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Section 2.3 Rational Knots and Conway Notation

Worksheet 2.3.1 A Fraction-Tangle Dictionary

Having established that the group of functions, whose operation is composition, generated by

\begin{align} t(x) &= x+1 \label{eq_tfunc}\tag{2.3.1}\\ r(x) &= -1/x\label{eq_rfunc}\tag{2.3.2} \end{align}

has all the same structure as the group of rational tangles, and furthermore that the functions in this group are uniquely determined by their values at \(x=0\text{,}\) we now have a recipe for translating between rational tangles and rational numbers.

1.

Write the fraction of the rational tangle

\begin{equation*} Q_1 = T^3\, R\, T\, R\, T^2\, R\, T \end{equation*}

as:

  1. A continued fraction, i.e., \(a + \frac{1}{b+\frac{1}{c+\frac{1}{d+\frac{1}{\cdots}}}}\)

  2. A fraction in lowest terms: \(F(Q_1) = \)

2.

Using the Shoes-and-socks theorem, determine a rational tangle word for the inverse of \(Q_1\text{:}\)

\begin{equation*} Q_1^{-1} = \hspace{2in} \end{equation*}
3.

Now, use the functions (2.3.1) and (2.3.2), applied to your answer to Worksheet Exercise 2.3.1.1, to determine a sequence of rational function applications which returns \(Q_1\) to an empty tangle (having fraction zero).

4.

Determine a word (in \(T\) and \(R\)) for a rational tangle \(Q_2\) that has fraction \(F(Q_2) = -\frac58\text{.}\)

5.

Now, use the functions (2.3.1) and (2.3.2), applied to your answer to Worksheet Exercise 2.3.1.4, to determine a sequence of rational function applications which returns \(Q_2\) to an empty tangle (having fraction zero).

6.

Make a conjecture: Given any rational number \(x=p/q\) in lowest terms, is it always possible to find a sequence of applications of the functions (2.3.1) and (2.3.2) that transforms \(x\) into zero?

If yes: Describe a constructive procedure for discovering this sequence, and give a proof that your procedure will always work (will always eventually result in zero).

If no: Find a counterexample, i.e., a rational number \(p^\prime/q^\prime\) for which no sequence of \(t(x)\) and \(r(x)\) functions will ever return it to zero.

Worksheet 2.3.2 “Coloring” a Tangle with Integers

If we have a (countably) infinite box of crayons, we can build a coloration for a rational tangle using the process outlined in this video excerpt. The result of this process is a \(2\times 2\) “color matrix” associated with a tangle \(Q\text{,}\) that we will call \(M(Q)\text{.}\)