ScalingStacks

In [9] to a plane diagram $D$ of an oriented tangle $T$ with $2n$ bottom and $2m$ top endpoints we associated a complex ${\mathcal{F}}(D)$ of $(H^{m},H^{n})$-bimodules, for certain rings $H^{n}.$ We proved that the isomorphism class of this complex in the homotopy category is an invariant of $T.$ In this paper we give a short argument that our construction yields an invariant of tangle cobordisms. To a diagram of an oriented cobordism between diagrams $D_{1}$ and $D_{2}$ of tangles $T_{1}$ and $T_{2}$ we assign a homomorphism of complexes ${\mathcal{F}}(D_{1})\to{\mathcal{F}}(D_{2})$ and then check that (in the homotopy category) this homomorphism depends on the choice of a diagram of the cobordism only up to the overall minus sign. The result follows from the basic properties of rings $H^{n}$ and $H^{n}$-bimodules assigned to tangle diagrams.

For link cobordisms this result was recently obtained by Magnus Jacobsson [6].

In a previous paper [10] we conjectured that such an invariant exists over the ring $\mathbb{Z}[c].$ This conjecture, which should be understood ”rel boundary” (as emphasized by Jacobsson [6]), remains open. Jacobsson established the $c=0$ specialization, and it also follows from this work.