Lecture 5 (12.10.2015)

Theorem 1.14 (Reidemeister’s Theorem): Two diagrams are equivalent if and only if they are related by a finite sequence of Reidemeister moves.

  • For a proof of this theorem, it is useful to represent geometrical links not by smooth curves in ${\mathbb R}^3$, but rather by points that are connected by straight lines (“stick figures”). One has to check several cases. The proof is not given here, see for example the lecture notes by Roberts.
  • If one considers oriented diagrams, and oriented Reidemeister moves, the theorem still holds.

 

To convince ourselves that the theorem is true, we consider a few other moves, each of which does not change the equivalence class of a diagram. According to the theorem, each of these moves can be represented as sequences of Reidemeister moves 0-III.

moremoves

 

  • Some of these look very similar but are in fact different from the Reidemeister moves listed above! The reason for singling out only a small number of distinguished moves (in fact, a minimal number) is that to check that some property of a geometrical link is invariant under ambient isotopy, all we have to do is to check that the property is invariant under the moves 0)-III).
  • Note that this theorem tells us that two topological links ${\cal L}_1$ and ${\cal L}_2$ are the same if and only if they have diagrams $D_1$ and $D_2$ that are related by a finite sequence of Reidemeister moves. In that case, the same is true for any diagram obtained from any geometrical representation of these links.
  • Move I) will sometimes play a different role than the moves 0), II), III). We therefore define that two diagrams are regularly isotopic if they can be related by a sequence of Reidemeister moves, but using only moves 0), II), III). This is clearly a special case of ambient isotopy, which for diagrams means being connected by a sequence of any of the Reidemeister moves, including move I).
  • We also point out that although Reidemeister’s Theorem tells us that any two equivalent diagrams are related by a finite sequence of Reidemeister moves, it does not tell us how to find such a sequence. It also does not tell us how many moves we need – it is a pure existence theorem.

For the first move in the above list, it was demonstrated in the lecture that

2kink

holds not only as an ambient isotopy, but even as a regular isotopy. That is, move I is not needed for this transformation, moves 0,II,III suffice.

To demonstrate the usefulness of Reidemeister’s Theorem, we now prove that linking number is an invariant.

Proposition 1.15: The linking number is an invariant of oriented 2-links.

Before showing the proof, note that this means in detail: Given an oriented link with two components $K_1$ and $K_2$, then the number $\ell(K_1,K_2)$, computed in any diagram of any projection of the link is always the same. It therefore is a property of the topological 2-link, not just an artefact of the projection or diagram.

Proof of Proposition 1.15: The proof amounts to showing that the linking number is unchanged under any Reidemeister move. Move 0 does not change crossings, and, since linking number is computed exclusively from the crossing, also linking number is unchanged under move 0.

Move I adds or deletes only intersections of one component of the link with itself. But such crossings are not counted in the definition of linking number. Hence also move I does not change linking number.

For Move II, one has to distinguish different cases, depending on the chosen orientations.

m-ii

Depicted here are two choices, first “both orientations up”, then “left up, right down”. In both rows, one checks that the contribution to the linking number (remember that we are only showing part of the link here) is zero. On both right hand sides, there are no crossings, and on the left hand side, we only have a contribution if the two strands belong to different components of the link. Also in that case, however, the contribution to the linking number is zero because the two signs of the crossings cancel. Taking into account the behaviour of linking number under changes of orientation, it is also clear that the same is true in the remaining two cases, where the orientations are chosen as “both down” or “left down, right up”.

For Move III, consider again

move3

Independent of how the orientations on the three strands are chosen, the following is true: If one picks any two of the three strands that are shown, they cross in both the left and right picture in precisely one crossing, and the sign of this crossing is the same on the left and right hand side. This implies that also type III moves do not change the linking number.

According to Reidemeister’s Theorem, when two oriented 2-links ${\cal L}_1\cong{\cal L}_2$ are equivalent, their diagrams are related by a finite sequence of Reidemeister moves. But we have checked that in any such move, linking number does not change. Thus linking number is the same for any diagram of ${\cal L}_1$ and ${\cal L}_2$, i.e. it is an invariant of oriented 2-links. $\square$

 

 

We already saw that the similar but different writhe, defined for any oriented link, is not an invariant — the example we used to show this amounted to demonstrating that type I moves do not leave writhe invariant, but change it by $\pm1 $.

However, the following is true:

Lemma 1.16: Writhe is a link invariant under regular isotopy (but not under ambient isotopy).

The proof is left as an exercise (Problem 4).

 

Exercise 1.17: (Problem 2) Compute the writhe and linking number of both the following oriented link diagrams. Are these diagrams equivalent?

link1

 

 

 

 

Chapter 2: Colourability, the Alexander Polynomial, and Colouring Groups

In this second chapter, we set out to find link invariants that will at least allow us to (finally) prove that the trefoil is inequivalent to the unknot, i.e. to show that different topological knots do exist. In fact, we will find invariants that will be able to distinguish very many knots.

The basic idea is quite ad hoc, and quite simple. It is based on the notion of a colouring of a diagram. We agree to call arc a connected curve in a link diagram that starts end ends at an undercrossing. Since the unknot has no crossings at all, we will also call arc a curve in a link diagram that is a closed loop. Note that in a diagram with no closed loops, there are precisely as many arcs as crossings. To see this, chose an orientation, and map each crossing to that arc that starts on the understrand at this crossing, in the direction given by the arrow. Then one easily checks that this is a 1:1 mapping of crossings to arcs. As an example for the exceptional case where the number of crossings and arcs is different, consider the unknot in its usual circle projection: Here we have one arc but zero crossings.

Starting with three colours (say red, green, blue, abbreviated R,G,B), we call a colouring (or, more precisely, a 3-colouring) an assignment of the colours R,G,B to arcs. That is, each arc is coloured in one solid colour.

Definition 2.1: A diagram is called 3-colourable if it has a colouring such that

  • at least two colours are used (in other words, the monochromatic colourings are not allowed)
  • at each crossing, either the three meeting strands have all the same colour, or they have all different colours.

We will show next time that the property of being 3-colourable is an invariant under ambient isotopy. Hence we can use it to distinguish topolical links. Our first example is the diagram of the unknot,

unknot

which is not 3-colourable because we cannot colour it with more than one colour, and thus have to violate rule 1 of Def. 2.1. In comparison, the trefoil is 3-colorable, as is clear from its diagram

tref-color

which does comply with the conditions in Def. 2.1.