Open Algebra and Knots: REDOAK Edition

Section 1.1 Anagram Algebra

What is algebra, and where (else) can we find it? Ask a high-school student this question. Then, ask a research mathematician. You may get two very different answers.

Both of the images above are taken from “algebra” texts. Believe it or not, both are part of the study of the same kinds of mathematical objects: here, polynomials. But until we have a better sense of what the word means, it is entirely unclear that these two excerpts are both speaking a common language of “algebra”.

The goal of this introductory set of activities is to broaden our horizons to the meaning of the word “algebra”, so that we will be able to recognize algebraic structure and algebraic reasoning in unfamiliar and unexpected contexts. We'll begin from simple anagram puzzles, and end with some fundamental understandings of the raw ingredients necessary for algebra.

The heart of Worksheet 1.1.1 is the explanations you provided in #3-5 and then again in #6. Here, you first described a process, namely, that turns one five-letter word into a different five-letter word. Then, you supplied another description for the opposite of that process. Finally, you investigated the result of composing that process with itself, that is, following it twice in succession.

Now, consider what we do in our first algebra lessons in middle- and high-school. We begin with a numerical quantity (say, \(2\)). We apply a process, for example, by adding seven (\(2 + 7\)). We understand that that process can be reversed: it has an opposite, that we call subtracting seven (\(2 + 7 - 7\)), and we know this is the opposite because \(2 + 7 - 7\) is the same as, simply, the original expression \(2\text{.}\) And, we know that this process can be repeated, followed up by itself, forming expressions like \(2 + 7 + 7\text{.}\)

What we mean by “algebra” lies in the commonalities between the previous two structures. We are engaging in algebraic reasoning any time we describe combinable, reversible processes. Using traditional algebra notation, we need our processes to combinable because in order to simplify the left-hand side of the equation

we need to know that the objects “\(X\)” and “\(Y\)” can indeed be combined together by the process “\(+\)” into a single object of the same type, “\(Z\)”. And, in order to solve this equation for \(X\text{,}\) say, we need the operation to be reversible: we need to have an opposite procedure “\(-Y\)” that we can apply on both sides to obtain

So it is that the two horsemen of high-school algebra, simplify and solve, necessitate the three types of structure that we will say characterizes all kinds of algebraic reasoning: combine-ability, reversibility, and “neutrality”, the latter being the ability of the “zero” in the above calculation to disappear. This brings us to our first definition.

The associative law, which makes its first appearance in this definition, is a convenience that permits us to do things like raise an element to an integer power, e.g., \(g^5 = g\cdot g\cdot g\cdot g\cdot g\text{,}\) without worrying about which “\(g\cdot g\)” in that product is meant to be evaluated first.

It is perhaps surprising that, given this relatively sparse set of requirements for a group, there are nevertheless many important properties that can be deduced from them.

Conspicuous by its absence is the “commutative law.” This is not an accidental omission, but rather a special property that some groups enjoy but many (most?) do not.

Eventually, when the context is clear, we will use simpler notation \(G\) to denote an arbitrary group rather than listing its elements and operations \(G=(X,\cdot)\text{.}\)

When \(G\) is a finite group, it is possible to explicitly write down a list of all its elements, and every pairwise product of those elements. This information can be arranged in abstract algebra's equivalent of a multiplication table, known as an “operation table” or Cayley table.

Since the associative law makes the expression unambiguous, we can evaluate any integer power of a group element:

This permits us to define another meaning of the word “order”, this time as it applies to an element of a group.

Let's revisit the anagrams in Worksheet 1.1.1 to conclude this section. How can we formalize the idea of anagrams being elements of a group?

We spoke of each anagram as being a “process” of rearrangement of letters, and in the activity you came up with descriptions of those rearrangements that did not call the letters by name. If each rearranging process is an element of a group, and that group's operation is to follow (compose) one process with the next, we can study rearrangements as a group.

To study the algebra of anagrams with more ease, we define a notation for the step-by-step directions you gave. This notation is called cycle notation.

You can use this interactive Scratch app to explore how cycle notation works. Click to open it in a new window.

1. Use the app to figure out: what is the result of following the cycle \((12)\) with the cycle \((24)\text{...}\)as an anagram? ...as a single cycle from elsewhere on the "keyboard"?

2. Repeat the previous, but reversing the order, \((24)\) followed by \((12)\) instead. What does your result tell you about whether the groups comprised of anagrams are abelian?

3. What is the difference between “\((14)\) followed by \((23)\)” and “\((1423)\)”? How would you describe what each of these is doing to the four positions in the word POST?

