Robert W. McGrail

Implementation of a solution to the conjugacy problem in Thompson's group F

We present an efficient implementation of the solution to the conjugacy problem in Thompsons group F. This algorithm checks for conjugacy by constructing and comparing directed graphs called strand diagrams. We provide a description of our solution algorithm, including the data structure that represents strand diagrams and supports simplifications.

Tricoloring as a corrective measure (abstract only)

This work presents a two-stage logic program to find tricolorings of a geometric braid [1]. This software is applied to data from the KnotData digital library of the Mathematica [5] symbolic computing system, revealing errors in that collection. The two algorithms presented below have been implemented in SWI-Prolog [4]. The first produces a crossing relation from a Knotdata braid word and the second searches for tricolorings given a crossing relation.

The Word Problem for Finitely Presented Quandles is Undecidable

This work presents an algorithmic reduction of the word problem for recursively presented groups to the word problem for recursively presented quandles. The faithfulness of the reduction follows from the conjugation quandle construction on groups. It follows that the word problem for recursively presented quandles is not effectively computable, in general. This article also demonstrates that a recursively presented quandle can be encoded as a recursively presented rack. Hence the word problem for recursively presented racks is also not effectively computable.

CSPs and Connectedness: P/NP Dichotomy for Idempotent, Right Quasigroups

In the 1990s, Jeavons showed that every finite algebra corresponds to a class of constraint satisfaction problems. Vardi later conjectured that idempotent algebras exhibit P/NP dichotomy: Every non NP-complete algebra in this class must be tractable. Here we discuss how tractability corresponds to connectivity in Cayley graphs. In particular, we show that dichotomy in finite idempotent, right quasi groups follows from a very strong notion of connectivity. Moreover, P/NP membership is first-order axiomatizable in involutory quandles.

Deciding Conjugacy in Thompson's Group F in Linear Time

We present an efficient implementation of the solution to the conjugacy problem in Thompsons group F, a certain infinite group whose elements are piecewise-linear homeomorphisms of the unit interval. This algorithm checks for conjugacy by constructing and comparing directed graphs called strand diagrams. We provide a comprehensive description of our solution algorithm, including the data structure that stores strand diagrams and methods to simplify them. We prove that our algorithm theoretically achieves a linear time bound in the size of the input, and we present a quadratic time working solution.

Tricolorable torus knots are NP-complete

This work presents a method for associating a class of constraint satisfaction problems to a three-dimensional knot. Given a knot, one can build a knot quandle, which is generally an infinite free algebra. The desired collection of problems is derived from the set of invariant relations over the knot quandle, applying theory that relates finite algebras to constraint satisfaction problems. This allows us to develop notions of tractable and NP-complete quandles and knots. In particular, we show that all tricolorable torus knots and all but at most 2 non-trivial knots with 10 or fewer crossings are NP-complete.