An unoriented skein exact triangle for unoriented grid homology.
Abstract: In a previous paper, the author gave a combinatorial proof of the unoriented skein exact triangle for knot Floer homology, first proven by Ciprian Manolescu, using the isomorphic grid homology. More recently, Ozsváth–Stipsicz–Szabó have defined unoriented knot Floer homology, corresponding to the t=1 case of modified knot Floer homology. Unoriented grid homology can be defined similarly. In this paper, we extend the unoriented skein exact triangle for grid homology to the case of unoriented grid homology.
An unoriented skein relation for tangle Floer homology, with Ina Petkova.
Abstract: In a previous paper, Vértesi and the first author used grid-like Heegaard diagrams to define tangle Floer homology, which associates to a tangle T a differential graded bimodule CT(T). If L is obtained by gluing together T_1, ..., T_m, then the knot Floer homology HFK(L) of L can be recovered from CT(T_1), . . . , CT(T_m). In another paper, the second author reproved the existence of an unoriented skein exact triangle for HFK combinatorially using grid diagrams. In the present paper, we prove that tangle Floer homology satisfies an unoriented skein relation, generalizing the skein exact triangle for HFK.
Grid diagrams and Manolescu's unoriented skein exact triangle for knot Floer homology.
Algebr. Geom. Topol. 17 (2017), no. 3, 1283--1321.
Abstract: We re-derive Manolescu's unoriented skein exact triangle for knot Floer homology over F_2 combinatorially using grid diagrams, and extend it to the case with Z coefficients by sign refinements. Iteration of the triangle gives a cube of resolutions that converges to the knot Floer homology of an oriented link. Finally, we re-establish the homological sigma-thinness of quasi-alternating links.
From self-similar structures to self-similar groups, with Daniel J. Kelleher and Benjamin A. Steinhurst.
Internat. J. Algebra Comput. 22 (2012), no. 7, 1250056, 16 pages.
Abstract: We explore the relationship between limit spaces of contracting self-similar groups and self-similar structures. We give the condition on a contracting group such that its limit space admits a self-similar structure, and also the condition such that this self-similar structure is p.c.f. We then give the necessary and sufficient condition on a p.c.f. self-similar structure such that there exists a contracting group whose limit space has an isomorphic self-similar structure; in this case, we provide a construction that produces such a contracting group. Finally, we illustrate our results with several examples.