Lecture 2 (29.09.2015)

When experimenting with knots, one gets soon convinced that a knot “stays the same” (in some way) when stretching, knotting, rotating, twisting it. We now want to capture this idea in a mathematical definition. To begin with, consider the demonstration in knotplot under DemoA > Nasty, which shows you examples of deformations of knots that do not change the essential character of the knot.

We want to declare certain deformations of knots/links as “allowed”, whereas others will be “not allowed”. Examples of allowed deformations are all movements that you can do with an infinitely flexible, infinitely thin, but unbreakable rope (with the ends joined as always). So, for example, rotating a knot in space, or stretching/twisting it in some arbitrary manner is “allowed”.

On the other hand, it is “not allowed” to rip the rope apart, erase or add components (new pieces of rope), move one strand through the other as if they were of thin air, or pull knots infinitely tight so that they vanish to points.

Every “allowed move” will be called an ambient isotopy. While this makes perfect sense intuitively, it is not yet a definition, which we give now.

Definition 1.5: Two geometrical links $L_1, L_2$ are called equivalent (or: ambient isotopic), if there exists a continuous map $h:{\mathbb R}^3\times[0,1]\to{\mathbb R}^3$, $(\vec{x},t)\mapsto h_t(\vec{x})$ (an ambient isotopgy), such that $h_0(L_1)=L_1$, $h_1(L_1)=L_2$, and each $h_t$ is bijective. We write $L_1\cong L_2$ if $L_1$ and $L_2$ are equivalent.

Although it sounds a bit technical, this definition precisely captures the intuition of continuously deforming $L_1$ to $L_2$. Not that in an ambient isotopy, the curve cannot pass through itself. See the knotplot animation DemoA>GuessKnot for a demonstration that passing the rope through itself can make drastic changes to knots, which are not considered equivalent.

On a mathematical level, ambient isotopy is an example of an equivalence relation. This means precisely that the following three properties are satisfied:

  1. Any link $L$ is equivalent to itself, $L\cong L$. (take the identity as ambient isotopy)
  2. If $L_1,L_2$ are two links such that $L_1$ is equivalent to $L_2$; i.e. $L_1\cong L_2$, then $L_2$ is also equivalent to $L_1$, i.e. $L_2\cong L_1$ (think of the movie showing the deformation running backwards)
  3. If $L_1,L_2,L_3$ are three links such that $L_1\cong L_2$ and $L_2\cong L_3$, then also $L_1\cong L_3$ (first deform $L_1$ to $L_2$, then proceed further to deform it to $L_3$.)

Given any link $L$, we denote its equivalence class under ambient isotopy by $$[L]:=\{L’\,:\,L’ \text{ is a link with } L’\cong L\}.$$

Such an equivalence class $[L]$ is a family of many (infinitely many) different geometrical knots, that are grouped together in one class because they have the property that they are all equivalent to each other (and in particular equivalent to $L$).

Definition 1.6: topological knot is an equivalence class of a geometrical knot under ambient isotopy. A topological link is an equivalence class of a geometrical link under ambient isotopy.

This definition implies a number of facts and formulations that match our intuition with actual physical knots/links:

  • If $\cal L$ is a topological link and $h$ an ambient isotopy, then $h(\cal L)=\cal L$. That is, if we deform, twist, stretch, wiggle (but don’t cut or tear) a topological link, it stays “the same”. Effectively, this definition means that a topological link $\cal L$ is made of infinitely flexible rubber band.
  • Every geometrical knot/link lies in precisely one equivalence class: On the one hand, each knot/link does lie in one class (namely, in its own), and on the other hand, two different equivalence classes don’t have any geometrical knots/links in common. Topological links therefore cleanly partition the huge family of all geometrical links into subsets, the equivalence classes under ambient isotopy.
  • Another frequently used name for topological knot/link is knot type and link type, respectively.
  • We say that a geometrical link $L$ represents a topological link $\cal L$ if $L\in\cal L$ — that is, if $L$ is one of the (infinitely many) equivalent configurations of $\cal L$.
  • We are interested primarily in topological links, not so much in the geometrical ones. We therefore often drop the term “topological”.
  • In our notation, we use “flexible, curly” letters such as $\cal L$ or $\cal K$ to denote topological (i.e., flexible) links or knots, and “straight, rigid” letters such as $L,K$ for geometrical links/knots.

