ScalingStacks

Definition 6.1. The category $\boldsymbol{\mathrm{TD}}$ of oriented tangle diagrams has objects given by finite words in the alphabet $\{\uparrow,\downarrow\}$, including the empty word. The morphisms are admissible words in the alphabet $\{\textrm{cup}_{i},\textrm{cap}_{i},\textrm{crossing}_{i},\textrm{crossing}^{-1}_{i}\}_{i\geq 0}$ of generating morphisms.

The realization $r(t)$ of a morphism $t\colon A\to B$ is a tangle diagram drawn in the square ${[0,1]}\times{[0,1]}$ by first placing the words $A$ and $B$ as collections of oriented tangle endpoints on ${[0,1]}\times\{0\}$ and ${[0,1]}\times\{1\}$ respectively, and then constructing an oriented tangle diagram starting from the bottom $A$ by attaching cups, caps, crossings or inverse crossings with $i$ parallel strands to the left, as specified by the $t$. The word $t$ is defined to be admissible if this procedure succeeds in generating an oriented tangle diagram. We will consider these diagrams up to individually rescaling the $x$- and $y$-coordinates in ${[0,1]}\times{[0,1]}$ by orientation-preserving diffeomorphisms of ${[0,1]}$. As such, every morphism in $\boldsymbol{\mathrm{TD}}$ has a unique realization, and we say that the morphism is the Morse data of the oriented tangle diagram.

The composition of morphisms in $\boldsymbol{\mathrm{TD}}$ is given by concatenating lists of generating morphisms.