Study examines how changing regions affects planar graphs.
problem Effect of region crossing change on planar trivalent graphs.
method Investigation of region crossing changes on planar trivalent graphs.
result Effect of region crossing change on planar trivalent graphs.
Introduces Niebrzydowski algebras for trivalent spatial graphs and handles.
problem Counting and distinguishing trivalent spatial graphs and handlebody-links.
method Defines Niebrzydowski algebras with ternary operation and partially defined multiplication, motivated by Reidemeister moves.
result Niebrzydowski algebras can distinguish some trivalent spatial graphs and handlebody-links.
A new invariant for knotted graphs defined by label bracket.
problem Defining an invariant for knotted trivalent graphs.
method Generalizing Akimova and Manturov's construction to define the label bracket.
result The label bracket defines an isotopy invariant of knotted trivalent graphs.
Proves Markov theorem for trivalent braids using L-move approach.
problem Proving Markov theorem for trivalent braids.
method Follows L-move approach to prove Markov theorem.
result Proves one-move Markov-type theorem and algebraic Markov-type theorem for trivalent braids.
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.
problem Creating an invariant for virtual trivalent spatial graphs.
method Colorings by virtual Niebrzydowski algebras.
result Generalization and computational implementation of invariants.
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.
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.
Enhanced trivalent tangles and handlebody-tangles invariants created.
problem Creating invariants for trivalent and handlebody-tangles.
method Using enhanced trivalent tangles and classical knot theory.
result Constructed invariants for trivalent and handlebody-tangles.
Study combinatorial analogues of Kronheimer-Mrowka theory for graphs.
problem No specific problem stated; focuses on theory development.
method Introduce combinatorial equivariant analogues of Kronheimer-Mrowka homology theory.
result Developed combinatorial analogues for planar trivalent graphs.
Kronheimer-Mrowka's instanton homology dimension equals Tait colorings.
problem Calculating the dimension of a specific homology group for plane trivalent graphs.
method Using SO(3) instanton Floer homology, the dimension is shown to be equal to the number of Tait colorings.
result The dimension of J#(G) is equal to the number of Tait colorings of G.
Classifies 1-connected 2-stratifolds using graph theory.
problem Classifying 1-connected trivalent 2-stratifolds.
method Uses associated labeled graphs and operations to construct all graphs representing 1-connected 2-stratifolds.
result Developed methods to construct all graphs representing 1-connected 2-stratifolds from a single vertex.
The paper extends foam theory to more complex trivalent graphs.
problem Extending foam theory to more complex trivalent graphs.
method Considering foams with singular vertices homeomorphic to cones over more general planar trivalent graphs.
result Modules associated with the dodecahedron graph are free of rank 60.
The paper classifies virtual knot polynomials and trivalent graph invariants using skein theory.
problem Classifying virtual knot polynomials and trivalent graph invariants with specific conditions.
method Skein-theoretic techniques applied to classify invariants with smallness conditions.
result Classification of all non-trivial invariants of trivalent graphs and skein theories of virtual tangles.
Proves Alexander- and Markov-type theorems for virtual trivalent braids.
problem Classifying virtual trivalent braids and graphs.
method Two versions of the Markov-type theorem: algebraic and L-move based.
result Established new theorems for virtual trivalent braids.
Minimal sets of moves for isotopic knots and trivalent graphs identified.
problem Identifying minimal sets of moves for isotopic knots and trivalent graphs.
method Provided and proved the existence of minimal generating sets of oriented Reidemeister moves for isotopic knots and spatial trivalent graphs.
result Twelve minimal generating sets of oriented Reidemeister moves for isotopic knots and ten for spatial trivalent graphs identified.
New signatures for knotted graphs linked to classical knot signatures.
problem Defining invariants for knotted trivalent graphs.
method Using branched covers to define and relate new signatures to classical knot signatures.
result Computable invariants for Kinoshita's knotted theta graph.
The paper refines 2-factor homology to a stable homotopy type for planar trivalent graphs with perfect matchings.
problem Developing a stable homotopy type for planar trivalent graphs with perfect matchings.
method Defining a cover functor from the 2-factor flow category to the cube flow category, realizing the 2-factor spectrum, and showing it's an invariant.
result The stable homotopy type of the 2-factor spectrum is an invariant of planar trivalent graphs with perfect matchings.
New theory defines invariants for framed 3-manifolds using trivalent graphs with self-loops.
problem Defining invariants for framed 3-manifolds with semi-simple Lie groups.
method Constructing graph complexes and cocycles, including self-loops, to define invariants.
result Higher-loop invariants can be defined by graph cocycles with or without self-loops.
Let Gg,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 Ag,b of all automorphisms of graphs in Gg,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 S3 is a L…
Python code constructs connected 2-stratifolds from graphs.
problem Creating models of connected trivalent 2-stratifolds.
method Developed operations on associated labeled graphs to construct connected 2-stratifolds.
result Implemented Python code to automate the construction process.
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…
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 …
Alexander polynomial defined for MOY graphs.
problem Defining an Alexander polynomial for MOY graphs.
method Introduced a refined Alexander polynomial for framed trivalent MOY graphs.
result Invariant satisfies MOY-type relations and defines Alexander polynomial of links.
