The document is structured as follows.
(I) Chapter1: In this chapter (which you are reading right now), we give two high-level overviews of the project, including one that has a more personal/chronological feel to it (Section1.2), as well as a more detailed “table of contents,” which includes, among other things, a recursive
reference to its own contents (Section 1.3).
The More Technical Overview 9
(II) PartI: Fundamentals of Knot Theory. Here, we give an overview of the big picture ideas in knot theory, and list some basic definitions. We encourage those more familiar with the topics to read Chapter 2 but skip the rest.
1) In Chapter 2, we discuss the knot equivalence problem, with particular focus on an analogy with determining equivalence of arithmetic expressions. We discuss the differences between these two contexts (e.g., the absence of a good simplification algorithm for knots), as well as what kinds of additional structure on the knot category might help get around some of these difficulties in the future. This serves to motivate the approach taken in Chapter 5
2) In Chapter3, we provide the standard definitions for knots, ambi- ent isotopy (highlighting the problems with choosing something like isotopy instead as our definition of equivalence), tameness, regular diagrams, orientation, and so on. This is targeted mainly at those who are new to the topic; experts will probably find nothing surprising. One thing of note is that we place emphasis on highlighting the fact that going from “working with knots” to “working with knot diagrams” should not be taken for granted. (III) PartII: Combinatorial Representations. Here, we discuss what we call
combinatorial representations, i.e. ways of abstracting information in
knot diagrams to strings that can be manipulated purely algebraically / combinatorially.
4) In Chapter 4, we introduce the signed Gauss code for a knot diagram. We discuss the Gauss code encoding of Reidemeister moves, especially the planarity constraint of Reidemeister II. A significant portion of the exposition is dedicated to discussing what we have called the diagram graph (Definition 4.3), which is a graph constructed from the Gauss code that is particularly natural for computational manipulation. In particular, we prove that it has a unique planar embedding, and thus can be used to verify proposed Reidemeister II moves (we give a sketch for a greedy algorithm performing this check).
Finally, we finish by introducing virtual knots as a way to avoid the planarity concerns of Reidemeister II, and work instead in a more purely combinatorial context. Discussion of the forbidden moves leads us to briefly mentioning unknotting moves, and the
10 Introduction
intriguing sense in which they encode “recipes” for how to build knots from the unknot.
5) In Chapter5, motivated by the discussion of unknotting moves, we build up basic definitions for our formalism that connects Gauss codes to actions of the symmetric group on a countable set. The desire here is to flesh out the idea of unknotting moves as “building” knots and “converting them into each other.” Un- fortunately, it seems like on a purely theoretical level, it’s not possible to reconcile knot equivalence with the group structure in a sensible way. Nonetheless, we did see some unexpected patterns in performing computer searches with sage.
Underpinning all of our results in this chapter is the example of an unknot with a countable number of crossings. To show this is valid, we use a slightly simpler version of the uniform convergence proof we give later (Theorem 7.4). To be extra sure that the result is valid (even if our proof turns out to secretly contain a flaw), we construct a C1 embedding for our example of interest (Section5.1.1), which suffices to guarantee it is tame (see AppendixA).
(IV) In PartIII, we shift our focus to general topological embeddings, withe the goal of getting a better understanding our examples of countable Reidemeister I moves.
6) In Chapter 6, we clarify common definitions for tameness and wildness that we have found in the literature, reconciling those that we can with the terms used when studying more general embeddings of m-manifolds into n-manifolds. We try to be par- ticularly cognizant of which category we are working in at all times.
7) In Chapter7, we begin building up machinery for our later work in studying ambient isotopy for general topological embeddings. We develop two tools (strand separation and uniform converence) which prove useful for working with wild knots whose wild points are topologically discrete. We do not build machinery for working with everywhere-wild knots.
8) In Chapter 8, we apply these tools in the case of R2, and show that all curves K : S1 ,→ R2 are ambient isotopic (this will be
The More Technical Overview 11
discuss the pathologies that can arise in diagrams in R2, which we refer as feral behavior.
9) In Chapter9, we use the techniques of Chapter 7and Chapter 8
to study ambient isotopies in R3. The loose idea is that as long as the crossing points in our diagrams are toplogically discrete, we have a bunch of strands that essentially act like they’re curves embedded inR2(because, after all, they don’t cross). This reduces the behavior to results covered by Chapter8. By then showing we can also constrain the behavior of our embeddings near crossing points, we can then show that if a wild knot has a diagram with topologically discrete crossings, then it is ambient isotopic to a representative comprised of a countable union of polygonal segments. We conclude with some directions for future work, and offer a very brief sketch of how one might build an analogue to Reidemeister’s theorem in this context.
10) Lastly, in Chapter 10, we summarize the results of the project, and discuss possible ways of turning our “crossing-discrete wild knots” into a category more directly analogous to the PL case. This concludes the main body of the document.
(V) Part IVis the appendix. Here, we include some miscellany that didn’t fit particularly well into any parts of the main document.
A) Appendix Acontains two extra feral knots that we have parame- terized by a C1 embedding.
B) AppendixBcontains a basic crash-course in PL Topology, with an emphasis on the “crash” part.
