52 research outputs found
Perturbative Expansion of the Colored Jones Polynomial
No full textBoth the Alexander polynomial and the colored Jones polynomial are well-known knot invariants. While the Jones polynomial seems similar to the Alexander polynomial, it lacks an interpretation in classical topology. Because the Alexander polynomial has a classical topological definition, exploring a relationship between the two polynomials offers the possibility of interpreting the Jones polynomial topologically. Melvin and Morton conjectured a relationship between the two through an expansion of the colored Jones polynomial. The conjecture was proven by Bar-Natan and Garoufalidis and Rozansky extended the result further. Rozansky proved an expansion of the colored Jones polynomial in h=q-1. At each power of h in the expansion, there is a rational expression with powers of the Alexander polynomial in the denominator and new polynomial knot invariants in the numerator. In this dissertation, we will describe how we used the quantum group of sl?2? and techniques from quantum field theory to calculate the first two of these polynomial invariants for all prime knots of up to nine crossings and present these results. Furthermore, we will provide evidence of the validity of a conjecture from Rozansky by calculating the first two polynomial invariants in the expansion for all amphicheiral knots of up to ten crossings.Doctor of Philosoph
Review about "A note on Gornik's perturbation of Khovanov-Rozansky homology" by A. Lobb
No full textIn the paper under review, the author starts from a spectral sequence defined by B. Gornik for Khovanov-Rozansky homology. The graded complex vector space H~i,jn(K) associated to Gornik's spectral sequence is supported in homological degree zero.
The author shows that the quantum degrees of the nonzero H~0,jn(K) are determined only by an even integer sn(K).
As a consequence sn(K) provides a lower bound for the smooth slice genus of K
Virtual crossings and filtrations in link homology
No full textIn 2006 Khovanov and Rozansky introduced a triply-graded link homology theory categorifying the HOMFLY-PT polynomial. Khovanov later gave an alternate construction of HOMFLY-PT homology using Rouquier's braid group action on the category of Soergel bimodules. Soergel bimodules can be filtered by submodules which are the images of virtual crossings in an action of the virtual braid group on the category of graded bimodules over polynomial rings. We conjecture that this filtration extends to HOMFLY-PT homology. We prove that the filtered version of HOMFLY-PT homology is invariant under Reidemeister I and II moves, and conjecture that Reidemeister III does the same. We show that Reidemeister III can violate filtration by at most two levels. This filtration would give a fourth grading on HOMFLY-PT homology, which has been suggested by experimental calculations in recent physics research. The use of filtrations allows us to replace proofs done by generators and relations for Soergel bimodules with more intuitive and diagrammatic proofs.Doctor of Philosoph
Categorification of braid group representations
No full textIn this dissertation, we define categorifications of two braid group representations: theBurau representation and a specialization of the Lawrence-Krammer-Bigelow representation.We first prove a modestly novel realization of each representation in terms of a basis ofTemperley-Lieb diagrams, relying on definitions of these representations in terms of rep-resentations of the quantum group Uq(sl2). We then categorify the basis diagrams usingBar-Natan’s categories BN of tangles and cobordisms from [BN] and the categorified Jones-Wenzl projectors from work of Cooper-Krushkal in [CK] and Rozansky in [Roz]. Finally, wegenerate certain subcategories of the dg category of complexes over BN that are closed undera dg-categorical braid group action. In an appropriate sense, these subcategories decategorifyto representations of the braid group isomorphic to Burau and (a specialization of) LKB.As far as we know, this is the first known categorification of any version of the LKBrepresentation. In addition, our categorification of Burau is a slight generalization of priorwork due to Khovanov-Seidel in [ KS ]: specifically, our categorification is equivalent to one interms of a dg category of dg modules over a dg algebra; and up to a small modification, itszeroth homology is equivalent to the algebraic construction in [KS].Doctor of Philosoph
Witten–Reshetikhin–Turaev Invariants of¶Seifert Manifolds
No full textFor Seifert homology spheres, we derive a holomorphic function of K whose value at integer K is the sl 2 Witten–Reshetikhin–Turaev invariant, Z K , at q= exp 2π i / K . This function is expressed as a sum of terms, which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the Witten–Chern–Simons path integral. The trivial connection contribution is found to have an asymptotic expansion in powers of K −1 which, for K an odd prime power, converges K -adically to the exact total value of the invariant Z K at that root of unity. Evaluations at rational are also discussed. Using similar techniques, an expression for the coloured Jones polynomial of a torus knot is obtained, providing a trivialconnection contribution which is an analytic function of the colour. This demonstrates that the stationary phase expansion of the Chern–Simons–Witten theory is exact for Seifert manifolds and for torus knots in S 3 . The possibility of generalising such results is also discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41998/1/220-205-2-287_92050287.pd
Three-dimensional topological field theory and symplectic algebraic geometry I
No full textWe study boundary conditions and defects in a three-dimensional topological sigma-model with a complex symplectic target space X (the Rozansky–Witten model). We show that boundary conditions correspond to complex Lagrangian submanifolds in X equipped with complex fibrations. The set of boundary conditions has the structure of a 2-category; morphisms in this 2-category are interpreted physically as one-dimensional defect lines separating parts of the boundary with different boundary conditions. This 2-category is a categorification of the Z_2-graded derived category of X; it is also related to categories of matrix factorizations and a categorification of deformation quantization (quantization of symmetric monoidal categories). In Appendix B we describe a deformation of the B-model and the associated category of branes by forms of arbitrary even degree
Monodromic model for Khovanov–Rozansky homology
No full textAbstract
We describe a new geometric model for the Hochschild cohomology of Soergel bimodules based on the monodromic Hecke category studied earlier by the first author and Yun. Moreover, we identify the objects representing individual Hochschild cohomology groups (for the zero and the top degree cohomology this reduces to an earlier result of Gorsky, Hogancamp, Mellit and Nakagane). These objects turn out to be closely related to explicit character sheaves corresponding to exterior powers of the reflection representation of the Weyl group. Applying the described functors to the images of braids in the Hecke category of type A we obtain a geometric description for Khovanov–Rozansky knot homology, essentially different from the one considered earlier by Webster and Williamson.</jats:p
The loop expansion of the Kontsevich integral, the null-move and S-equivalence
No full textAbstractThe Kontsevich integral of a knot is a graph-valued invariant which (when graded by the Vassiliev degree of graphs) is characterized by a universal property; namely it is a universal Vassiliev invariant of knots. We introduce a second grading of the Kontsevich integral, the Euler degree, and a geometric null-move on the set of knots. We explain the relation of the null-move to S-equivalence, and the relation to the Euler grading of the Kontsevich integral. The null-move leads in a natural way to the introduction of trivalent graphs with beads, and to a conjecture on a rational version of the Kontsevich integral, formulated by the second author and proven in Geom. Top 8 (2004) 115 (see also Kricker, preprint 2000, math/GT.0005284)
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