Lecture 4 (06.10.2015)

We begin with some remarks regarding signs of crossings and linking number, defined last time.

  • Intuitively speaking, $\ell(K_1,K_2)$ measures “how often $K_1$ winds around $K_2$”.
  • A topological 2-link is called splittable if it is equivalent to a link with one component on each side of the plane $\{x=0\}\subset{\mathbb R}^3$ (That is, the link “falls apart”). If a 2-link is splittable, then its linking number is zero. But if the linking number of a 2-link $\cal L$ is zero, that does not necessarily mean that the linking number of $\cal L$ vanishes. An example of this is given by the Whitehead link:whitehead
  • For a topological knot $\cal K$, let $r\cal K$ denote the same knot, but with the opposite orientation. We then have $$w(r{\cal K})=w({\cal K})$$ by the definition of writhe, and $$\ell(r{\cal K}_1,r{\cal K}_2)=\ell({\cal K}_1,{\cal K}_2)=-\ell(r{\cal K}_1,{\cal K}_2)=-\ell({\cal K}_1,r{\cal K}_2)$$ for the 2-link with components ${\cal K}_1$ and ${\cal K}_2$. This follows directly by checking how the sign of a crossing changes when one or both orientations of the crossing strands is flipped.

 


 

Last time, we introduced diagrams as a tool to describe knots and links without drawing three-dimensional pictures. We saw that a topological link $\cal L$ has a multitude of diagrams, but can be recovered from any of these uniquely. What we have not yet clarified is how the various diagrams of $\cal L$ are related – this is what we do now.

To begin with, we discuss four different kinds of changes that can be applied to a diagram without changing its link. These moves — in particular, moves I-III — are called the Reidemeister moves.

0) The first (or rather, the “zeroth” one in the usual notation) move is called planar isotopy. It amounts to moving, stretching, curling a diagram without tearing the line and without changing any crossings of the diagram. Here are examples of some trefoil diagrams that are all related by type 0 moves:
move0I) The first Reidemeister move is usually depicted as
move1Here the dashed circle is not part of the link diagram, it indicates that only a small part of the diagram is shown (the one contained in the circle), and the rest of the diagram, lying outside the circle, is unchanged in the move.
Note that this move, as well as the others below, can be performed either from the left to the right, or from the right to the left. Also note that in combination with move 0, the precise shape of the curve can be deformed.

II) With the same notational conventions as for move I, move II is,
move2III) and move III is
move3

 

By looking at these pictures, it is clear that none of the moves 0)-III) changes the equivalence class of a diagram. Hence if we perform some (finite) sequence of various Reidemeister moves on any diagram $D$, the equivalence class of $D$ is not changed because it remains the same in any single step.