Work

Infinitely Many Knots Admitting the Same Integer Surgery and a Four-Dimensional Extension

Year: 2015

Type: article

Abstract: We prove that for any integer |$n$| there exist infinitely many different knots in |$S^3$| such that |$n$|-surgery on those knots yields the same |$3$|-manifold. In particular, when |$|n|=1$| homology spheres arise from these surgeries. This answers Problem 3.6(D) on the Kirby problem list. We construct two families of examples, the first by a method of twisting along an annulus and the second by a generalization of this procedure. The latter family also solves a stronger version of Problem 3.6(D), that for any integer |$n$|⁠, there exist infinitely many mutually distinct knots such that 2-handle addition along each with framing |$n$| yields the same |$4$|-manifold.

Source: International Mathematics Research Notices

DOI: 10.1093/imrn/rnv008

Authors: T. Abe, I. D. Jong, J. Luecke, J. Osoinach

Institutions: Tokyo Institute of Technology, Kindai University, The University of Texas at Austin, University of Dallas

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Japan Society for the Promotion of Science

Award: ラスムッセン不変量とスライス・リボン予想について