Related papers: Alexander polynomial and spanning trees
We find that Alexander polynomial of a ribbon knot in $ \mathbb{Z}HS^3 $ is determined by the intrinsic singularity information of its ribbon, and give a formula to calculate Alexander polynomial of a ribbon knot by that. We define half…
We consider the polyhedral properties of two spanning tree problems with additional constraints. In the first problem, it is required to find a tree with a minimum sum of edge weights among all spanning trees with the number of leaves less…
We prove that a polynomial fraction of the set of $k$-component forests in the $m \times n$ grid graph have equal numbers of vertices in each component, for any constant $k$. This resolves a conjecture of Charikar, Liu, Liu, and Vuong, and…
We give combinatorial criteria for predicting the transcendental weight of Feynman integrals of certain graphs in $\phi^4$ theory. By studying spanning forest polynomials, we obtain operations on graphs which are weight-preserving, and a…
Recently, Bigelow defined a diagrammatic method for calculating the Alexander polynomial of a knot or link by resolving crossings in a planar algebra. I will present my multivariate version of Bigelow's calculation. The advantage to my…
The solution of the classical field equation generates the sum of all tree graphs. We show that the classical equation reduces to an easily solved ordinary differential equation for certain multiparticle threshold amplitudes and compute…
Fox's conjecture from 1962, that the absolute values of the coefficients of the Alexander polynomial of an alternating link are trapezoidal, has remained stubbornly open to this date. Recently Fox's conjecture was settled for all special…
We observe the twisted Alexander polynomial for metabelian representations of knot groups into SL(2,C) and study relations to the characterizations of metabelian representations in the character varieties. We give a factorization of the…
We define twisted Alexander polynomials of a complex hypersurface with arbitrary singularities. These generalize the classical Alexander polynomials of high dimensional hypersurfaces and the twisted Alexander polynomial of plane curves. We…
We introduce a version of the Alexander polynomial for singular knots and tangles and show how it can be strengthened considerably by introducing a perturbation. For singular long knots, we also prove that our Alexander polynomial agrees…
We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link of any number of components, and by infinitely many such links of at least 4 components. As a consequence, a Mahler measure minimizing…
In the 1920's Artin defined the braid group in an attempt to understand knots in a more algebraic setting. A braid is a certain arrangement of strings in three-dimensional space. It is a celebrated theorem of Alexander that every knot is…
The avalanche polynomial on a graph captures the distribution of avalanches in the abelian sandpile model. Studied on trees, this polynomial could be defined by simply considering the size of the subtrees of the original tree. In this…
We prove that the Alexander polynomials of certain families of alternating 4-braid knots satisfy Fox's Trapezoidal Conjecture. Moreover, we give explicit formulas for the signature and for the first 4 coefficients of the Alexander…
The Alexander polynomial (1928) is the first polynomial invariant of links devised to help distinguish links up to isotopy. Fox's conjecture (1962) -- stating that the absolute values of the coefficients of the Alexander polynomial for any…
We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes $\Delta$, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree…
Twisted torus knots are a generalization of torus knots, obtained by introducing additional full twists to adjacent strands of the torus knots. In this article, we present an explicit formula for the Alexander polynomial of twisted torus…
In this paper, we address the Ehrenborg's conjecture which proposes that for any bipartite graph the number of spanning trees does not exceed the product of the degrees of the vertices divided by the product of the sizes of the graph…
We construct new invariant polynomial for long virtual knots. It is a generalization of Alexander polynomial. We designate it by $\zeta$ meaning an analogy with $\zeta$-polynomial for virtual links. A degree of $\zeta$-polynomial estimates…
Given a complete graph with positive weights on its edges, we define the weight of a subset of edges as the product of weights of the edges in the subset and consider sums (partition functions) of weights over subsets of various kinds:…