Abstract. The Jones-Wenzl projectors p n play a central role in quantum topology, underlying the construction of SU.2/ topological quantum field theories and quantum spin networks. We construct chain complexes P n , whose graded Euler characteristic is the "classical" projector p n in the Temperley-Lieb algebra. We show that the P n are idempotents and uniquely defined up to homotopy. Our results fit within the general framework of Khovanov's categorification of the Jones polynomial. Consequences of our construction include families of knot invariants corresponding to higher representations of U q sl.2/ and a categorification of quantum spin networks. We introduce 6j -symbols in this context.
This paper introduces a conceptual framework, in the context of quantum topology and the algebras underlying it, for analyzing relations obeyed by the chromatic polynomial .Q/ of planar graphs. Using it we give new proofs and substantially extend a number of classical results concerning the combinatorics of the chromatic polynomial. In particular, we show that Tutte's golden identity is a consequence of level-rank duality for SO.N / topological quantum field theories and BirmanMurakami-Wenzl algebras. This identity is a remarkable feature of the chromatic polynomial relating . C 2/ for any triangulation of the sphere to . . C 1// 2 for the same graph, where denotes the golden ratio. The new viewpoint presented here explains that Tutte's identity is special to these values of the parameter Q. A natural context for analyzing such properties of the chromatic polynomial is provided by the chromatic algebra, whose Markov trace is the chromatic polynomial of an associated graph. We use it to show that another identity of Tutte's for the chromatic polynomial at Q D C 1 arises from a Jones-Wenzl projector in the Temperley-Lieb algebra. We generalize this identity to each value Q D 2 C 2 cos.2 j =.n C 1// for j < n positive integers. When j D 1, these Q are the Beraha numbers, where the existence of such identities was conjectured by Tutte. We present a recursive formula for this sequence of chromatic polynomial relations.57M15; 05C15, 57R56, 81R05
We present a new, more elementary proof of the Freedman-Teichner result that the geometric classification techniques (surgery, s-cobordism, and pseudoisotopy) hold for topological 4-manifolds with groups of subexponential growth. In an appendix Freedman and Teichner give a correction to their original proof, and reformulate the growth estimates in terms of coarse geometry. AMS Classification numbers 407The disk embedding theorem for 4-manifolds with "good" fundamental group is the key ingredient of the classification theory: it is used in the proof of the 4-dimensional surgery theorem, and the 5-dimensional s-cobordism theorem and pseudoisotopy theorems. The homotopy hypotheses of the theorem always allow one to find a 2-stage immersed capped grope. If one can find such a grope so that loops in the image are nullhomotopic in the ambient manifold, then Freedman's theorem [1,3] shows there is a topologically flat embedded disk. The current focus, therefore, is on obtaining this π 1 -nullity condition.Freedman [2] showed this is possible if the fundamental group of the manifold is poly-(finite or cyclic). This was extended to groups of polynomial growth in [7]. The current best result is for groups of subexponential growth.The disk theorem for subexponential groups was stated by Freedman and Teichner [4]. However the "key point" of [4] page 521, line 17, is incorrect. In the Appendix Freedman and Teichner show how to modify their construction to correct this. The present paper sidesteps the issue by using a different and more elementary construction developed by the first author (see [5]). It displays particularly clearly how the proof fails in the general (exponential growth) case, and suggests that an infinite construction may be necessary to make further progress.The following result is the input needed for the disk embedding theorem for manifolds with subexponential fundamental groups. For a full statement of the disk embedding theorem and applications to surgery and s-cobordism, see [4]. For the application to pseudoisotopy see [6]. Theorem Suppose G → M 4 is a properly immersed (capped) grope of height ≥ 2, and ρ: π 1 M → π is a homomorphism with π of subexponential growth. Then the total contraction of G is regularly homotopic rel boundary to an immersion whose double point loops have trivial image in π .This slightly extends the usual immersion-improvement formulation in that we do not require the total contraction to be a disk, and the output immersion is regularly homotopic to the input. Neither extension has new consequences, but they come for free in the proof and they simplify applications. Capped gropes, contractions, and subexponential growth are all reviewed in the text.In rough outline the proof goes as follows: The images of the double point loops of G give a finite subset of π . Subexponential growth implies that in a large collection of words of fixed length in the finite subset, somewhere there is a subword whose product is trivial. We organize the data so this subword can be realized ...
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