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7 papers · 2 filters
Homology of dihedral quandles
Maciej Niebrzydowski, Jozef H. Przytycki
We solve the conjecture by R. Fenn, C. Rourke and B. Sanderson that the rack homology of dihedral quandles satisfies H_3^R(R_p) = Z \oplus Z_p for p odd prime. We also show that H_…
3-coloring and other elementary invariants of knots
Jozef H. Przytycki
This paper has two-fold goal: it provides gentle introduction to Knot Theory starting from 3-coloring, the concept introduced by R. Fox to allow undergraduate students to see that…
On the first group of the chromatic cohomology of graphs
Milena D. Pabiniak, Jozef H. Przytycki, Radmila Sazdanovic
The algebra of truncated polynomials A_m=Z[x]/(x^m) plays an important role in the theory of Khovanov and Khovanov-Rozansky homology of links. We have demonstrated that Hochschild…
Behavior of knot invariants under genus 2 mutation
Nathan M. Dunfield, Stavros Garoufalidis, Alexander Shumakovitch +1
Genus 2 mutation is the process of cutting a 3-manifold along an embedded closed genus 2 surface, twisting by the hyper-elliptic involution, and gluing back. This paper compares ge…
On colored quandle longitudes and its applications to tangle embeddings and virtual knots
Maciej Niebrzydowski
We define new invariants of knots by means of quandle colorings and longitudinal information. These invariants can be applied to a tangle embedding problem and recognizing non-clas…
Skein modules
Jozef H. Przytycki
We describe in this chapter (Chapter IX) the idea of building an algebraic topology based on knots (or more generally on the position of embedded objects). That is, our basic build…