nLab
writhe

Context

Knot theory

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Idea

Writhe is a measure of how much a knot or link writhes around itself. As a 1-dimensional line cannot actually twist, this is not a knot invariant but is an invariant of framed knots (and links). Since a link diagram can be given a natural framing (the blackboard framing), it is possible to compute the writhe of a specific diagram. One place where this is used very neatly is to convert the Kauffman bracket?, which is an invariant of framed links, into the Jones polynomial, being an invariant of ordinary links.

Definition

Recall that a framed link can be thought of as a link together with a normal direction along each component, which we call the framing direction.

Definition

The writhe of a framed link is the linking number of the link with its infinitesimal displacement in the framing direction.

For an oriented link diagram, the writhe is defined using the orientation of the crossings.

Definition

The writhe of an oriented link diagram is defined to be the sum of the orientations of its crossings.

Revised on June 28, 2017 14:52:32 by Anonymous (73.70.160.118)