C) AppendixC includes misc. data from the project. This includes tables for the cycle representations computed in Chapter 5.
Part I
Fundamentals of Knot
Theory
Chapter 2
Motivation
That very night in Max’s room a forest grew and grew– and grew until his ceiling hung with vines and the walls became the world all around —Maurice Sendak, Where the Wild Things Are
One of the most fascinating things about knot theory is the disconnect between the relative ease of posing a question and the great difficulty of providing a rigorous answer to it. Granted, many mathematical fields are like this — but knot theory is somewhat curious in the extremity of the mismatch. Many of the most fundamental problems in the field can be boiled down to ideas that are accessible to any lay-person, and yet are quite challenging to approach mathematically.
Understanding why is the goal of this chapter of the document. As a motivating example, we discuss the problem of knot equality through analogy with equality of arithmetic strings. This analogy ends up also serving as the motivation for our proposal that studying unknotting moves could offer new insights into the structure of the knot category.
In Chapter3, we give background definitions, with a focus on emphasizing the fact that relationships between knots and knot diagrams are subtle, and should not be taken for granted. As we will see later on, loosening what we call “an admissible diagram” can give us some valuable tools for understanding
Knots.
Lastly, in Section3.3we offer a brief discussion of polygonal knots. Mainly, this is a gallery of some pictures; rigorous treatment is left to the section
16 Motivation
in the appendix about PL Topology and our examination of tameness in PartIII.
The remainder of the discussion in this chapter is presented at a high level, with just a few (optional) formal definitions.1 We hope the ideas remain both〔accessible to a non-technical audience〕and〔interesting for experts〕.
2.1
The Big Picture
Picture
Figure 2.1 A Bad Joke
In mathematics, a knot is an embedding of a circle into another space.2 Intuitively, think of taking a rope and twisting it around in space in all sorts of ways, finally fusing the ends together so that we get a closed loop:
Figure 2.2 Constructing a knot
1We hope the more rigor-oriented readers will be patient in tolerating some imprecision for the time being, but if not, formal definitions can be found in Chapter3, Page31.
2In loose terms, “embedding” means that (a) if we zoom in closely enough to our knot, it looks like a line, and (b) if we walk all the way around in one direction, we get back to where we started. This is made formal in Section3.1.
The Big Picture 17
Note, our loop does not need to have any twists to be considered a knot — a regular old circle is a perfectly valid knot! We call this the unknot, and we’ll see that it has some interesting properties later (e.g., it acts like the number 0 for a knot “addition” operation).
We say two knots K1, K2 are equivalent (denoted K1 ∼= K2) if we can
deform K1 into K2 without cutting the rope and gluing it back together. For
example, the left two knots in the diagram below are equivalent, and both are distinct from the knot on the right.3
Figure 2.3 Two equivalent knots and one inequivalent one
One of the central questions in knot theory is “given diagrams D1 and
D2 how do we determine whether they represent the same knot?” If starting
from first principles, this question is HARD to approach mathematically; in fact it requires a bit of topological knowledge to even formulate the question properly. Thankfully, in practice we don’t usually need to think about any of that because of a theorem proven byReidemeister (1927) and, independently, Alexander and Briggs (1926). In essence, they were able to show that two “well-behaved” diagrams4D
1, D2represent the same knot iff D1can be turned
into D2 by a sequence of the following so-called Reidemeister moves:
∼
∼
∼
which are creatively referred to (in left-to-right order) as “Reidemeister I,” “Reidemeister II,” and “Reidemeister III,” respectively.5 Note, while Reidemeister III might look complicated, it’s really just saying that we can move one strand between the crossing formed by two other strands.
3At this point, it is worth noting that there’s a technical distinction between a diagram for a knot and the actual knot itself. We’ll return to this later when we define things rigorously, but gist is that knots live in R3 and diagrams are projections onto R2.
4Again, we’ll discuss what “well-behaved” means in great detail during Chapter3, but it boils down to “the string only crosses itself in an X shape.”
5We also include another move, which allows us to bend the string arbitrarily as long as we don’t introduce a crossing.
18 Motivation
On a theoretical level, this is a very elegant characterization of knot equivalence. However, in practice, determining equivalence is still quite challenging. Even if D1, D2 are relatively simple and both represent the same knot, the sequence of moves relating the two can be quite long. It’s even worse when K1 6∼= K2, because then we have to prove a negative result: namely,
that there does not exist a sequence of Reidemeister moves takes D1 to D2!
Again, this is usually quite hard. Though Algorithms deciding the problem do exist, they are currently far too inefficient to be practical. For those who are familiar with Complexity Theory: It has been proven (Hass et al. (1998)) that a special case of knot equality is at least NP. Reidemeister-based
algorithms for the general problem have runtimes like O(k ↑↑ n) (Lackenby (2016)), which makes even an NP solution seem out of reach for now.
But why? This seems like it should be easy! Our objects are very tangible, the space we’re working in (R3) is well-behaved, and we aren’t asking for
anything too fancy — just a simple way to determine equality. How can we understand the source of this difficulty? Here, an analogy with something more familiar will be helpful.