Alexander representation of tangles

A tangle is an oriented 1-submanifold of the cylinder whose endpoints lie on the two disks in the boundary of the cylinder. Using an algebraic tool developed by Lescop, we extend the Burau representation of braids to a functor from the category of oriented tangles to the category of Z[t,t^{-1}]-modules. For (1,1)-tangles (i.e., tangles with one endpoint on each disk) this invariant coincides with the Alexander polynomial of the link obtained by taking the closure of the tangle. We use the notion of plat position of a tangle to give a constructive proof of invariance in this case.Comment: 13 pages, 5 figure

The critical Ising model via Kac-Ward matrices

The Kac-Ward formula allows to compute the Ising partition function on any finite graph G from the determinant of 2^{2g} matrices, where g is the genus of a surface in which G embeds. We show that in the case of isoradially embedded graphs with critical weights, these determinants have quite remarkable properties. First of all, they satisfy some generalized Kramers-Wannier duality: there is an explicit equality relating the determinants associated to a graph and to its dual graph. Also, they are proportional to the determinants of the discrete critical Laplacians on the graph G, exactly when the genus g is zero or one. Finally, they share several formal properties with the Ray-Singer \bar\partial-torsions of the Riemann surface in which G embeds.Comment: 30 pages, 10 figures; added section 4.4 in version

The critical Z-invariant Ising model via dimers: locality property

We study a large class of critical two-dimensional Ising models, namely critical Z-invariant Ising models. Fisher [Fis66] introduced a correspondence between the Ising model and the dimer model on a decorated graph, thus setting dimer techniques as a powerful tool for understanding the Ising model. In this paper, we give a full description of the dimer model corresponding to the critical Z-invariant Ising model, consisting of explicit expressions which only depend on the local geometry of the underlying isoradial graph. Our main result is an explicit local formula for the inverse Kasteleyn matrix, in the spirit of [Ken02], as a contour integral of the discrete exponential function of [Mer01a,Ken02] multiplied by a local function. Using results of [BdT08] and techniques of [dT07b,Ken02], this yields an explicit local formula for a natural Gibbs measure, and a local formula for the free energy. As a corollary, we recover Baxter's formula for the free energy of the critical Z-invariant Ising model [Bax89], and thus a new proof of it. The latter is equal, up to a constant, to the logarithm of the normalized determinant of the Laplacian obtained in [Ken02].Comment: 55 pages, 29 figure

Oral Fluid–Based Biomarkers of Alveolar Bone Loss in Periodontitis

Periodontal disease is a bacteria-induced chronic inflammatory disease affecting the soft and hard supporting structures encompassing the teeth. When left untreated, the ultimate outcome is alveolar bone loss and exfoliation of the involved teeth. Traditional periodontal diagnostic methods include assessment of clinical parameters and radiographs. Though efficient, these conventional techniques are inherently limited in that only a historical perspective, not current appraisal, of disease status can be determined. Advances in the use of oral fluids as possible biological samples for objective measures of current disease state, treatment monitoring, and prognostic indicators have boosted saliva and other oral-based fluids to the forefront of technology. Oral fluids contain locally and systemically derived mediators of periodontal disease, including microbial, host-response, and bone-specific resorptive markers. Although most biomarkers in oral fluids represent inflammatory mediators, several specific collagen degradation and bone turnover-related molecules have emerged as possible measures of periodontal disease activity. Pyridinoline cross-linked carboxyterminal telopeptide (ICTP), for example, has been highly correlated with clinical features of the disease and decreases in response to intervention therapies, and has been shown to possess predictive properties for possible future disease activity. One foreseeable benefit of an oral fluid–based periodontal diagnostic would be identification of highly susceptible individuals prior to overt disease. Timely detection and diagnosis of disease may significantly affect the clinical management of periodontal patients by offering earlier, less invasive, and more cost-effective treatment therapies.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/73247/1/annals.1384.028.pd