nLab
singular knot

Context

Knot theory

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Theorems

Idea

A singular knot is a smooth map (or isotopy class of smooth maps) S 1 3S^1 \to \mathbb{R}^3 which is not an embedding. That is, it is not a knot. Singular knots come in various flavours according to the ways in which it is possible to fail to be an embedding. One of the simplest type of singular knot is where the only failures allowed are simple double points. That is, places where the curve intersects itself transversally, and at the intersection only two threads intersect.

Formally, f:S 1 3f \colon S^1 \to \mathbb{R}^3 has a (simple) double point at x 3x \in \mathbb{R}^3 if f 1(x)={t 1,t 2}f^{-1}(x) = \{t_1, t_2\} with t 1t 2t_1 \ne t_2 and {f(t 1),f(t 2)}\{f'(t_1), f'(t_2)\} are linearly independent.

Singular knots are an important piece of the theory of Vassiliev finite type invariants of knots.

category: knot theory

Revised on August 27, 2015 11:01:27 by Noam Zeilberger (81.253.56.60)