| Summary: | In his 1947 work "Theory of Braids" Emil Artin asked whether the braid
group remained unchanged when one considered classes of braids under linkhomotopy,
allowing each strand of a braid to pass through itself but not
through other strands. We generalize Artin's question to string links over
orientable surface M and show that under link-homotopy surface string links
form a group PBn(M), which is isomorphic to a quotient of the surface pure
braid group PBn(M). Surface braid groups and their properties are an area
of active research by González-Meneses, Paris and Rolfsen, Goçalves and
Guaschi, and our work explores the geometric and visual beauty of this
subject. We compute a presentation of PBn(M) in terms of the generators
and relations and discuss the orderability of the group in the case when the
surface in question is a unit disk D. === Science, Faculty of === Mathematics, Department of === Graduate
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