Spin Link Homology

Abstract

We put a new spin on Khovanov--Rozansky homology. That is, we equip $\hat{I}{›}^n$-colored ${\mathfrak{sl}}_{2n}$ Khovanov--Rozansky homology with an involution whose $\pm 1$-eigenspaces are link invariants. When $n=1,2,3$ (and assuming technical conjectures for $n\ge 4$), we prove that this refined invariant categorifies the spin-colored ${\mathfrak{so}}_{2n+1}$ quantum link polynomial. Along the way, we partially develop the theory of quantum ${\mathfrak{so}}_{2n+1}$ webs and make contact with ${\hat{I}}^1$quantum groups.

137 pages, including 3 appendices; many diagrams; best viewed in color; comments welcome!