arXiv research

Study examines how changing regions affects planar graphs.

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.

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.

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

We show that given a trivalent graph in $S^3$, either the graph complement contains an essential almost meridional planar surface or thin position for the graph is also bridge position. This can be viewed as an extension of a theorem of Thompson to graphs. It follows that any graph complement always contains a useful p…

We introduce a new cohomology theory for planar trivalent graphs with perfect matchings. The graded Euler characteristic of the cohomology is a one variable polynomial called the 2-factor polynomial that, if nonzero when evaluated at one, implies that the perfect matching is even and therefore the graph is 4-face color…

We introduce and study combinatorial equivariant analogues of the Kronheimer--Mrowka homology theory of planar trivalent graphs.

The paper extends foam theory to more complex trivalent graphs.

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

Moduli space linked to Tait colorings of planar graphs.

We introduce two invariants called sl(3) Khovanov module and pointed sl(3) Khovanov homology for spatial webs (bipartite trivalent graphs). Those invariants are related to Kronheimer-Mrowka's instanton invariants $J^\sharp$ and $I^\sharp$ for spatial webs by two spectral sequences. As an application of the spectral seq…

The paper generalizes virtual knot theory using multiple types of virtual crossings.

Murakami-Ohtsuki-Yamada introduced an evaluation of certain oriented planar trivalent graphs with colored edges. This evaluation plays a key role in the evaluation of the colored HOMFLY polynomial of a link in 3-space and its Khovanov-Rozansky categorification. Our goal is is to give a generating series formula for the…

We associate a moduli problem to a colored trivalent graph; such graphs, when planar, appear in the state-sum description of the quantum sl(N) knot polynomial due to Murakami, Ohtsuki, and Yamada. We discuss how the resulting moduli space can be thought of a representation variety. We show that the Euler characteristic…

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…

Graph coloring is explained using a topological field theory with defects.

Invariants for trivalent graphs using algebraic colorings.

Paper describes a state sum formula for a graph coloring polynomial.

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.

The SO(3) instanton homology recently introduced by the authors associates a finite-dimensional vector space over the field of two elements to every embedded trivalent graph (or "web"). The present paper establishes a skein exact triangle for this instanton homology, as well as a realization of the octahedral axiom. Fr…

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…

We generalize the construction of Akimova and Manturov, define the label bracket for knotted trivalent graphs in $\mathbb{R}^3$ and show it defines an isotopy invariant of such 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…

The L-move for classical braids extends naturally to trivalent braids. We follow the L-move approach to the Markov Theorem, to prove a one-move Markov-type theorem for trivalent braids. We also reformulate this L-Move Markov theorem and prove a more algebraic Markov-type theorem for trivalent braids. Along the way, we …

Computes Khovanov homology for 2-strand braids via graph relations.

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

Lectures introduce evaluation of SL(3) foams and link homology.

New theory defines invariants for framed 3-manifolds using trivalent graphs with self-loops.

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.

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

Let $G_{g,b}$ be the set of all uni/trivalent graphs representing the combinatorial structures of pant decompositions of the oriented surface of genus $g$ with $b$ boundary components. We describe the set $A_{g,b}$ of all automorphisms of graphs in $G_{g,b}$ showing that, up to suitable moves changing the graph within …

The generalized volume conjecture and the AJ conjecture (a.k.a. the quantum volume conjecture) are extended to $U_q(\fraksl_2)$ colored quantum invariants of the theta and tetrahedron graph. The $\SL(2,\bC)$ character variety of the fundamental group of the complement of a trivalent graph with $E$ edges in $S^3$ is a L…

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…

Trivalent $2$-stratifolds are a generalization of $2$-manifolds in that there are disjoint simple closed curves where three sheets meet. We develop operations on their associated labeled graphs that will effectively construct from a single vertex all graphs that represent $1$-connected $2$-stratifolds. We describe an i…

We construct an extension of the Kontsevich integral of knots to knotted trivalent graphs, which commutes with orientation switches, edge deletions, edge unzips, and connected sums. In 1997 Murakami and Ohtsuki [MO] first constructed such an extension, building on Drinfel'd's theory of associators. We construct a step …

The paper introduces a quantum state system to count perfect matchings in graphs.

Study detects non-trivial elements in diffeomorphism groups via trivalent graphs.

Graph potentials link to topological QFTs, with computational methods.

In 1965, E. C. Zeeman proved that the (+/-)-twist spin of any knotted sphere in (n-1)-space is unknotted in the n-sphere. In 1991, Y. Marumoto and Y. Nakanishi gave an alternate proof of Zeeman's theorem by using the moving picture method. In this paper, we define a knotted 2-dimensional foam which is a generalization …

Characterizes minor-minimal separating projective planar graphs and their generalizations.

The paper extends graph signatures to Klein graphs and foams, linking signatures to knot properties.

It had been known since old times [MO, Da] that there exists a universal finite type invariant ("an expansion") Z^{old} for Knotted Trivalent Graphs (KTGs), and that it can be chosen to intertwine between some of the standard operations on KTGs and their chord-diagrammatic counterparts (so that relative to those operat…

$2$-stratifolds are a generalization of $2$-manifolds that occur as objects in applications such as in TDA. These spaces can be described by an associated bicoloured labelled graph. In previous papers we obtained a classification of 1-connected trivalent $2$-stratifolds. In this paper we classify trivalent $2$-stratifo…

Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.

We define some signature invariants for a class of knotted trivalent graphs using branched covers. We relate them to classical signatures of knots and links. Finally, we explain how to compute these invariants through the example of Kinoshita's knotted theta graph.

Study on planar graph braid groups' second homology.