In this article, we introduce a fixed parameter tractable algorithm for computing the Turaev-Viro invariants TV4,q , using the dimension of the first homology group of the manifold as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter. The computation of TV4,q is known to be #P-hard in general; using a topol...

In this paper, we develop the local theory of elliptic operators with a mind to proving the Hodge Decomposition Theorem. We then deduce a few of its corollaries including, for compact, oriented manifolds, Poincaré Duality and finite-dimensionality of the de Rham cohomology groups.

The central Delannoy numbers, (dn)n≥0 = 1, 3, 13, 63, 321, 1683, 8989, 48639, . . . (A001850 of The On-Line Encyclopedia of Integer Sequences) will be defined so that dn counts the lattice paths running from (0, 0) to (n, n) that use the steps (1, 0), (0, 1), and (1, 1). In a recreational spirit we give a collection of 29 configurations that these numbers count.

Let (m, b) be a pair of natural numbers. For m even (resp. m odd and b ≥ 2) we show that if there is an m-dimensional non-formal compact oriented manifold with first Betti number b1 = b, there is also a symplectic (resp. contact) manifold with these properties.

Weitzenböck formulas are an important tool in relating local differential geometry to global topological properties by means of the so–called Bochner method. In this article we give a unified treatment of the construction of all possible Weitzenböck formulas for all irreducible, non–symmetric holonomy groups. The resulting classification is two–fold, we construct explicitly a basis of the space...

This paper is concerned with Wilmes’ conjecture regarding abelian sandpile models, as presented in [12]. We introduce the concept of boundary divisors and use this to prove the conjecture for the first Betti number. Further results suggest that this method could be used to tackle the general case of the problem.

In this paper we consider the problem of bounding the Betti numbers, bi(S), of a semi-algebraic set S ⊂ R k defined by polynomial inequalities P1 ≥ 0, . . . , Ps ≥ 0, where Pi ∈ R[X1, . . . , Xk] , s < k, and deg(Pi) ≤ 2, for 1 ≤ i ≤ s. We prove that for 0 ≤ i ≤ k − 1, bi(S) ≤ 1 2 + (k − s) + 1 2 · min{s+1,k−i}

The main objective of this paper is to analyze the geometry of the modular diagonal quotient surface ZN,ε = ∆ε\(X(N) × X(N)) which classifies pairs of elliptic curves E1 and E2 together with an isomorphism of “determinant ε” between their associated modular representations mod N . In particular, we calculate some of the numerical invariants of its minimal desingularization Z̃N,ε such as its Bett...

It is known that perturbative invariants of rational homology 3-spheres can be formulated by using arithmetic perturbative expansion of quantum invariants of them. However, we could not make arithmetic perturbative expansion of quantum invariants for 3-manifolds with positive Betti numbers by the same method. In this paper, we explain how to make arithmetic perturbative expansion of quantum SO(...

We propose a refinement of the Betti numbers and of the homology with coefficients in a field of a compact ANR X, in the presence of a continuous real valued function on X.The refinement of Betti numbers consists of finite configurations of points with multiplicities in the complex plane whose total cardinality are the Betti numbers and the refinement of homology consists of configurations of v...