We give a generators-and-relations description of differential graded algebras recently introduced by Ozsváth and Szabó for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal. IntroductionHeegaard Floer homology [OSz04c, OSz04b] is a powerful family of invariants for 3-and 4-manifolds. It originated from the study of Seiberg-Witten theory and Donaldson theory, although its methods involve holomorphic curves rather than gauge theory, and it shares these theories' applicability to the exotic world of smooth 4-manifolds. Compared with its gauge-theoretic relatives, Heegaard Floer homology is often the easiest for computations, and many forms of Heegaard Floer homology have now been given combinatorial definitions.One form of Heegaard Floer homology, called knot Floer homology (or HF K), assigns graded abelian groups to knots and links in 3-manifolds [OSz04a, Ras03]. Like Khovanov homology [Kho00], HF K is especially well-adapted to the study of problems in knot theory with a 4-dimensional character, such as the structure of the knot concordance group. There are many interesting similarities between HF K and Khovanov homology; for example, while the Euler characteristic of Khovanov homology is the Jones polynomial, the Euler characteristic of HF K is the Alexander polynomial.Combined with constructions of the Jones and Alexander polynomial from the representation theory of U q (sl(2)) and U q (gl(1|1)) respectively, this analogy suggests a close link between Heegaard Floer homology and categorifications of the Witten-Reshetikhin-Turaev topological quantum field theory (TQFT) invariants, see e.g. [Ras05,DGR06]. Indeed, both Donaldson-Floer theories in 4 dimensions and Witten-Reshetikhin-Turaev theories in 3 dimensions were initial motivations for the mathematical study of TQFTs, and Heegaard Floer homology offers a promising framework for understanding the relationship between these two types of theories.Among Heegaard Floer theories, HF K admits an especially wide variety of combinatorial descriptions, some allowing very fast computations. In particular, Ozsváth-Szabó have a computer program [OSzb] that can compete with Bar-Natan's fast Khovanov homology program [BN07]. Ozsváth-Szabó's program can quickly compute HF K for most knots with up to around 40 or 50 crossings, and can even handle the larger 90+ crossing examples from the paper [FGMW10].Ozsváth-Szabó's program is based on an exciting new description of HF K [OSz18, OSz17, OSza, OSzc] in the algebraic language of bordered Floer homology, an extended TQFT approach to Heegaard Floer homology.
The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. This paper provides three basic structural results on the critical group of a line graph.• The first deals with connected graphs containing no cut-edge. Here the number of independent cycles in the graph, which is known to bound the number of generators for the critical group of the graph, is shown also to bound the number of generators for the critical group of its line graph. • The second gives, for each prime p, a constraint on the p-primary structure of the critical group, based on the largest power of p dividing all sums of degrees of two adjacent vertices. • The third deals with connected graphs whose line graph is regular. Here known results relating the number of spanning trees of the graph and of its line graph are sharpened to exact sequences which relate their critical groups. The first two results interact extremely well with the third. For example, they imply that in a regular nonbipartite graph, the critical group of the graph and that of its line graph determine each other uniquely in a simple fashion. Contents Functoriality. Suppose that one has two rational orthogonal decompositions. When does f induce a homomomorphism f : K 1 → K 2 between their critical groups?It is natural to assume that f carries the integer lattice Z m1 into Z m2 , that is, f is represented by a matrix in Z m2×m1 . Note that this already implies that the adjoint map f t : R m2 → R m1 with respect to the standard inner products will also satisfy f t (Z m2 ) ⊂ Z m1 , since this map is represented by the transposed Z m1×m2 matrix.What one needs further to induce homomorphisms of critical groups is that f (B 1 ) ⊂ B 2 and f (Z 1 ) ⊂ Z 2 . The following proposition gives a useful reformulation. Proposition 2.3. For a linear map f : R m1 → R m2 satisfying f (Z m1 ) ⊂ Z m2 , one has f (B 1 ) ⊂ B 2 ⇐⇒ f t (Z 2 ) ⊂ Z 1 ⇐⇒ f (Z 1 ) ⊂ Z # 35 References 35
Abstract. The 3-strand pretzel knots and links are a well-studied source of examples in knot theory. However, while there have been computations of the Khovanov homology of some sub-families of 3-strand pretzel knots, no general formula has been given for all of them. We give a general formula for the unreduced Khovanov homology of all 3-strand pretzel links, over the rational numbers.
We relate decategorifications of Ozsváth-Szabó's new bordered theory for knot Floer homology to representations of Uq(gl(1|1)). Specifically, we consider two subalgebras Cr(n, S) and C l (n, S) of Ozsváth-Szabó's algebra B(n, S), and identify their Grothendieck groups with tensor products of representations V and V * of Uq(gl(1|1)), where V is the vector representation. We identify the decategorifications of Ozsváth-Szabó's DA bimodules for elementary tangles with corresponding maps between representations. Finally, when the algebras are given multi-Alexander gradings, we demonstrate a relationship between the decategorification of Ozsváth-Szabó's theory and Viro's quantum relative A 1 of the Reshetikhin-Turaev functor based on Uq(gl(1|1)).
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