arXiv research

To study embeddings of tangles in knots, we use quandle cocycle invariants. Computations are carried out for the tables of knots and tangles, to investigate which tangles may or may not embed in knots in the tables.

The paper examines when 2-string tangles can be embedded into specific link types.

New obstructions for embedding one compact oriented 3-manifold in another are given. A theorem of D. Krebes concerning 4-tangles embedded in links arises as a special case. Algebraic and skein-theoretic generalizations for 2n-tangles provide invariants that persist in the corresponding invariants of links in which they…

We define new invariants of knots by means of quandle colorings and longitudinal information. These invariants can be applied to a tangle embedding problem and recognizing non-classical virtual knots.

Paper defines new topological invariants for DP tangles.

A genus-1 tangle G is an arc properly embedded in a standardly embedded solid torus S in the 3-sphere. We say that a genus-1 tangle embeds in a knot K in S^3 if the tangle can be completed by adding an arc exterior to the solid torus to form the knot K. We call K a closure of G. An obstruction to embedding a genus-1 ta…

A result about spanning forests for graphs yields a short proof of Krebes's theorem concerning embedded tangles in links.

We reprove and extend a result of David Krebes (J. Knot Theory Ramif. 8 (1999), 321-352) giving an obstruction to embedding a tangle T into a link L. Closing the tangle up in the two obvious ways gives rise to two links, the numerator and denominator links n(T) and d(T). Applying a homological argument to the 2-fold br…

We produce embeddings of knots in thin position that admit compressible thin levels. We also find the bridge number of tangle sums where each tangle is high distance.

In this paper we study how to distinguish two embeddings of a finite collection of disjoint circles into the plane up to planar isotopy. We adopt the spirit of the approach by V. Turaev, Operator Invariants of Tangles, Math. USSR-Izv. 35 (1990), 411--444, by considering a category of planar tangles and representing it …

The paper extends a knot invariant to graphs and connects it to homology cylinders.

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 …

Khovanov homology for pro-tangles and spectral sequences

Random hyperbolic surfaces are mostly tangle-free, with geometric implications.

Paper defines and analyzes mathematical equivalence of periodic tangles.

Formula removes geometric patterns from random hyperbolic surfaces.

Extends tangle theory to include undetermined crossings in periodic structures.

We consider the ways in which a 4-tangle T inside a unit cube can be extended outside the cube into a knot or link L. We present two links n(T) and d(T) such that the greatest common divisor of the determinants of these two links always divides the determinant of the link L. In order to prove this result we give a two-…

A $3$-$tangle$ $T$ is the disjoint union of $3$ properly embedded arcs in the unit 3-ball; it is called rational if there is a homeomorphism of pairs from $(B^3,T)$ to $(D^2\times I,\{x_1,x_2,x_3\}\times I)$. Two rational 3-tangles $T$ and $T'$ are isotopic if there is an orientation-preserving self-homeomorphism $h: (…

We consider a class of topological objects in the 3-sphere $S^3$ which will be called {\it $n$-punctured ball tangles}. Using the Kauffman bracket at $A=e^{πi/4}$, an invariant for a special type of $n$-punctured ball tangles is defined. The invariant $F$ takes values in $PM_{2\times2^n}(\mathbb Z)$, that is the set of…

Study of 3-manifolds in 5-sphere using bridge decompositions.

New method to compute tangle sums in character varieties.

Given a compact oriented 3-manifold M in S^3 with boundary, an (M,2n)-tangle T is a 1-manifold with 2n boundary components properly embedded in M. We say that T embeds in a link L in S^3 if T can be completed to L by a 1-manifold with 2n boundary components exterior to M. The link L is called a closure of T. We define …

We consider a class of topological objects in the 3-sphere $S^3$ which will be called $n$-punctured ball tangles. Using the Kauffman bracket at $A=e^{i π/4}$, an invariant for a special type of $n$-punctured ball tangles is defined. The invariant $F^n$ takes values in $PM_{2\times2^n}(\mathbb Z)$, that is the set of $2…

New method for knot closures from 1-tangles and annulus twists.

We describe the first part of a gluing theory for the bigraded Khovanov homology with integer coefficients. This part associates a type D structure to a tangle properly embedded in a half-space and proves that the homotopy class of the type D structure is an invariant of the isotopy class of the tangle. The constructio…

New diagrams classify triply periodic entanglements.

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

Ribbon tangles are proper embeddings of tori and cylinders in the $4$-ball~$B^4$, "bounding" $3$-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathsf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. Th…

Virtual links are generalizations of classical links that can be represented by links embedded in a ``thickened'' surface $Σ\times I$, product of a Riemann surface of genus $h$ with an interval. In this paper, we show that virtual alternating links and tangles are naturally associated with the $1/N^2$ expansion of an i…

In this paper we describe three geometric applications of quandle homology. We show that it gives obstructions to tangle embeddings, provides the lower bound for the 4-move distance between links, and can be used in determining periodicity of links.

Classifies prime algebraic tangles up to 14 crossings.

Simpler method detects trivial rational 3-tangle.

An enhanced trivalent tangle is a trivalent tangle with some of its edges labeled. We use enhanced trivalent tangles and classical knot theory to provide a recipe for constructing invariants for trivalent tangles, and in particular, for knotted trivalent graphs. Our method also yields invariants of, what we refer to as…

Introduces XC-tangles for quantum tangle invariants.

Study connects taffy pulling, fractions, and rational tangles.

Method for computing Khovanov homology of tangles.

This article addresses persistent tangles. These are tangles whose presence in a knot diagram forces that diagram to be knotted. We provide new methods for constructing persistent tangles. Our techniques rely mainly on the existence of non-trivial colorings for the tangles in question. Our main result in this article i…

We introduce a generalization of oriented tangles, which are still called tangles, so that they are in one-to-one correspondence with the sutured manifolds. We define cobordisms between sutured manifolds (tangles) by generalizing cobordisms between oriented tangles. For every commutative algebra A over Z/2Z, we define …

Paper explores conditions for constructing new quasi-alternating links.

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

We show that we can release the rigidity of the skew Howe duality process for ${\mathfrak sl}_n$ knot invariants by rescaling the quantum Weyl group action, and recover skein modules for web-tangles. This skew Howe duality phenomenon can be extended to the affine ${\mathfrak sl}_m$ case, corresponding to looking at tan…

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.

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…

New method uses mosaics to study wild knots.