arXiv research

The paper calculates minimum crossing numbers for essential tangles.

The study shows knots can have infinitely many string decompositions.

L-space knots lack essential Conway spheres, proven with Floer theory.

A persistent lamination for a knot K is an essential lamination in the complement of the K, which remains essential after every non-trivial Dehn surgery along K. In particular, this implies that all of the Dehn surgery manifolds have universal cover R^3. This paper presents a method for building tangles T with the prop…

In this paper, we characterize all links in the 3-sphere with bridge number at least three that have a bridge sphere of distance two. We show that a link L has a bridge sphere of distance at most two then it falls into at least one of three categories: (1) The exterior of L contains an essential meridional sphere. (2) …

We show that essential punctured spheres in the complement of links with distance three bridge spheres have bounded complexity. We define the operation of tangle product, a generalization of both connected sum and Conway product. Finally, we use the bounded complexity of essential punctured spheres to show that the bri…

If a tangle, K, in the 3-ball has no planar, meridional, essential surfaces in its exterior then thin position for K has no thin levels.

In the present paper, we will show that for any integer n>0 there are infinitely many twisted torus knots with n-string essential tangle decompositions.

Classifies essential annuli in genus two handlebody-knots, determining hyperbolicity and constructing obstructions.

The study finds infinite meridional essential surfaces in hyperbolic knot exteriors of various genera.

Study counts specific surfaces in Montesinos knots with 4 rational tangles.

The preceding paper constructed tangle machines as diagrammatic models, and illustrated their utility with a number of examples. The information content of a tangle machine is contained in characteristic quantities associated to equivalence classes of tangle machines, which are called invariants. This paper constructs …

In this paper, we study the hyperbolicity of arborescent tangles and arborescent links. We will explicitly determine all essential surfaces in arborescent tangle complements with non-negative Euler characteristic, and show that given an arborescent tangle T, the complement X(T) is non-hyperbolic if and only if T is a r…

Proves conjectures for pretzel knots using polynomial degrees.

The paper uses bordered Floer homology to detect incompressible surfaces and tangles.

The paper explores knots' height, trunk, and representativity, finding gaps and bounds.

We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological pictures a little longer than in other articles on the subject, the required extension becomes essentially tautological. A…

Define web algebras for annular SL(2) and SL(3) using foam TQFTs.

Wu has shown that if a link or a knot $L$ in $S^3$ in thin position has thin spheres, then the thin sphere of lowest width is an essential surface in the link complement. In this paper we show that if we further assume that $L \subset S^3$ is prime, then the thin sphere of lowest width also does not have any vertical c…

Boring is an operation which converts a knot or two-component link in a 3--manifold into another knot or two-component link. It generalizes rational tangle replacement and can be described as a type of 2--handle attachment. Sutured manifold theory is used to study the existence of essential spheres and planar surfaces …

New tools for Brouwer mapping classes identified.

In this paper, we characterize closed incompressible surfaces of genus two in the complements of 3-bridge knots and links. This characterization includes that of essential 2-string tangle decompositions for 3-bridge knots and links.

The universal Vassiliev-Kontsevich invariant is a functor from the category of tangles to a certain graded category of chord diagrams, compatible with the Vassiliev filtration and whose associated graded is an isomorphism. The Vassiliev filtration has a natural extension to tangles in any thickened surface $M\times I$ …

Using Hirasawa-Murasugi's classification of fibered Montesinos knots we classify the L-space Montesinos knots, providing further evidence towards a conjecture of Lidman-Moore that L-space knots have no essential Conway spheres. In the process, we classify the fibered Montesinos knots whose open books support the tight …

Study character varieties of arborescent knots and hyperbolic knots.

Enhanced trivalent tangles and handlebody-tangles invariants created.

Classifies prime algebraic tangles up to 14 crossings.

This is the second in a series of papers dedicated to studying w-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc.). These are classes of knotted objects that are wider but weaker than their "usual" counterparts. To get (say) w-knots from usual knots (or u-knots), one has to allow non-planar "virt…

Paper defines new topological invariants for DP tangles.

Simpler method detects trivial rational 3-tangle.

Generalizes tangle Floer homology to A-tangles with A-colorings.

New methods found persistent tangles in knots.

Introduces XC-tangles for quantum tangle invariants.

Study connects taffy pulling, fractions, and rational tangles.

Method for computing Khovanov homology of tangles.

Paper explores conditions for constructing new quasi-alternating links.

Invariants for virtual rational tangles defined using Kauffman's bracket.

Tangle machines describe network information flow, solving covariance intersection problems.

The paper extends knot contact homology to tangles and proves a gluing formula.

New homotopy refinements for tangle invariants defined.

Paper defines and analyzes mathematical equivalence of periodic tangles.

This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof defines the tangle fraction from the canonical form and uses flyping to prove in…

Updated polynomial for virtual tangles, compatible with decompositions.

Spatial graphs study tangle replacement with equivalence classes.

New tangles found for 21 ribbon knots with 10 or fewer crossings.

Paper defines linking numbers for periodic tangles.

We show that for a tangle $T$ with $-\partial^0T \cong \partial^1 T$ the Hochschild homology of the tangle Floer homology $\widetilde{\mathit{CT}}(T)$ is equivalent to the link Floer homology of the closure $T' = T/(-\partial^0T \sim \partial^1 T)$ of the tangle, linked with the tangle axis. In addition, we show that t…

Computes Kauffman bracket polynomial for specific 2-tangle shadows.