MathDB

Wrapping number conjecture

For every annular link LL in a solid torus, the Kauffman bracket skein element ⟨L⟩\langle L\rangle has a nonzero homogeneous component in annular degree wrap⁡(L)\operatorname{wrap}(L); equivalently, deg⁡ann⟨L⟩=wrap⁡(L)\deg_{\mathrm{ann}}\langle L\rangle=\operatorname{wrap}(L), where wrap⁡(L)\operatorname{wrap}(L) is the wrapping number of LL.

References

Primary source

arXiv

Additional references

knot-theorylow-dimensional-topology
Posted · edited

Progress summary

Refreshed
Claimed solved

A September 2026 unrefereed preprint claims a concrete annular-knot counterexample, so the conjecture is false, but the result has not been independently verified.

Hoste and Przytycki posed the conjecture in 1995: the Kauffman-bracket skein module of an annular link should be nonzero in annular degree wrap⁡(L)\operatorname{wrap}(L).

Known results

Hoste and Przytycki proved the assertion for adequately wrapped annular links. A preprint dated August 9, 2026, proves a categorified analogue for links admitting a perfectly wrapped uniform resolution, including alternating annular links, but does not settle the original conjecture.

September 2026 counterexample

Qiuyu Ren's preprint, reported September 10, 2026, claims an explicit annular knot whose wrapping number differs from the annular degree of its Kauffman bracket. If correct, this disproves the proposed equality and identifies a concrete obstruction to recovering wrapping number from bracket degree.

Current status (as of September 2026): The conjecture has a claimed counterexample, but that unrefereed preprint remains unverified; the categorified analogue is proved only in stated classes.

Sources

Solutions 0

No solutions have been posted yet.