nLab longitude of a link component

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

1) Begin at $p$ with the empty word.

2) Walk along $L$ following the orientation of $L$ until one reaches a crossing of $L$ 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 $a$ to the end of the word obtained thus far.

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

The arc $a$ is not required to, and may not, belong to $C$.

3) Repeat Step 2) until we return to $p$.

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 $p$ and orientation as shown below. Labels have been chosen for the arcs.

The longitude of the trefoil with respect to $p$ and this orientation is then $c a b$.

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

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 $p$ is a rotation permutation of the longitude obtained using any other choice of $p$.

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