Singular TQFTs, Foams and Type D Arc Algebras

Abstract

We combinatorially describe the 2-category of singular cobordisms, called (rank one) foams, which governs the functorial version of Khovanov homology. As an application we topologically realize the type D arc algebra using this singular cobordism construction.

On the functoriality of 𝔰𝔩2 tangle homology

“…It is well defined, because the rank of the morphism space depends only on the images of webs in the Grothendieck group. Theorem 2.11 and the isomorphism (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) imply together that hOE! ; OE!…”

“…is a cylinder with a red membrane, the canonical normal vector of which is oriented upwards. The membrane can be removed with (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) and (2)(3)(4)(5)(6)(7)(8). This gives no difference for the cup with no dot…”

“…The middle foam is again a cup foam, but not the right one: the projection has double points -the shaded region outside of the cupcoming from the disk to the side of the vertical plane. Indeed, the left moves in (5-3) and are realized by detaching red cylinders with (2-9), whereas the right ones are by eliminating red caps with (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16). Likewise, the relations (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) and (2)(3)(4)(5)(6)(7)(8)(9)(10) give the signs for the left and right sides respectively of both (5-5) and .…”

“…each consisting of the identity foams .c C / Š OE0; 1 and a C OE0; 1 glued to the halfrotation of b C around the boundary line. These foams take a particularly nice form when B is a red-over-blue basis, as they involve then only three types of moves: merging (5-10) and splitting (5-11) blue loops at points outside of all red circles, merging unnested red loops (5)(6)(7)(8)(9)(10)(11)(12)(13)(14), and removing bigons external to the projection of red disks (5-1).…”

“…In order to obtain a strictly functorial quantized annular homology, we wanted first to understand the Ehrig-Stroppel-Tubbenhauer isomorphism between Khovanov's arc algebras and their web algebras, and then reconstruct the Chen-Khovanov functor in the framework of webs and foams. However, after a chain of simplifications of their arguments, especially replacing the foam basis used in [13] with another one, more natural from the topological perspective, we understood the real reason why all the isomorphisms popped out: foams and cobordisms constitute equivalent bicategories. Despite sounding as a natural thing, the result is by no means obvious -compare eg Lauda, Queffelec and Rose [23,Section 3.1.2] or [14, page 186].…”