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An RBG construction of integral surgery homeomorphisms

Qianhe Qin

Abstract

We generalize the RBG construction of Manolescu and Piccirillo to produce pairs of knots with the same $n$-surgery, and investigate the possibility of constructing exotic definite four-manifolds using $n$-surgery homeomorphisms.

An RBG construction of integral surgery homeomorphisms

Abstract

We generalize the RBG construction of Manolescu and Piccirillo to produce pairs of knots with the same -surgery, and investigate the possibility of constructing exotic definite four-manifolds using -surgery homeomorphisms.
Paper Structure (11 sections, 26 theorems, 11 equations, 15 figures)

This paper contains 11 sections, 26 theorems, 11 equations, 15 figures.

Table of Contents

  1. Introduction
  2. n-special RBG links
  3. n-peculiar RBG links
  4. $\langle n\rangle$-surgery homeomorphisms
  5. $n$-slice knots from $n$-surgery homeomorphisms
  6. Extendability over $n$-traces
  7. $n$-special RBG links
  8. potential exotica
  9. $n$-special RBG link and the $s$-invariant
  10. Experiments
  11. $n$-peculiar RBG links

Key Result

Theorem 1.3

Any $|n|$-RBG link $L=\{(R,r),(B,b),(G,g)\}$ has an associated knot pair $(K_B, K_G)$ and a homeomorphism $\phi_L: S^3_{f_b}(K_B) \rightarrow S^3_{f_g}(K_G)$ with $f_b,f_g\in\{n,-n\}$. Conversely, given a homeomorphism $\phi: S_l^3(K)\rightarrow S^3_{m}(J)$ with $l,m\in\{n,-n\}$, there exists an $|n

Figures (15)

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  • ...and 10 more figures

Theorems & Definitions (68)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 1.8
  • Definition \ref{def:RBG}
  • Remark \ref{def:RBG}
  • ...and 58 more