Mohamed Elhamdadi

Cocycle knot invariants from quandle modules and generalized quandle homology

Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Grana. We specialize that theory to the case when there is a group action on the coefficients. First, quandle modules are used to generalize Burau representations and Alexander modules for classical knots. Second, 2-cocycles valued in non-abelian groups are used in a way similar to Hopf algebra invariants of classical knots. These invariants are shown to be of quantum type. Third, cocycles with group actions on coefficient groups are used to define quandle cocycle invariants for both classical knots and knotted surfaces. Concrete computational methods are provided and used to prove non-invertibility for a large family of knotted surfaces. In the classical case, the invariant can detect the chirality of 3-colorable knots in a number of cases.

Twisted quandle homology theory and cocycle knot invariants

The quandle homology theory is generalized to the case when the coecient groups admit the structure of Alexander quandles, by includ- ing an action of the innite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to group extension theories for group cocycles. Ex- plicit formulas for cocycles corresponding to extensions are given, and used to prove non-triviality of cohomology groups for some quandles. The cor- responding generalization of the quandle cocycle knot invariants is given, by using the Alexander numbering of regions in the denition of state- sums. The invariants are used to derive information on twisted cohomology groups. AMS Classication 57N27, 57N99; 57M25, 57Q45, 57T99

Extensions of Quandles and Cocycle Knot Invariants

Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and is extended to links component-wise.

Quandle colorings of knots and applications

We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of 23 finite quandles. We study the colorings of these 2977 knots by all of the 431 connected quandles of order at most 35 found by Vendramin. Among other things, we collect information about quandles that have the same number of colorings for all of the 2977 knots. For example, we prove that if Q is a simple quandle of prime power order then Q and the dual quandle Q* of Q have the same number of colorings for all knots and conjecture that this holds for all Alexander quandles Q. We study a knot invariant based on a quandle homomorphism f : Q1 → Q0. We also apply the quandle colorings we have computed to obtain some new results for the bridge index, the Nakanishi index, the tunnel number, and the unknotting number. In an appendix we discuss various properties of the quandles in Vendramins list. Links to the data computed and various programs in C, GAP and Maple are provided.

Augmented biracks and their homology

We introduce augmented biracks and define a (co)homology theory associated to augmented biracks. The new homology theory extends the previously studied Yang–Baxter homology with a combinatorial formulation for the boundary map and specializes to N-reduced rack homology when the birack is a rack. We introduce augmented birack 2-cocycle invariants of classical and virtual knots and links and provide examples.

Connected quandles associated with pointed abelian groups

A quandle is a self-distributive algebraic structure that appears in quasigroup and knot theories. For each abelian group A and c2 A, we define a quandle G. A; c/ on Z3 A. These quandles are generalizations of a class of nonmedial Latin quandles defined by V. M. Galkin, so we call them Galkin quandles. Each G. A; c/ is connected but not Latin unless A has odd order. G. A; c/ is nonmedial unless 3 AD 0. We classify their isomorphism classes in terms of pointed abelian groups and study their various properties. A family of symmetric connected quandles is constructed from Galkin quandles, and some aspects of knot colorings by Galkin quandles are also discussed.

VIRTUAL KNOT INVARIANTS FROM GROUP BIQUANDLES AND THEIR COCYCLES

A group-theoretical method, via Wada’s representations, is presented to distinguish Kishino’s virtual knot from the unknot. Biquandles are constructed for any group using Wada’s braid group representations. Cocycle invariants for these biquandles are studied. These invariants are applied to show the non-existence of Alexander numberings and checkerboard colorings. The purposes of this paper include defining biquandle structures on groups, and giving a grouptheoretic proof that Kishino’s virtual knot is non-trivial. A biquandle structure or a birack structure is related to solutions to the set-theoretic Yang-Baxter equation (SYBE). Given an invertible solution that satisfies an additional condition (corresponding to a Reidemeister type I move), we obtain a biquandle, and every biquandle gives a solution to the SYBE. Most examples that were known up to this point came from generalizations of the Burau representation. The principal examples that we consider in this paper come from Wada’s representations of braid groups as free group automorphisms. Our first example indicates that one of these representations can be used to distinguish Kishino’s virtual knot from the unknot. By abelianizing such groups, we recover an analog of Burau matrix. Using these examples, we construct and calculate cocycle invariants that come from the homology theory of biquandles. As applications we give obstructions to checkerboard colorability and (mod 2)-Alexander numberings of arcs of virtual knots. In Section 2 we examine Wada’s group invariants for virtual knots. The biquandle structures are defined on any group in Section 3 using Wada’s representations, and colorings of virtual knot diagrams by such biquandles are studied. Cocycle invariants are constructed and applied in Section 4.

Cohomology of the adjoint of Hopf algebras

A cohomology theory of the adjoint of Hopf algebras, via deformations, is presented by means of diagrammatic techniques. Explicit calculations are provided in the cases of group algebras, function algebras on groups, and the bosonization of the super line. As applications, solutions to the YBE are given and quandle cocycles are constructed from groupoid cocycles.

Foundations of topological racks and quandles

We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then introduced providing a first step towards a general continuous cohomology theory for topological racks and quandles.

Tangle Embeddings and Quandle Cocycle Invariants

To study embeddings of tangles in knots, we use quandle cocycle invariants. Computations are carried out for tables of knots and tangles to investigate which tangles may or may not embed in knots in the tables.