Foam evaluation and Kronheimer–Mrowka theories

Anchored foams and annular homology

“…This connection was made the most transparent by the work of Robert and Wagner [RW20], who wrote down a combinatorial formula for GL(N ) closed foam evaluation that allows to build GL(N ) link homology from the ground up, bypassing categorical approaches to the latter. A variation of their formula was used to evaluate unoriented SL(3) foams [KR21], giving a combinatorial approach to some of the structures discovered by Kronheimer and Mrowka [KM19b].…”

“…Our construction complements earlier work [QR18,QW21] on the subject. The same approach allows to define state spaces for unoriented SL(3) annular webs, extending the construction in [KR21].…”

“…We show that this modified evaluation, with anchor points contributing δ i,j ±f (x i ), perfectly matches the structure of state spaces of annular homology, in SL(2) and SL(3) cases. The construction also allows us to define unoriented SL(3) homology for annular trivalent graphs, extending [KR21] to the annular framework.…”

“…With state spaces at hand, it is straightforward to define annular SL(2) and SL(3) link homology, by analogy with [Kho00, APS04, BN05, Akh20] in the SL(2) setting, with [KR21] in the unoriented SL(3) setting, and with [Kho04, MV07,RW20] in the oriented SL(3) setting. State spaces and link homology carry additional gradings coming from intersection points of foams with the z-axis.…”

“…Section 3 constructs the state spaces for the annular unoriented SL(3) foam theory, extending the construction of [KR21]. We start with the evaluation (Section 3.1), followed by skein relations on annular foams (Section 3.2) and properties of state spaces (Section 3.3).…”