Arrow ribbon graphs

We introduce an additional structure on ribbon graphs, arrow structure. We extend the Bollob\'as-Riordan polynomial to ribbon graph with this structure. The extended polynomial satisfies the contraction-deletion relations and naturally behaves with respect to the partial duality of ribbon graphs. We construct an arrow ribbon graph from a virtual link whose extended Bollob\'as-Riordan polynomial specializes to the arrow polynomial of the virtual link recently introduced by H.Dye and L.Kauffman. This result generalizes the classical Thistlethwaite theorem to the arrow polynomial of virtual links.Comment: to appear in Journal of Knot Theory and Its Ramification

No Escape

The initial idea for the film came from an inspiring performance of Chaplin's Easy Street (1917) accompanied by Donald MacKenzie, resident organist at the Odeon Leicester Square, which led me into researches of early cinema (c1895-1907), a period described by Tom Gunning as the ‘cinema of attractions’. James Lastra points out that during this time competition between cinemas was based on the success of various sound strategies all emphasising the ‘liveness’ of the film experience and films were made to motivate particular types of sound accompaniment. Particularly intriguing was the use of live sound effects performed by a skilled troupe from behind the film screen to produce ‘realistic’ sound effects. This is translated in No Escape into the manipulation of on-screen diegetic sound, also inspired by Pierre Schaeffer's musique concrète and his notions of the sound object and reduced listening. The interaction between the live piano and the onscreen sound is crucial to No Escape as is that of the piano and images, which exist alone together for long stretches. The visual content and structure of the film draws on the city symphonies of Walter Ruttman and especially Dziga Vertov whose formal experimentation, startling juxtaposition of images and very rapid editing is important to No Escape’s non-narrative and at times complex montage of British rural and urban vistas. Vertov’s Man with a Movie Camera (1929) is by and partially about the man with the camera as is No Escape, the title of which refers to the idea that though we may travel to get away from something, there is no escape from the inner life. This is represented by the piano music, which varies but within fairly restricted limits. It does respond or drive image choice and editing but the overall sense should be that one cannot escape and these responses are temporary and fleeting Extrapolating from Tom Gunning's cinema of attractions, James Beattie's concept of ‘documentary display’ - a poetic, sensual and subjective approach which encourages listening and looking rather than cognitive understanding - underpins the aesthetic of No Escape, as is a belief in the supremacy of sound and of film as a performative event

Introduction to Vassiliev Knot Invariants

This book is a detailed introduction to the theory of finite type (Vassiliev) knot invariants, with a stress on its combinatorial aspects. It is intended to serve both as a textbook for readers with no or little background in this area, and as a guide to some of the more advanced material. Our aim is to lead the reader to understanding by means of pictures and calculations, and for this reason we often prefer to convey the idea of the proof on an instructive example rather than give a complete argument. While we have made an effort to make the text reasonably self-contained, an advanced reader is sometimes referred to the original papers for the technical details of the proofs. Version 3: some typos and inaccuracies are corrected.Comment: 512 pages, thousands picture

The Structure of W-graphs Arising in Kazhdan-Lusztig Theory.

This thesis is primarily about the combinatorial aspects of Kazhdan-Lusztig theory. Central to this area is the notion of a W-graph, a certain weighted directed graph which encodes a representation of the Iwahori-Hecke algebra of a Coxeter group. The most important examples were given in the original work of Kazhdan and Lusztig in 1979; from these graphs the Kazhdan-Lusztig polynomials are obtained via a weighted path count. In the first part, we consider ``parallel transport'' relations among edge weights. Some of these relations, namely those coming from simply-laced Weyl groups, appeared in the same paper of Kazhdan and Lusztig. We introduce additional ones corresponding to doubly-laced Weyl groups, and, as an application, prove Green's 0-1 conjecture in type B. In the second part we clarify the structure of W-graphs corresponding to minuscule and quasi-minuscule quotients of finite Weyl groups. The W-graphs for minuscule quotients can be deduced, on a case-by-case basis, from previous work on the associated Kazhdan-Lusztig polynomials; we give a type-independent proof of a weaker result that these graphs can be characterized by simple combinatorial rules. For quasi-minuscule quotients, we compute the graphs for all finite Weyl groups except for Lie type D (where we give a conjectural answer). We also compute the parabolic Kazhdan-Lusztig polynomials for the type A quasi-minuscule quotient. The last part concerns the conjecture that in Lie type A, the only strongly connected W-graphs which satisfy a weak set of conditions known as ``admissibility'' are the Kazhdan-Lusztig examples. We prove a partial result that the symmetrically weighted edges of such a graph are the same as the symmetrically weighted edges of some Kazhdan-Lusztig examples.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/108802/1/mchmutov_1.pd

Unsigned state models for the Jones polynomial

It is well a known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs and show that the Jones polynomial of any link can be expressed as a vertex model of an unsigned embedded graph. In the process of deriving this result, we show that for every diagram of a link in the 3-sphere there exists a diagram of an alternating link in a thickened surface (and an alternating virtual link) with the same Kauffman bracket. We also recover two recent results in the literature relating the Jones and Bollobas-Riordan polynomials and show they arise from two different interpretations of the same embedded graph.Comment: Minor corrections. To appear in Annals of Combinatoric

Expansions for the Bollobas-Riordan polynomial of separable ribbon graphs

We define 2-decompositions of ribbon graphs, which generalise 2-sums and tensor products of graphs. We give formulae for the Bollobas-Riordan polynomial of such a 2-decomposition, and derive the classical Brylawski formula for the Tutte polynomial of a tensor product as a (very) special case. This study was initially motivated from knot theory, and we include an application of our formulae to mutation in knot diagrams.Comment: Version 2 has minor changes. To appear in Annals of Combinatoric

Ribbon Graph Minors and Low-Genus Partial Duals

We give an excluded minor characterisation of the class of ribbon graphs that admit partial duals of Euler genus at most one

Bipartite partial duals and circuits in medial graphs

It is well known that a plane graph is Eulerian if and only if its geometric dual is bipartite. We extend this result to partial duals of plane graphs. We then characterize all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph.Comment: v2: minor changes. To appear in Combinatoric

Generalization of the Bollob\'as-Riordan polynomial for tensor graphs

Tensor models are used nowadays for implementing a fundamental theory of quantum gravity. We define here a polynomial $\mathcal T$ encoding the supplementary topological information. This polynomial is a natural generalization of the Bollob\'as-Riordan polynomial (used to characterize matrix graphs) and is different of the Gur\uau polynomial, (R. Gur\uau, "Topological Graph Polynomials in Colored Group Field Theory", Annales Henri Poincare {\bf 11}, 565-584 (2010)) defined for a particular class of tensor graphs, the colorable ones. The polynomial $\mathcal T$ is defined for both colorable and non-colorable graphs and it is proved to satisfy the contraction/deletion relation. A non-trivial example of a non-colorable graphs is analyzed.Comment: 22 pages, 20 figure

Explicit computation of Drinfeld associator in the case of the fundamental representation of gl(N)

We solve the regularized Knizhnik-Zamolodchikov equation and find an explicit expression for the Drinfeld associator. We restrict to the case of the fundamental representation of $gl(N)$. Several tests of the results are presented. It can be explicitly seen that components of this solution for the associator coincide with certain components of WZW conformal block for primary fields. We introduce the symmetrized version of the Drinfeld associator by dropping the odd terms. The symmetrized associator gives the same knot invariants, but has a simpler structure and is fully characterized by one symmetric function which we call the Drinfeld prepotential.Comment: 14 pages, 2 figures; several flaws indicated by referees correcte