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Fundamental groupoids of digraphs and graphs

Alexander Grigor'yan, Rolando Jimenez, Yuri Muranov (2018)

Czechoslovak Mathematical Journal

We introduce the notion of fundamental groupoid of a digraph and prove its basic properties. In particular, we obtain a product theorem and an analogue of the Van Kampen theorem. Considering the category of (undirected) graphs as the full subcategory of digraphs, we transfer the results to the category of graphs. As a corollary we obtain the corresponding results for the fundamental groups of digraphs and graphs. We give an application to graph coloring.

Generating series and asymptotics of classical spin networks

Francesco Costantino, Julien Marché (2015)

Journal of the European Mathematical Society

We study classical spin networks with group SU 2 . In the first part, using Gaussian integrals, we compute their generating series in the case where the edges are equipped with holonomies; this generalizes Westbury’s formula. In the second part, we use an integral formula for the square of the spin network and perform stationary phase approximation under some non-degeneracy hypothesis. This gives a precise asymptotic behavior when the labels are rescaled by a constant going to infinity.

Intrinsic linking and knotting are arbitrarily complex

Erica Flapan, Blake Mellor, Ramin Naimi (2008)

Fundamenta Mathematicae

We show that, given any n and α, any embedding of any sufficiently large complete graph in ℝ³ contains an oriented link with components Q₁, ..., Qₙ such that for every i ≠ j, | l k ( Q i , Q j ) | ≥ α and | a ₂ ( Q i ) | ≥ α , where a ₂ ( Q i ) denotes the second coefficient of the Conway polynomial of Q i .

Khovanov-Rozansky homology for embedded graphs

Emmanuel Wagner (2011)

Fundamenta Mathematicae

For any positive integer n, Khovanov and Rozansky constructed a bigraded link homology from which you can recover the 𝔰𝔩ₙ link polynomial invariants. We generalize the Khovanov-Rozansky construction in the case of finite 4-valent graphs embedded in a ball B³ ⊂ ℝ³. More precisely, we prove that the homology associated to a diagram of a 4-valent graph embedded in B³ ⊂ ℝ³ is invariant under the graph moves introduced by Kauffman.

ℓ²-homology and planar graphs

Timothy A. Schroeder (2013)

Colloquium Mathematicae

In his 1930 paper, Kuratowski proves that a finite graph Γ is planar if and only if it does not contain a subgraph that is homeomorphic to K₅, the complete graph on five vertices, or K 3 , 3 , the complete bipartite graph on six vertices. This result is also attributed to Pontryagin. In this paper we present an ℓ²-homological method for detecting non-planar graphs. More specifically, we view a graph Γ as the nerve of a related Coxeter system and construct the associated Davis complex, Σ Γ . We then use a...

Light paths with an odd number of vertices in polyhedral maps

Stanislav Jendroľ, Heinz-Jürgen Voss (2000)

Czechoslovak Mathematical Journal

Let P k be a path on k vertices. In an earlier paper we have proved that each polyhedral map G on any compact 2 -manifold 𝕄 with Euler characteristic χ ( 𝕄 ) ≤ 0 contains a path P k such that each vertex of this path has, in G , degree ≤ k 5 + 49 - 24 χ ( 𝕄 ) 2 . Moreover, this bound is attained for k = 1 or k ≥ 2 , k even. In this paper we prove that for each odd k ≥ 4 3 5 + 49 - 24 χ ( 𝕄 ) 2 + 1 , this bound is the best possible on infinitely many compact 2 -manifolds, but on infinitely many other compact 2 -manifolds the upper bound can be lowered to ( k - 1 3 ) 5 + 49 - 24 χ ( 𝕄 ) 2 .

New categorifications of the chromatic and dichromatic polynomials for graphs

Marko Stošić (2006)

Fundamenta Mathematicae

For each graph G, we define a chain complex of graded modules over the ring of polynomials whose graded Euler characteristic is equal to the chromatic polynomial of G. Furthermore, we define a chain complex of doubly-graded modules whose (doubly) graded Euler characteristic is equal to the dichromatic polynomial of G. Both constructions use Koszul complexes, and are similar to the new Khovanov-Rozansky categorifications of the HOMFLYPT polynomial. We also give a simplified definition of this triply-graded...

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