Knot cobordism maps and functoriality — Examples and Counterexamples

Let $\mathcal{A}=\mathbb{Z}[X]\mathrm{/}(X^2)$ be the Frobenius algebra with unit $\eta (1)=1$ and multiplication $m$. In the algebraic TQFT, consider the movie that first creates a disjoint circle and then attaches a saddle band merging that circle with one existing circle. The induced local map is

$$\mathcal{A}\stackrel{{\mathrm{id}}_{\mathcal{A}}\otimes \eta }{\to }\mathcal{A}\otimes \mathcal{A}\stackrel{m}{\to }\mathcal{A},$$

and this composite is the identity on $\mathcal{A}$.

The diagram records the two successive surface events; it is only a description of the movie, while the verification below is algebraic.

diagram failed to render:
\begin{tikzcd}
\mathcal A
\arrow[r, "{\operatorname{id}\otimes\eta}"]
& \mathcal A\otimes\mathcal A
\arrow[r, "m"]
& \mathcal A
\end{tikzcd}

The integral Frobenius algebra and local cobordism maps

Khovanov, A categorification of the Jones polynomial, §7

Bar-Natan, Khovanov's homology for tangles and cobordisms, §§7–8

Kauffman, An Introduction to Khovanov Homology, §5

The assertion that a birth of a disjoint circle followed by a death of that same circle induces the identity up to an overall sign on an unaffected circle is false. Let $\mathcal{A}=\mathbb{Z}[X]\mathrm{/}(X^2)$ with unit $\eta (1)=1$ and counit $ϵ(1)=0,ϵ(X)=1$. On a movie with one unaffected circle, birth a disjoint circle and then cap that same circle by a death. The composite operator is

$$\mathcal{A}\stackrel{{\mathrm{id}}_{\mathcal{A}}\otimes \eta }{\to }\mathcal{A}\otimes \mathcal{A}\stackrel{{\mathrm{id}}_{\mathcal{A}}\otimes ϵ}{\to }\mathcal{A},$$

which is zero and therefore is not the identity, even after allowing multiplication by an overall sign.

diagram failed to render:
\begin{tikzcd}
\mathcal A
\arrow[r, "{\operatorname{id}\otimes\eta}"]
& \mathcal A\otimes\mathcal A
\arrow[r, "{\operatorname{id}\otimes\epsilon}"]
& \mathcal A
\end{tikzcd}

The movie is a cylinder on the original circle disjoint union a born-and-capped sphere; the diagram specifies the local birth and death sequence and is not used as a proof.

The unit, counit, and integral Frobenius algebra

Khovanov, A categorification of the Jones polynomial, §7

Bar-Natan, Khovanov's homology for tangles and cobordisms, §§7–8

Kauffman, An Introduction to Khovanov Homology, §5