One of the aspects of knot theory that makes knots challenging is that knots must be understood in their wholeness: not on the basis of looking just at some of their crossings but looking at how all those crossings fit together to paint a global picture of the knot. Ultimately, the knot invariants we study later in the semester will help us to take this perspective. For now, though, we'll take the global questions out of the picture by slicing apart our knots and pinning down the cut ends, much like an entomologist might study a mounted butterfly. The objects we obtain for study in this process are called tangles, and the class of them that are easiest to study are the so-called rational tangles.
Subsection1.2.1Constructing Rational Tangles
We begin by thinking of rational tangles as the result of applying combinations of two geometric operations to an "empty" horizontal tangle:
Twists of the rightmost strands, \(T\text{,}\) and
Quarter-turn rotations of the entire tangle, \(R\text{.}\)
So for instance, we might describe a tangle as a "word" built from these letters such as
Ideally, what we'd like to find is a way to represent each of these operations as an operation happening to rational numbers and not just to rational tangles. In this process what we're hoping to obtain is:
A group \(\Gamma\) of operations on tangles, and
A (right-)group action of this group on the rational numbers, \(\rho \colon \Gamma \to {\rm Func}(\mathbb{Q}\to\mathbb{Q})\text{.}\)
The latter definition might be new to you, but we'll hopefully see by example that it's pretty natural. To make it precise:
Definition1.2.1Group action
Let \(G\) be a group and \(S\) be a set. A (right) action of \(G\) on \(S\) is a function
\begin{equation*}
\rho : G \to \{\text{bijections } S\to S \}
\end{equation*}
that respects the group operation. That is, for all \(g,h \in G\) we have
\(\bigl( \rho(g) \circ \rho(h) \bigr)(x) = \bigl(\rho(h\cdot g)\bigr)(x)\) and
In effect, what a group action does is assign to each element \(g\in G\) an invertible function \(\rho(g) : S \to S\) in such a way that the product of two elements \(h\cdot g \in G\) is assigned to the composition of the functions \(\rho(g)\circ \rho(h)\text{.}\) The reversal of this order is what makes this a right action, and permits us to read a product of group elements as acting from left to right, for example,
\begin{align*}
TTRTR \amp\amp \text{twist, then twist, then rotate, then twist, then rotate}\\
\downarrow\\
r\bigl( t\bigl( r\bigl( t\bigl(t(x)\bigr)\bigr)\bigr)\bigr).
\end{align*}
Subsection1.2.2How the Tangle Group Operates
Determining the structure of the tangle group will help us to understand what rational tangles can look like after all the twisting and rotating is done. We can observe how each operation interacts with itself:
Iterated twists \(T, T^2, T^3, T^4, \ldots\) continue to add new crossings and change the tangle, so \(T\) is an element of infinite order in the tangle group.
Iterated rotations \(R, R^2, R^3, R^4,\ldots\) undo themselves. It's easy to see that since \(R\) is a quarter-turn, \(R^4 = I\) will be the identity. What's more surprising, and will take until Section 1.3 to understand, is that for rational tangles, only two rotations are needed to return to the identity (\(R^2 = I\)). So \(R\) is an element of order 2 in the tangle group.
The final step is to understand the inverses of each of these basic operations (generators) in the tangle group. What's clear from point (2) above is that \(R\) is its own inverse. What's less clear is how to "un-twist a twist," i.e. how to construct an inverse twist using only more of the same twist (combined with possibly some rotations). The video above does the best justice to the observation that
\begin{equation*}
T \underbrace{RTRTR}_{T^{-1}} = I
\end{equation*}
and the discovery of this inverse completes our presentation of the tangle group.
Definition1.2.2Tangle group
The tangle group \(\Gamma\text{,}\) defined as the set of all operations that carry one rational tangle into another, has the presentation
\begin{equation*}
\Gamma = \bigl\langle T,R \; | \; R^2 = I = (TR)^3 \bigr\rangle.
\end{equation*}
When we attempt to understand this group more concretely, we might try, as in the video above, to examine its action on the set of four vertices of the tangle (the four people holding the ropes). This serves to represent each tangle group operation by the permutation that it induces on the four people holding the ropes. But it doesn't tell the whole story because, for example, the permutation induced by the twist \(T\) has infinite order in the tangle group (where successive twists always compound on each other) but order two in the permutation group (which only looks at the fact that the two people holding the twisting ropes have returned back to their starting places after two twists).
Subsection1.2.3The Tangle Group Action on Rational Numbers
What we ultimately want when we determine an action of the tangle group on the rational numbers is that the above situation not happen: we need the functions operating on the rational numbers to reflect all the structure revealed in the presentation of Definition 1.2.2. So what we need are functions \(t : \mathbb{Q}\to\mathbb{Q}\) and \(r : \mathbb{Q}\to\mathbb{Q}\) which satisfy all of the following.
\(t\) has infinite order, i.e. \(t^n(x) = t(t(t(\cdots t(x) \cdots )))\) must never equal the identity function \(i(x) = x\) unless \(n=0\text{.}\)
\(r\) has order two, i.e. \(r^2(x) = r(r(x)) = i(x) = x\) for all \(x\in\mathbb{Q}\text{.}\)
\((r\circ t)\) has order three, i.e. \((r\circ t)^3(x) = r(t(r(t(r(t(x)))))) = i(x)=x\) for all \(x \in\mathbb{Q}\text{.}\)
Beginning by positing that the simple "add one" function \(t(x) = x+1\) is a suitable choice to represent the infinite-order operation \(T\text{,}\) we then go in search of another function \(r\) which is an "involution" (i.e., it is equal to its own inverse function) and which also interacts with \(t\) in the manner specified by (3) above. We find that the opposite-reciprocal function miraculously fits the bill, and arrive at a right action of \(\Gamma\) on \(\mathbb{Q}\) defined by
The video above concludes with an explanation of how this permits us to compute a rational number, called the tangle number, for a rational tangle once we assign to the empty tangle the number 0.
Subsection1.2.4Exercises
1
Find an example of a knot diagram with an equal number of "over" crossings as "under" crossings, but the knot which it represents is not amphicheiral, i.e., \(K \neq -K\text{.}\)
2
(Adams, 2.12) Draw a sequence of pictures that show that the following two rational tangles are equivalent.
3
Find an example of a 2-tangle which is not a rational tangle, and explain how you can tell that it is not rational. (Adams' text and the Kauffman/Lambropoulou paper are good places to start looking.)
