Abstract. Partition functions for dimers on closed oriented surfaces are known to be alternating sums of Pfaffians of Kasteleyn matrices. In this paper, we obtain the formula for the coefficients in terms of discrete spin structures.

Abstract. In this paper, we use 'generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in S 3 . This yields an integral valued function on the µ-dimensional torus, where µ is the number of colors of the link. The case µ = 1 corresponds to the Levine-Tristram signature. We show that many remarkable properties of the latter invariant extend to this µ-variable generalization: it vanishes for achiral colored links, it is 'piecewise continuous', and the places of the jumps are determined by the Alexander invariants of the colored link. Using a 4-dimensional interpretation and the Atiyah-Singer Gsignature theorem, we also prove that this signature is invariant by colored concordance, and that it provides a lower bound for the 'slice genus' of the colored link.

Abstract. In a previous paper [3], we showed how certain orientations of the edges of a graph Γ embedded in a closed oriented surface Σ can be understood as discrete spin structures on Σ. We then used this correspondence to give a geometric proof of the Pfaffian formula for the partition function of the dimer model on Γ. In the present article, we generalize these results to the case of compact oriented surfaces with boundary. We also show how the operations of cutting and gluing act on discrete spin structures and how they change the partition function. These operations allow to reformulate the dimer model as a quantum field theory on surface graphs.

We provide a concise exposition with original proofs of combinatorial formulas for the 2D Ising model partition function, multi-point fermionic observables, spin and energy density correlations, for general graphs and interaction constants, using the language of Kac-Ward matrices. We also give a brief account of the relations between various alternative formalisms which have been used in the combinatorial study of the planar Ising model: dimers and Grassmann variables, spin and disorder operators, and, more recently, s-holomorphic observables. In addition, we point out that these formulas can be extended to the double-Ising model, defined as a pointwise product of two Ising spin configurations on the same discrete domain, coupled along the boundary.2000 Mathematics Subject Classification. 82B20.

Abstract. We give a geometric construction of the multivariable Conway potential function for colored links. In the case of a single color, it is Kauffman's definition of the Conway polynomial in terms of a Seifert matrix. Mathematics Subject Classification (2000). 57M25.