Related papers: Alternating Knots
This paper gives new and elementary combinatorial topological proofs of the classification of unoriented and oriented rational knots and links. These proofs are based on the known classification of alternating knots through flyping, and the…
We show that one can interweave an unknot into any non-alternating connected projection of a link so that the resulting augmented projection is alternating.
This article gives a brief survey of the theory and applications of anomalies.
We introduce a new numerical invariant of knots and links from the descending diagrams. It is considered to live between the unknotting number and the bridge number.
We show that, for an alternating knot, the ratio of the diameter of the set of boundary slopes to the crossing number can be arbitrarily large.
This an expository article on Givental's axiomatic Gromov--Witten theory and some of its applications.
The formulation of the alternative theory of neutrino oscillations is presented. Also the application of that theory to a system of neutrinos produced by a source is formulated and some basic formulae are derived.
It was previously shown by the second author that every knot in $S^3$ is ambient isotopic to one component of a two-component, alternating, hyperbolic link. In this paper, we define the alternating volume of a knot $K$ to be the minimum…
This is an expository paper discussing various versions of Khovanov homology theories, interrelations between them, their properties, and their applications to other areas of knot theory and low-dimensional topology.
There is an infinitely generated free subgroup of the smooth knot concordance group with the property that no nontrivial element in this subgroup can be represented by an alternating knot. This subgroup has the further property that every…
We investigate the negative definite spin fillings of branched double covers of alternating knots. We derive some obstructions for the existence of such fillings and find a characterization of special alternating knots based on them.
This is an expository paper discussing some parallels between the Khovanov and knot Floer homologies. We describe the formal similarities between the theories and give some examples which illustrate a somewhat mysterious correspondence…
We study knots in $\mathbb{S}^3$ obtained by the intersection of a minimal surface in $\mathbb{R}^4$ with a small 3-sphere centered at a branch point. We construct examples of new minimal knots. In particular we show the existence of…
We discuss the possibility of the existence of finite algorithms that may give distinct knot classes. In particular we present two attempts for such algorithms which seem promising, one based on knot projections on a plane, the other on…
As a supplement to the authors' article "Prime knots with arc index up to 11 and an upper bound of arc index for non-alternating knots", to appear in the Journal of Knot Theory and its Ramifications, we present minimal arc presentations of…
We construct a new type of geometric knot theory, plumbers' knots, and solve the problems of distinguishing and enumerating such knots at a fixed level of complexity. (v2) Minor edits, added theorem 3.18. (v3) Substantial revisions,…
This is an expository article of our work on analogies between knot theory and algebraic number theory. We shall discuss foundational analogies between knots and primes, 3-manifolds and number rings mainly from the group-theoretic point of…
In this paper, we introduce a new type of relation between knots called the descendant relation. One knot $H$ is a descendant of another knot $K$ if $H$ can be obtained from a minimal crossing diagram of $K$ by some number of crossing…
Minimum braids are a complete invariant of knots and links. This paper defines minimum braids, describes how they can be generated, presents tables for knots up to ten crossings and oriented links up to nine crossings, and uses minimum…
We construct two distinct yet related M-theory models that provide suitable frameworks for the study of knot invariants. We then focus on the four-dimensional gauge theory that follows from appropriately compactifying one of these M-theory…