4. Can you give step-by-step instructions for how to apply the cycle \((243)\) to the word \(PLEA\text{?}\)

Worksheet 1.1.3 Teasing Anagrams Apart

In the anagram \(SHALE \to LEASH\text{,}\) you noticed that two pairs of letters (positions) trade places: the first and fourth, which we may denote \(r=(14)\) in cycle notation, and the second and fifth, \(s=(25)\text{.}\)

If we treat each of these two-place swaps (called transpositions) as its own anagram, then we can study the algebraic structure formed by these anagrams in several ways.

1.

What anagram is discovered for \(SHALE\) defined by each of the following?

1. The identity \(e=()\text{:}\) \(\quad SHALE \to\underline{\hspace{5ex}}\)

   The transposition \(r=(14)\text{:}\) \(\quad SHALE \to \underline{\hspace{5ex}}\)

   The transposition \(s=(25)\text{:}\) \(\quad SHALE \to \underline{\hspace{5ex}}\)

   The permutation \(rs=(14)(25)\text{:}\) \(\quad SHALE \to \underline{\hspace{5ex}}\)

   The permutation \(sr=(25)(14)\text{:}\) \(\quad SHALE \to \underline{\hspace{5ex}}\)

   The permutation \(s^2=(25)(25)\text{:}\) \(\quad SHALE \to \underline{\hspace{5ex}}\)

2.

Explain: How do we know that each of the (a) closure, (b) identity, and (c) inverses axioms of a group is satisfied by \(X=\{ e, r, s, rs \}\) with the operation of composition?

3.

Let \(G\) be the group whose elements are \(X=\{e,r,s,rs\}\) with the operation of composition. Create a Cayley table for \(G\) using these elements' names. (For fun, also create a Cayley table using their actions on the word \(SHALE\text{.}\))

4.

Refer again to the Four Fours Worksheet 1.1.2. Where does this group \(G\) belong among the four groups in that activity? How do you know?

A visual way to represent the permutation that turns \(SOBER \to ROBES\) is to arrange the words above one another, and draw a curve connecting each letter's starting position with its ending position.

You might try constructing this diagram using your loom and string.

5.

How would you express the anagram \(SOBER\to ROBES\) using cycle notation? \(u = \bigl( \qquad \qquad \bigr)\)

6.

Cover up this diagram with a piece of opaque paper. Slowly pull the paper down until you reveal the first crossing of two lines. Write the anagram and the cycle notation it corresponds to, like so:

Continue, stopping after every new crossing you reveal to write both the new anagram and the transposition that occurred at that crossing, using cycle notation. Repeat until you've arrived at the final word, \(ROBES\text{.}\)

Hint

Tracing the curves back to the original word shows you the anagram. But remember that the cycle notation comes from which two positions are trading places at that moment -- so don't let the original positions of the letters throw you off!

7.

Use your answer to the previous question to predict: how can the permutation \(u\) that you found in #5 be written as a composition of adjacent (“neighboring”) transpositions?

8.

Redraw this figure (or manipulate your loom and strings) so that the topmost crossing and the bottom-most crossing both represent the same transposition \((12)\text{.}\) Use this new diagram to discover a different answer to #7.

9.

Can you redraw the figure so that the top- and bottom-most crossings are both \((45)\text{?}\) What about \((23)\text{?}\)

10.

Can you redraw the figure so that the composition of crossings is a palindrome reading the same in forward order as reverse order? How, or why not?

11.

What conjecture(s) can you make about anagrams in general, based on the results of this worksheet? Test one of your conjectures on the anagram \(STONE\to ONSET\text{.}\)

Hint

Start exploring by asking, for each of #6-9, “do I think this always works?” If you think the answer is yes, how could you phrase a general statement of which this \(SOBER\to ROBES\) example is a special case?

You can use the Sage Math worksheet below to experiment: Does changing the order in which the seven crossings are encountered in the \(SOBER \to ROBES\) anagram change the result?

Worksheet 1.1.3 gives us our first taste of how the study of algebra -- group theory, in particular, motivated at first by the study of simple anagrams of words -- can give rise to a physical object that resembles a set of tangled strings! This bodes well for the future of the relationship between algebraic structures on the one hand, and physical knots on the other. As happened in this worksheet, the crossings, those places where one string passes through another, will play a crucial role in uncovering algebraic structure from knot-like objects.

Note also that, in this example, it did not matter how one string passed through another, that is, it did not matter whether one passed over the other in three-dimensional space, or vice versa, or if somehow these incorporeal strings literally transected one another. We'll see how that makes our algebra simpler, at the cost of sacrificing a question that will become much more critical if we want to study physical knots. Indeed, we'll see in this chapter how, as soon as we care which string passes over which at a crossing, the structure we get, called a braid, has a rich algebraic life indeed.