Linear Bound for Kakimizu Complex Diameter of Hyperbolic Knots

Speaker: 陈啸 (YMSC)

Time: 2025-12-04 14:00-16:00
Location: 610, SIMIS
Zoom Meeting ID: 827 1983 6412 Passcode: SIMIS

摘要：

Given a knot K (or a link L) in S^3, a connected, orientable spanning surface of minimal genus is called a minimal-genus Seifert surface. The complex MS(K) introduced by Kakimizu is a simplicial complex whose 0-simplices are isotopy classes of minimal-genus Seifert surfaces, and whose n-simplices are collections of n distinct disjoint such classes.

Recent work by Agol and Zhang developed a related concept called guts, revealing deep connections between the structure of MS(K), the hyperbolic volume of the knot complement, and how far a non-fibered knot is from being fibered.

Over a decade ago, Sakuma–Shackleton and Pelayo established a quadratic upper bound for the diameter of the Kakimizu complex of an atoroidal knot in terms of the knot genus. Sakuma–Shackleton asked whether this bound could be improved to a linear one.

We show that the upper bound of the diameter of the Kakimizu complex of an atoroidal knot grows linearly with the knot genus g . Specifically, the diameter is at most 2 when g = 1, at most 6 when g = 2, and at most 4g −3 for g ≥ 3. This confirms a conjecture of Sakuma–Shackleton. This work is joint with Wujie SHEN.