Mazur's knot and the Octahedron

Abstract

Mazur's knot exterior in $S^1\times S^2$ admits a geometric description using a single regular ideal octahedron. The resulting hyperbolic structure is closely related to the Whitehead link exterior through Adams' theorem on thrice-punctured spheres. The same octahedral framework applies to the family of Jester manifolds introduced by Sparks. Using hyperbolic geometry, hyperbolic Dehn filling, and recent results on systolic geodesics, we prove that the boundaries of all Mazur and Jester manifolds are pairwise nonhomeomorphic, regardless of orientation. Consequently, the corresponding compact, contractible $4$-manifolds are pairwise nonhomeomorphic.