New relations for the vertex polynomial
We extend the vertex polynomial to graphs of arbitrary degree and prove local relations that hold when a graph contains a digon, triangle, quadrilateral or pentagon.
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arXiv subjects
Scott Baldridge
Publications and source records attributed to Scott Baldridge.
A counterexample for the polar conjecture of Spencer-Brown
In 1976, George Spencer-Brown announced a proof of the four color theorem, using operations on Tait colorings for trivalent plane graphs. In subsequent work he formulated these operations in terms of an algorithm that he called a parity-pass and claimed that when the parity pass algorithm is performed on a non-polar pentagon region, it necessarily terminates in an edge coloring that is extendable to the entire graph. We provide here a counterexample to show that this claim is false. We then raise questions related to the existence of this sort of counterexample.
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New relations for the Penrose polynomial
We introduce two new relations involving the pentagon and the quadrilateral for the evaluation of the Penrose polynomial at $n=4$ that is proven using a new type of ribbon graph polynomial. Additionally, we extend several relations for the evaluation of the Penrose polynomial at $n=3$ to all $n$.
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A new way to prove configuration reducibility using gauge theory
We show how ideas coming out of gauge theory can be used to prove configurations in the list of ``633 unavoidable configurations" are reducible. In this paper, we prove the smallest nontrivial example, the Birkhoff diamond, is reducible using our filtered $3$- and $4$-color homology. This is a new proof of a 111-year-old result that is a direct consequence of a special (2+1)-dimensional topological quantum field theory. As part of the proof, we introduce the idea of a state-reducible configuration. Because state-reducibility does not involve Kempe switches, this leads to an independent way to verify the proof of the four color theorem. We conjecture that these gauge theoretic ideas could also lead to a non-computer-based proof of it.
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Quantum state systems that count perfect matchings
In this paper we show how to categorify the $n$-color vertex polynomial, which is based upon one of Roger Penrose's formulas for counting the number of $3$-edge colorings of a planar trivalent graph. Using topological quantum field theory (TQFT), we introduce a quantum state system to build a new bigraded theory called the bigraded $n$-color vertex homology. The graded Euler characteristic of this homology is the $n$-color vertex polynomial. We then produce a spectral sequence whose $E_\infty$-page is a filtered theory called filtered $n$-color vertex homology and show that it is generated by certain types of face colorings of ribbon graphs. For $n=2$, we show that the filtered $n$-color vertex homology is generated by face colorings that correspond to perfect matchings. Finally, we introduce and give meaning to what the vertex polynomial counts when $n \geq 2$. This polynomial is a new abstract graph invariant that can be inferred from certain formulas of Penrose.
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A state sum for the total face color polynomial
The total face color polynomial is based upon the Poincaré polynomials of a family of filtered $n$-color homologies. It counts the number of $n$-face colorings of ribbon graphs for each positive integer $n$. As such, it may be seen as a successor of the Penrose polynomial, which at $n=3$ counts $3$-edge colorings (and consequently $4$-face colorings) of planar trivalent graphs. In this paper we describe a state sum formula for the polynomial. This formula unites two different perspectives about graph coloring: one based upon topological quantum field theory and the other on diagrammatic tensors.
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A topological quantum field theory approach to graph coloring
In this paper, we use a topological quantum field theory (TQFT) to define families of new homology theories of a $2$-dimensional CW complex of a smooth closed surface. The dimensions of these homology groups can be used to count the number of ways that each face of the CW complex can be colored with one of $n$ colors so that no two adjacent faces have the same color. We use these homologies to define new invariants of graphs, give new characterizations of well-known polynomial invariants of graphs, and rephrase and offer new approaches to famous conjectures about graph coloring. In particular, we show that the TQFT has the potential to generate $4$-face colorings of a bridgeless planar graph, leading to a constructive approach to the four color theorem. The TQFT has ramifications for the study of smooth surfaces and provides examples of new types of Frobenius algebras.
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A Cohomology Theory for Planar Trivalent Graphs with Perfect Matchings
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 colorable. We also define several new polynomials invariants of graphs with and without perfect matchings that are invariants of abstract tensors systems and spin networks defined by Roger Penrose in the 1970s. We show how some of these polynomials can be ``categorified'' into their own homology theories.