 

Note that equivalent geometrical links can appear to be very different — see for example the knotplot animation DemoA>GuessKnot mentioned earlier, which demonstrates the equivalence of the knot shown in the first frame of the movie with knot shown in the final frame of the movie.

Having introduced the basic concept of a (topological) knot/link, we can now pose the main questions of knot theory:

  • Given two links (or just knots) $\cal L_1$, $\cal L_2$, how can one decide if ${\cal L}_1={\cal L}_2$?
  • Given a (geometrical) representative $K$ of a (topological) knot $\cal K$, how can one decide if $K$ is equivalent to the unknot? In other words, how can we decide if $\cal K$ is the topological unknot?
  • How can geometrical knots/links be classified up to ambient isotopy? In other words, how can topological knots/links be classified?

The last question amounts to identifying properties of a knot that do not change under ambient isotopy. Thinking about actual physical knots, it soon becomes clear that this is a rather tricky question — familar concepts like, for example, the length of the rope, or its curvature, are useless here. We need to find the topological properties of knots to distinguish them. The following concept is central:

Definition 1.7: 
link invariant is a function $I$ from the set of all geometrical links to some set $S$ such that $I(L_1)=I(L_2)$ whenever $L_1\cong L_2$. A complete link invariant is a link invariant $I$ such that $I(L_1)=I(L_2)$ if and only if $L_1\cong L_2$.

  • Analogously, one defines knot invariants, and complete knot invariants.
  • link invariant is nothing but a well-defined function from topological links to certain other objects — one assigns the value $I(L)$ to the equivalence class $[L]$, which is independent of the choice of geometrical knot $L$ representing it.
  • Each link invariant $I$ can be used as a test to check if two given links $L_1$, $L_2$ are equivalent: If $I(L_1)\neq I(L_2)$, then $L_1$ and $L_2$ are not equivalent. In this case, we say that $L_1$ and $L_2$ are distinguished by the invariant $I$. However, if $I(L_1)=I(L_2)$, then usually we can not be sure if $L_1$ is equivalent to $L_2$ — only in the case of a complete link invariant.

Example 1.8: 
Let us define a function $I$ from the set of all geometrical links to the set of integers, $S=\mathbb N$, by setting $$I(L):=\text{number of components of } L.$$ Then it is easy to check that this is a link invariant — continuously deforming a link does not delete or add any components. However, all links with the same number of components have the same value $n(L)$ — for example, on knots this invariant always returns the value 1. But there exist many inequivalent knots (as you might believe right away, and as we will prove later). Therefore this invariant ist not complete.

  • It is challenging to find link invariants that distinguish many knots and are also easily computable. We will give many examples in the lectures to follow.
  • The set $S$ in the definition of a link invariant is arbitrary, i.e. we can in principle use all kinds of objects $I(L)$ to distinguish knots/links. In particular, we will use integers, polynomials, matrices, and groups.
  • At present, no useful complete link (or knot invariant) invariant is known.

Exercise 1.9: Equivalence of Matrices
This is an exercise independent of knot theory, meant to illustrate the problem of characterising equivalent objects. It requires some previous knowledge on matrices (matrix multiplication, determinant, eigenvalues …) and equivalence relations.

Let us consider square matrices $A,B,…$, and call two such matrices equivalent if there exists an invertible square matrix $V$ such that $A=VBV^{-1}$. Show that this is an equivalence relation. Find invariant properties of matrices that do not change when going from $A$ to some equivalent matrix $B$. Can you find a complete invariant (maybe only for a subfamily of certain matrices)?