nLab
longitude of a link component

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Introduction

The longitude of a knot, or more generally of a component of a link, plays a crucial role in the link-theoretic approach to 3-manifolds by means of the Lickorish-Wallace theorem? and the Kirby calculus. It can be defined either geometrically or combinatorially.

Combinatorial definition

The longitude of a knot or of a link component can be defined purely within diagrammatic knot theory. We describe this in this section.

Definition

Let CC be a component of a link diagram LL, viewed as defining a link with the blackboard framing?. Pick an orientation? of LL, and pick a point pp of CC. The longitude of CC with respect to pp and the chosen orientation of LL is the word in the free group of the set of arcs? of LL defined inductively as follows.

1) Begin at pp with the empty word.

2) Walk along LL following the orientation of LL until one reaches a crossing of LL which one approaches by means of an under-edge (one does not stop at crossings which one approaches and leaves by means of over-edges). If the orientations of the crossings are as follows, add aa to the end of the word obtained thus far.

If the orientations of the arcs of the crossing are instead as follows, add a 1a^{-1} to the end of the word obtained thus far.

The arc aa is not required to, and may not, belong to CC.

3) Repeat Step 2) until we return to pp.

Remark

There is an alternative definition if one works with framed link diagrams?, which involves first replacing one’s original link diagram with one to which a certain number of twists (i.e. R1 moves) have been applied according to the framing, and then using Definition .

Example

Consider the trefoil with a chosen point pp and orientation as shown below. Labels have been chosen for the arcs.

The longitude of the trefoil with respect to pp and this orientation is then cabc a b.

Example

Consider the Hopf link, in which both components have been equipped with a chosen point and an orientation as shown below. Labels have been chosen for the arcs.

The longitude of the component to which the arc aa belongs is b 1b^{-1}. The longitude of the component to which the arc bb belongs is a 1a^{-1}.

Remark

The longitude of a component of an oriented link diagram is unique up to rotation permutation: the longitude obtained using any one choice of pp is a rotation permutation of the longitude obtained using any other choice of pp.

When it comes to 3-manifolds, for example when describing the fundamental group, everything is typically invariant under rotation permutation. Thus it is usual to speak of the longitude of a component of an oriented link diagram.

category: knot theory

Last revised on January 18, 2019 at 08:21:20. See the history of this page for a list of all contributions to it.