ScalingStacks

A $(1,1)$-tangle is a proper smooth embedding $e:T\hookrightarrow\mathbb{R}^{2}\times[0,1]$ of a finite collection $T$ of circles and one interval $[0,1]$ into $\mathbb{R}^{2}\times[0,1]$ such that the boundary points of $[0,1]$ go to the corresponding boundary component of $\mathbb{R}^{2}\times[0,1]:$

Two $(1,1)$-tangles are called equivalent if they are isotopic via an isotopy that fixes the boundary. Oriented $(1,1)$-tangles are $(1,1)$-tangles with a chosen orientation of each component, orientation of $[0,1]$ always chosen in the direction from $0$ to $1.$

Define a marked oriented link (in $\mathbb{R}^{3}$) as an oriented link with a marked component. Obviously, there is a natural one-to-one correspondence between marked oriented links and oriented $(1,1)$-tangles: the closure of a $(1,1)$-tangle is a marked oriented link. Denote this map from oriented $(1,1)$-tangles to oriented marked links by $\mbox{cl}.$