^q, refining the slice genus metric, on the…

In this sequel to my previous paper, "Is String Theory in Knots?" I explore ways of constructing symmetries through an algebraic stepping process using knotted graphs. The hope is that this may lead to an algebraic formulation of string…