Algorithmic Problems in the Braid Group

The study of braid groups and their applications is
a field which has attracted the interest of
mathematicians and computer scientists alike. We
begin with a review of the notion of a braid group.
We then discuss known solutions to decision problems
in braid groups. We then prove new results in braid
group algorithmics. We offer a quick solution to the
generalized word problem in braid groups, in the
special case of cyclic subgroups. We illustrate this
solution using a multitape Turing machine. We then
turn to a discussion of decision problems in cyclic
amalgamations of groups and solve the word problem
for the cyclic amalgamation of two braid groups. We
then turn to a more general study of the conjugacy
problem in cyclic amalgamations. We revise and prove
some theorems of Lipschutz and show their
application to cyclic amalgamations of braid groups.
We generalize this application to prove a new
theorem regarding the conjugacy problem in cyclic
amalgamations.
We then discuss some application of braid groups,
culminating in a section devoted to the discussion
of braid group cryptography.

Autorentext

Dr. Elie Feder received his Ph.D. from the Graduate Center of
City University of New York (CUNY) in 2003. He is currently an
assistant professor at Kingsborough Community College of CUNY.
His areas of research include: graph theory, combinatorics and
combinatorial group theory.

Klappentext

The study of braid groups and their applications is
a field which has attracted the interest of
mathematicians and computer scientists alike. We
begin with a review of the notion of a braid group.
We then discuss known solutions to decision problems
in braid groups. We then prove new results in braid
group algorithmics. We offer a quick solution to the
generalized word problem in braid groups, in the
special case of cyclic subgroups. We illustrate this
solution using a multitape Turing machine. We then
turn to a discussion of decision problems in cyclic
amalgamations of groups and solve the word problem
for the cyclic amalgamation of two braid groups. We
then turn to a more general study of the conjugacy
problem in cyclic amalgamations. We revise and prove
some theorems of Lipschutz and show their
application to cyclic amalgamations of braid groups.
We generalize this application to prove a new
theorem regarding the conjugacy problem in cyclic
amalgamations.
We then discuss some application of braid groups,
culminating in a section devoted to the discussion
of braid group cryptography.