Home > Math > Legendrian Knot Theory: Lecture 1

Legendrian Knot Theory: Lecture 1

-Joan Licata. IAS

Knot theory is an elegant union of topology and geometry. We all know what a knot is.  Let’s look at the mathematical definition.

Defn. A knot is a smooth embedding K: S^1 \rightarrow \mathbb{R}^3

We are only interested in classes of equivalent knots. By equivalent I mean that knots obtained on translating or stretching each other should all be equivalent. This naturally leads to defining an isotopy.

Defn.  K_0, K_1 are isotopic if there is a homotopy  K: S^1X [0,1] \rightarrow \mathbb{R}^3 such that

  • \forall t, H(S^1Xt) is an embedding
  • H(S^1X0)=K_0
  • H(S^1X1)=K_1

This simply means that you can go from one knot to the other by a smooth transformation. Think of ‘t’ as some sort of a parametrisation for the ‘path’ between the two knots.

Projections

On paper, knots are represented by 2-D diagrams called knot diagrams. Of course, we need to show the over and undercrossings to completely describe a knot. This is called a knot diagram. A projection is a diagram without this information. It is the ‘shadow’ of a knot.

Here is a common knot called ‘trefoil’ and its realisation as a physical model.

Fig 1: Isotopic knots: From Left to Right: a, b, c.

Fig 2: Models of the trefoil corresponding to a and b in Fig 1

In Fig 1, a and b are isotopic. One could deform a to get b without cutting the string. c is the Knot projection of b.

So far we have spoken of general knots. Now we come to a very important additional structure that makes Legendrian knots so special.

The standard contact structure

Defn. The standard contact structure \xi_{std} on \mathbb{R}^3 is the 2 plane field such that at (x,y,z),

  • the normal vector is \left[ \begin{array}{c} -y \\ 0\\1\end{array} \right]
  • (equivalent defn) The plane is spanned by \left[ \begin{array}{c} 0 \\1\\ 0\end{array} \right] and \left[ \begin{array}{c} 1\\ 0\\y\end{array} \right]

By a ‘plane field’ we mean that there is a plane associated with each point is space, just like an electric field associates a vector with each point in space. Try sketching these planes in \mathbb{R}^3 .  The planes don’t change along X! It will always be convenient to be an observer on the -Y axis, very far away from the origin, such that we are facing the X-Z plane. This is how the plane field looks.

Fig 3: Standard contact structure

Now we come to the definition on a Legendrian knot.

Defn. K is Legendrian (wrt \xi_{std} if at every point on K, the tangent vector to K lies in \xi_{std}.

Note that this will qualify only certain special knots as Legendrian, of the infinite possibilities. Two Legendrian knots are isotopic if one can be deformed into another while always preserving the Legendrian condition.

See figure 4 for an  example of a Legendrian knot showing the contact structure.

Fig 4: Example of a Legendrian knot showing the contact structure

Defn The projection of a Legendrian knot on the XZ plane is called the front projection.

It would seem that projection would lose out a lot of information (in the Y direction) and make it impossible to reconstruct the knot simply by looking at its shadow. But Legendrian knots are special.  It  turns out that a projection  is enough to reconstruct a Legendrian knot. Let us see why.

Consider a point P on a the front projection of Legendrian knot K. This corresponds to the point P on the actual knot K. Consider the line L tangent to K at P. This line, by definition, must belong to the XZ plane. Moreover, the slope of the line, dz/dx, is nothing but the y-coordinate of P! Therefore the information lost by projecting is retrieved from the slope.

Observe the way the panes twist as one moves along the Y axis in Fig 3. A line on the +Y side of space will be seen having a positive slope in the front projection, a line on the -Y side will be seen to have a negative slope. Hence, just looking at the tangent lines of our front projection, we can tell which how the strands are oriented, which strands are in front and which go behind. You must have noticed that there seems to be a problem when the slope goes to infinity, i.e. for vertical tangents. It’s not really a problems since these appear as cusps in our projection.

Fig 5: Reconstructing a Legendrian knot. From left to right: a, b, c.

Observe figure 5. Let’s start with c. How do we know which of the strands goes behind and which is in the front? Use the thumb rule that the more negative slope is in the front (remember that our observer is at -\infty on the Y axis, facing the XZ plane. Now you can easily see how a and b follow.

In the next lecture I will write about knot invariants.

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