Linear Bound for Kakimizu Complex Diameter of Hyperbolic Knots

Abstract: 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 $g \geq 3$. This confirms a conjecture of Sakuma--Shackleton.