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On ribbon graphs and virtual links
We introduce a new equivalence relation on decorated ribbon graphs, and show that its equivalence classes directly correspond to virtual links. We demonstrate how this correspondence can be used to convert any invariant of virtual links into an invariant of ribbon graphs, and vice versa.
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Unoriented Virtual Khovanov Homology
The Jones polynomial and Khovanov homology of a classical link are invariants that depend upon an initial choice of orientation for the link. In this paper, we give a Khovanov homology theory for unoriented virtual links. The graded Euler characteristic of this homology is proportional to a similarly-defined unoriented Jones polynomial for virtual links, which is a new invariant in the category of non-classical virtual links. The unoriented Jones polynomial continues to satisfy an important property of the usual one: for classical or even virtual links, the unoriented Jones polynomial evaluated at one is two to the power of the number of components of the link. As part of extending the main results of this paper to non-classical virtual links, a new framework for computing integral Khovanov homology is described that can be efficiently and effectively implemented on a computer. We define an unoriented Lee homology theory for virtual links based upon the unoriented version of Khovanov homology.
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The 2-Factor Polynomial Detects Even Perfect Matchings
In this paper, we prove that the 2-factor polynomial, an invariant of a planar trivalent graph with a perfect matching, counts the number of 2- factors that contain the the perfect matching as a subgraph. Consequently, we show that the polynomial detects even perfect matchings.
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Lifting Lagrangian immersions in $\mathbb{C} P^{n-1}$ to Lagrangian cones in $\mathbb{C}^n$
In this paper we show how to lift Lagrangian immersions in $\mathbb{C} P^{n-1}$ to produce Lagrangian cones in $\mathbb{C} ^n$, and use this process to produce several families of examples of Lagrangian cones and special Lagrangian cones. Moreover we show how to produce Lagrangian cones, isotopic to the Harvey-Lawson and trivial cones, whose projections to $\mathbb{C} P^{n-1}$ are immersions with few transverse double points.
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On the rotation class of knotted Legendrian Tori in $\mathbb{R}^5$
In this paper we show how to combinatorically compute the rotation class of a large family of embedded Legendrian tori in $\mathbb{R}^5$ with the standard contact form. In particular, we give a formula to compute the Maslov index for any loop on the torus and compute the Maslov number of the Legendrian torus. These formulas are a necessary component in computing contact homology. Our methods use a new way to represent knotted Legendrian tori called Lagrangian hypercube diagrams.
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Cube diagrams and 3-dimensional Reidemeister-like moves for knots
In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under two types of cube diagram operations. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Floer homology.
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Coisotropic Luttinger surgery and some new symplectic 6-manifolds with vanishing canonical class
We introduce a surgery operation on symplectic manifolds called coisotropic Luttinger surgery, which generalizes Luttinger surgery on Lagrangian tori in symplectic 4-manifolds. We use it to produce infinitely many distinct symplectic non-Kahler 6-manifolds $X$ with $c_1(X)=0$ which are not of the form $M\times F$ for $M$ a symplectic 4-manifold and $F$ a closed surface.
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Embedded and Lagrangian Knotted Tori in $\BR^4$ and Hypercube Homology
In this paper we introduce a representation of a embedded knotted (sometimes Lagrangian) tori in $\BR^4$ called a hypercube diagram, i.e., a 4-dimensional cube diagram. We prove the existence of hypercube homology that is invariant under 4-dimensional cube diagram moves, a homology that is based on knot Floer homology. We provide examples of hypercube diagrams and hypercube homology, including using the new invariant to distinguish (up to cube moves) two "Hopf linked" tori. We also give examples of a "Trefoil" torus and an immersed knotted torus that is an amalgamation of the $5_2$ knot and a trefoil knot.
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Small examples of cube diagrams of knots
In this short note we highlight some of the differences between cube diagrams and grid diagrams. We also list examples of small cube diagrams for all knots up to 7 crossings and give some examples of links.
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Simply connected minimal symplectic 4-manifolds with signature less than --1
For each pair $(e,σ)$ of integers satisfying $2e+3σ\ge 0$, $σ\leq -2$, and $e+σ\equiv 0\pmod{4}$, with four exceptions, we construct a minimal, simply connected symplectic 4-manifold with Euler characteristic $e$ and signature $σ$. We also produce simply connected, minimal symplectic 4-manifolds with signature zero (resp. signature -1) with Euler characteristic $4k$ (resp. $4k+1$) for all $k\ge 46$ (resp. $k\ge 49$).
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