arXiv research

Enhanced trivalent tangles and handlebody-tangles invariants created.

The paper classifies virtual knot polynomials and trivalent graph invariants using skein theory.

Extends $sl(3)$ invariant to tangles via webs and functors.

We define a category $v\mathcal{T}$ of tangles diagrams drawn on surfaces with boundaries. On the one hand we show that there is a natural functor from the category of virtual tangles to $v\mathcal{T}$ which induces an equivalence of categories. On the other hand, we show that $v\mathcal{T}$ is universal among ribbon c…

Knotted trivalent graphs (KTGs) form a rich algebra with a few simple operations: connected sum, unzip, and bubbling. With these operations, KTGs are generated by the unknotted tetrahedron and Moebius strips. Many previously known representations of knots, including knot diagrams and non-associative tangles, can be tur…

It is known that every surface-link can be presented by a marked graph diagram, and such a diagram presentation is unique up to moves called Yoshikawa moves. G. Kuperberg introduced a regular isotopy invariant, called the quantum A_2 invariant, for tangled trivalent graph diagrams. In this paper, a polynomial for a mar…

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…

Proves Markov theorem for trivalent braids using L-move approach.

Study examines how changing regions affects planar graphs.

We construct a state model for the two-variable Kauffman polynomial using planar trivalent graphs. We also use this model to obtain a polynomial invariant for a certain type of trivalent graphs embedded in three-dimensional space.

Invariants for trivalent graphs using algebraic colorings.

Method counts connected 2D stratifolds with singular curves and components.

We propose to generalize the volume conjecture to knotted trivalent graphs and we prove the conjecture for all augmented knotted trivalent graphs. As a corollary we find that for any link L there is a link containing L for which the volume conjecture holds.

A new invariant for knotted graphs defined by label bracket.

We prove Alexander- and Markov-type theorems for virtual spatial trivalent graphs and virtual trivalent braids. We provide two versions for the Markov-type theorem: one uses an algebraic approach similar to the case of classical braids and the other one is based on L-moves.

We give a description of local and global moves on a class of locally planar trivalent graphs and we show that it contains $λ$-Scale calculus, therefore in particular untyped lambda calculus. Surprisingly, the beta reduction rule comes from a local "sewing" transformation of trivalent locally planar graphs.

Kronheimer-Mrowka's instanton homology dimension equals Tait colorings.

We introduce \textit{Niebrzydowski algebras}, algebraic structures with a ternary operation and a partially defined multiplication, with axioms motivated by the Reidemeister moves for $Y$-oriented trivalent spatial graphs and handlebody-links. As part of this definition, we identify generating sets of $Y$-oriented Reid…

We study configuration space integral formulas for Milnor's homotopy link invariants, showing that they are in correspondence with certain linear combinations of trivalent trees. Our proof is essentially a combinatorial analysis of a certain space of trivalent "homotopy link diagrams" which corresponds to all finite ty…

Minimal sets of moves for isotopic knots and trivalent graphs identified.

Study of Penner's cocycle on fatgraph complex.

Study combinatorial analogues of Kronheimer-Mrowka theory for graphs.

The paper extends foam theory to more complex trivalent graphs.

Classifies prime algebraic tangles up to 14 crossings.

The paper refines 2-factor homology to a stable homotopy type for planar trivalent graphs with perfect matchings.

Paper defines new topological invariants for DP tangles.

Simpler method detects trivial rational 3-tangle.

New methods found persistent tangles in knots.

Uniformly branching trees are equivalent to certain metric spaces.

Introduces XC-tangles for quantum tangle invariants.

Study connects taffy pulling, fractions, and rational tangles.

Method for computing Khovanov homology of tangles.

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

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.

Random trees found in Outer space boundary.

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

Trivalent $2$-stratifolds are a generalization of $2$-manifolds in that there are disjoint simple closed curves where three sheets meet. We obtain a classification of $1$-connected $2$-stratifolds in terms of their associated labeled graphs and develop operations that will construct from a single vertex all graphs that…

In this paper we show that via the configuration space integral construction a non-trivalent graph cocycle can also yield a non-zero cohomology class of the space of higher (and even) codimensional long knots. This simultaneously proves that the Browder operation induced by the operad action defined by R. Budney is not…

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.

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.

New homotopy refinements for tangle invariants.

Generalized Wriggle polynomial for virtual tangles.