Categorified Wrapping Number Conjecture

The conjecture, posed by Eli Grigsby in 2010, predicts that an annular link $L$ has a nonzero annular Khovanov-homology class in grading $\mathrm{wrap}(L)$. It remains a question for arbitrary annular links.

Known results

• Grigsby’s proposed 2010 proof was undermined by examples found independently by Matt Hedden and Stephan Wehrli soon afterward.
• The conjecture holds for alternating annular links (Daniels–Zhang, 2025).
• It holds whenever a diagram admits a perfectly wrapped, uniform resolution, including adequately wrapped links and further families (Daniels–Zhang, 2025).

January 2025 theorem; August 2026 publication report

Benjamin Daniels and Melissa Zhang proved the conjecture for links admitting perfectly wrapped, uniform resolutions and derived the alternating-link case. The later journal-publication report describes broad annular-link classes, but the work explicitly leaves open whether every annular link admits such a resolution; therefore it is substantial partial progress, not a general solution.

Current status (as of August 2026): The conjecture is settled for alternating links and other broad families, while the general case for all annular links remains open.