TY - JOUR
T1 - A presentation for the pure Hilden group
AU - Tawn, Stephen
PY - 2013
Y1 - 2013
N2 - Consider the half ball, B3 +, containing n unknotted and unlinked arcs a1, a2, . . . , an such that the boundary of each ai lies in the plane. The Hilden (or Wicket) group is the mapping class group of B3 + fixing the arcs a1 an setwise and fixing the half sphere S2 + pointwise. This group can be considered as a subgroup of the braid group on 2n strands. The pure Hilden group is defined to be the intersection of the Hilden group and the pure braid group. In a previous paper, we computed a presentation for the Hilden group using an action of the group on a cellular complex. This paper uses the same action and complex to calculate a finite presentation for the pure Hilden group. The framed braid group acts on the pure Hilden group by conjugation and this action is used to reduce the number of cases.
AB - Consider the half ball, B3 +, containing n unknotted and unlinked arcs a1, a2, . . . , an such that the boundary of each ai lies in the plane. The Hilden (or Wicket) group is the mapping class group of B3 + fixing the arcs a1 an setwise and fixing the half sphere S2 + pointwise. This group can be considered as a subgroup of the braid group on 2n strands. The pure Hilden group is defined to be the intersection of the Hilden group and the pure braid group. In a previous paper, we computed a presentation for the Hilden group using an action of the group on a cellular complex. This paper uses the same action and complex to calculate a finite presentation for the pure Hilden group. The framed braid group acts on the pure Hilden group by conjugation and this action is used to reduce the number of cases.
UR - http://handle.uws.edu.au:8081/1959.7/534134
UR - http://arxiv.org/pdf/0902.4840.pdf
U2 - 10.4310/MRL.2013.v20.n1.a15
DO - 10.4310/MRL.2013.v20.n1.a15
M3 - Article
SN - 1073-2780
VL - 20
SP - 181
EP - 202
JO - Mathematical Research Letters
JF - Mathematical Research Letters
IS - 1
ER -