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A Non-classification Result for Wild Knots

Vadim Kulikov

Abstract

Using methods of descriptive theory it is shown that the classification problem for wild knots is strictly harder than that for countable structures.

A Non-classification Result for Wild Knots

Abstract

Using methods of descriptive theory it is shown that the classification problem for wild knots is strictly harder than that for countable structures.

Paper Structure

This paper contains 4 sections, 26 theorems, 110 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Embedding $\cong_{{\operatorname{LO}}}$ into $E_K$
  4. Reducing a Turbulent Relation into $E_K$

Key Result

Theorem 2.4

$\cong_{{\operatorname{LO}}}$ is $\leqslant_B$-maximal among all isomorphism relations, i.e. $\cong_{S(L)}\ \leqslant_B \ \cong_{{\operatorname{LO}}}$ holds for any vocabulary $L$. ∎

Figures (2)

  • Figure 1: The singular arc whose copies are concatenated to form the knot $F(L)$. Figure produced by MetaPost.
  • Figure 2: An illustration of Proposition \ref{['prop:SequencesOfEmbeddings']}.

Theorems & Definitions (55)

  • Definition 2.1
  • Example 2.3
  • Theorem 2.4: H. Friedman and L. Stanley FrSt
  • Definition 2.5: Hjorth
  • Theorem 2.6: Hjorth
  • Definition 2.7
  • Theorem 2.8
  • Definition 2.9
  • Definition 2.10
  • Theorem 3.1
  • ...and 45 more