ScalingStacks

Remark 4.3. It is instructive to evaluate the inverse to the canonical isomorphisms from Corollary 4.2 on surfaces of revolution generated by links. Any framed, oriented link $K\subset B^{3}$ or $S^{3}$ defines a vegetarian¹¹ 1 A lasagna filling consisting only of a surface, without input meat balls. lasagna filling $S^{1}\times K$ of $S^{1}\times B^{3}$, which evaluates to a scalar multiple of the empty lasagna filling. It follows from the proofs of Lemma 4.1 and Corollary 4.2 that this scalar is the trace of the identity map on $\operatorname{KhR}_{N}(K)$. Here it is important to take the Koszul signs in the symmetric monoidal structure on (homologically and quantum) bigraded vector spaces into account. The trace is thus $\operatorname{tr}(\operatorname{Id}_{\operatorname{KhR}_{N}(K)})=\chi_{q=1}(\operatorname{KhR}_{N}(K))=\pm N^{|\pi_{0}(K)|}$ , i.e. the $\mathfrak{gl}_{N}$ quantum link polynomial of $K$, specialized at $q=1$. More generally, any endocobordism of $K$ defines a lasagna filling of $S^{1}\times B^{3}$ that is a multiple of the empty filling, with coefficient given by the graded trace of the induced endomorphism of $\operatorname{KhR}_{N}(K)$; see e.g. [16, Section 6], [3, Section 10.1], [7, Theorem D] for related discussions of Lefschetz traces in the case of Khovanov homology.