Boundary Theta Curves in $\mathbf S^3$

Abstract

It is introduced a split extension of groups $1\rightarrow P_2\rightarrow C_{1,2}\rightarrow \Theta\rightarrow 1$, where $P_2$ is the group of pure braids in 2 strings, $C_{1,2}$ is the group of cobordism classes of (pure) 2-string links and $\Theta$ is the group of cobordism classes of theta curves. The concept of boundary theta curve is introduced and it is proved that the group of boundary cobordism classes of boundary theta curves is isomorphic to the group of boundary cobordism classes of boundary string links in 2 strings.