Minimal hyperbolic surface diameter grows logarithmically with genus.
problem Finding the smallest possible diameter of hyperbolic surfaces.
method Random construction, lattice point counting, and exploration of random trivalent graphs.
result Minimal diameter is asymptotic to log(g) as genus g approaches infinity.
Study detects non-trivial elements in diffeomorphism groups via trivalent graphs.
problem Detecting non-trivial elements in homotopy groups of diffeomorphism spaces.
method Using Kontsevich classes and trivalent graphs, we lift elements from one moduli space to another.
result Non-trivial elements in π∗(BDiff∂(Dd))⊗Q are lifted to π∗(BDiff⊔(DdimesI))⊗Q and π∗(M∂psc(Dd)h0)⊗Q. Graph potentials link to topological QFTs, with computational methods.
problem Defining a topological quantum field theory using graph potentials.
method Using colored trivalent graphs and birational type to define a topological QFT.
result Graph potentials' birational type depends on the graph's homotopy type.
The paper generalizes virtual knot theory using multiple types of virtual crossings.
problem Generalizing virtual knot theory to include multiple types of virtual crossings.
method Starting with graph theory, the paper reviews previous work and then constructs multi-virtual knots and links.
result The multiplicity of virtual crossings allows for a broader application of the Penrose evaluation to all trivalent graphs.
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 …
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…
Classifies 2-stratifolds with fundamental group Z.
problem Classifying 2-stratifolds with specific fundamental groups.
method Using bicoloured labelled graphs and algorithms.
result Efficient algorithm to determine fundamental group Z.
New cohomology theory for planar graphs with perfect matchings.
problem Understanding cohomology of planar trivalent graphs with perfect matchings.
method Introducing a cohomology theory and defining new polynomials.
result 2-factor polynomial can indicate 4-face colorability.
Diagrammatic method characterizes non-split surfaces in 3-sphere.
problem Characterizing non-split compact surfaces in the 3-sphere.
method Using diagrams of spatial trivalent graphs with signs and Reidemeister moves.
result Two diagrams of embedded surfaces are related by Reidemeister moves if and only if the surfaces are ambient isotopic.
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…
Counting periodic geodesics of bounded length and commutator structure on hyperbolic surfaces.
problem Counting periodic geodesics with specific commutator structure.
method Reduction to counting critical realizations of trivalent graphs.
result Asymptotic count of geodesics with bounded length and commutator structure.
Generalizes Hodge correlators using quantum master equation concepts.
problem Developing a mathematical framework for non-acyclic Chern-Simons theory.
method Introduces a DG Lie algebra of uni-trivalent graphs with loops satisfying a Maurer-Cartan equation.
result Arithmetic analogue of effective action and quantum master equation.
New dg-algebras link graph colorings to sheaves.
problem Linking graph colorings to sheaves for Legendrian surfaces.
method Generalized Casals-Murphy dg-algebra to non-commutative coefficients and computed Legendrian contact dg-algebra.
result Rank r representations of dg-algebras correspond to colorings of faces in Grassmannian.
The tail of a sequence {Pn(q)}n∈N of formal power series in Z[[q]] is the formal power series whose first n coefficients agree up to a common sign with the first n coefficients of Pn. This paper studies the tail of a sequence of admissible trivalent graphs with edges colored n o…
Study of multi-moment maps on specific six-manifolds.
problem Understanding multi-moment maps on nearly Kähler six-manifolds.
method Explicit derivation of multi-moment maps and analysis of fixed-points and orbits.
result Explicit expression and configuration of fixed-points and orbits derived.
We study trivalent graphs in S3 whose closed complement is a genus two handlebody. We show that such a graph, when put in thin position, has a simple (i. e. non-loop) level edge.
A clover is a framed trivalent graph with some additional structure, embedded in a 3-manifold. We define surgery on clovers, generalizing surgery on Y-graphs used earlier by the second author to define a new theory of finite-type invariants of 3--manifolds. We give a systematic exposition of a topological calculus of c…
We show that given a trivalent graph in S3, 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…
Graph coloring is explained using a topological field theory with defects.
problem Graph coloring as a combinatorial problem is quantum in nature.
method Topological field theory with defects to interpret graph coloring.
result Graph coloring is related to sections of a certain bundle.
The paper proves infinite-dimensional rational homotopy groups for a specific embedding space.
problem The study of rational homotopy groups of the space of long embeddings of codimension 2.
method Utilizing hairy graphs, the authors construct elements in the homotopy groups and prove their nontriviality.
result The rational homotopy groups of the space of long embeddings are infinite-dimensional in infinitely many degrees.
We call a 3-manifold Platonic if it can be decomposed into isometric Platonic solids. Generalizing an earlier publication by the author and others where this was done in case of the hyperbolic ideal tetrahedron, we give a census of hyperbolic Platonic manifolds and all of their Platonic tessellations. For the octahedra…
A formula connects discrete harmonic surfaces to holomorphic functions.
problem Creating smooth discrete harmonic surfaces from holomorphic data.
method Weierstrass representation formula for discrete harmonic surfaces.
result Smooth converging sequence of discrete harmonic surfaces converges to a minimal surface.
A closed hyperbolic surface of genus g≥2 can be decomposed into pairs of pants along shortest closed geodesics and if these curves are sufficiently short (and with lengths uniformly bounded away from 0), then the geometry of the surface is essentially determined by the combinatorics of the pants decomposition. The…