COUNTING FUNDAMENTAL PATHS IN CERTAIN GARSIDE SEMIGROUPS
Christopher R. Cornwell, Stephen P. Humphries · Journal of Knot Theory and Its Ramifications · 2008
For elements a, b of a monoid, define the word pk(a,b) = abab⋯ of length k. We find the number of words in a, b which are equal to pk(a,b)n in the Artin semigroup . This number is related to counting certain paths in the ℕ × ℕ lattice. These Artin groups are examples of two generator Garside groups. We also define other examples of Garside groups G on more than two generators, having fundamental word Δ, and similarly find the number of words equal in G to Δn.