Abbot and Costello Meet Low-Dimensional Topology

My graduate training was (at least partially) in mathematical knot theory.  I guess because of that I’ve had knot puns running around my head for some time.  These finally crystallized into the following routine, which I wrote last night.  Besides, I figured we need a little break from the hard-core recreational math of partitions of graphs.

[Costello enters the room where Abbott is seated reading a book whose title, "Knot Theory," is visible to the audience, but not to Costello.]

Costello: Whatcha readin’ Abbott?

Abbott: Knot Theory.

Costello: Oh good.  I’m glad it ain’t theory.  Every time you read somethin’ theoretical, you go and explain it to me and try to get me all turned around and confused.

Abbott: Mmmph…[not really paying attention to Costello.]

Costello: So whatcha readin’ Abbott?

Abbott: Knot Theory.

Costello: Glad to hear it ain’t theory, but just tell me what it’s about.

Abbott: [Looking up at Costello now] Look, it’s a theoretical book.  Not the kind of stuff you’d be interested in.

Costello:  It is theoretical?

Abbott: Yes.

Costello: It’s about theory…

Abbott: Yeah.

Costello: You just said it wasn’t!

Abbott: No I didn’t.

Costello:  Did so!

Abbott: Hmph.

Costello: OK, then just tell me what kind of theory it’s about.

Abbott: It’s knot theory.

Costello: Abbott!  You’re doing it!  You’re tryin’ to run me around in circles!

Abbott: I am not–

Costello: I ask you what your book’s about–

Abbott: You want to know what my book’s about?

Costello: –And you won’t give me a straight answer.  “It is about theory. It ain’t about theory.”

Abbott: Look.  Calm down.  I’ll tell you what my book’s about.

Costello: –You’re gonna run me around in circles…

Abbott: No I won’t.  Now look.  I’ll make it very simple.  Every morning you wake up.

Costello: Yeah.

Abbott: You put on your shoes.

Costello: Yeah.  Your book talks about that?

Abbott: Now listen.  You put on your shoes, and you tie them, right?  Tie them in what?

Costello: In a little bow.

Abbott:  Yeah.  You tie them in a knot.

Costello: I do not!  I tie them in a little bow.

Abbott: Fine.  You tie them in a little bow.  Did you ever think about the theoretical aspects of that?

Costello: The theoretical aspects of puttin’ on my shoes?

Abbott: …of tying your shoes.

Costello: The theory behind puttin’ on my shoes is I don’t wanna get my feet wet when I go outside.

Abbott: I’m talking about tying your shoes.

Costello: The theoretical aspects of tying my shoes?

Abbott: Yes.

Costello: There’s some kinda theory there?

Abbott: Certainly.  It’s theoretical.

Costello: What kind of “theoretical” is it?

Abbott: It’s knot-theoretical.

Costello: there you go again, Abbott!  It is theoretical.  It ain’t theoretical.

Abbott: Now I didn’t say that.

Costello: You were tellin’ me about your book.–

Abbott: –About my book.

Costello: It talks about my shoes in there.

Abbott:Well…

Costello: What size are they?

Abbott: What size are what?

Costello: My shoes.

Abbott: How should I know?

Costello: Your book doesn’t tell ya?

Abbott: It doesn’t say anything about your shoes!

Costello: Are you gonna tell me what’s in that book ?

Abbott: I’m trying to tell you.

Costello: You tell me it is in there and it isn’t in there . . .

Abbott: You’re confusing yourself.

Costello: Somebody’s confusing me, but it ain’t me!

Abbott: I’m trying to tell you what’s in this book.

Costello: I’m tryin’ to figure out what’s in that book!

Abbott: OK, look.  It’s knot theory.

Costello: You said that.  So what’s it about?

Abbott: It’s about a theory…

Costello: Abbott! you’re doin’ it again!

Abbott:Will you calm down?  I havn’t said anything yet.

Costello: You can say that again.

Abbott: Look.  I’ll give you another example.

Costello: OK.

Abbott: Were you ever a Boy Scout?

Costello: I sure was.

Abbott:You were.

Costello: Boy Scout, Second Class.

Abbott: OK.  Did you ever do a unit on sailing in Boy Scouts?

Costello: Sure did.

Abbott: And you learned to tie sailors’ knots?

Costello: Sure did.  I learned my bowline.  I got my half hitches.

Abbott: That’s right, and you have a Lark’s head.  Now did you know you only have one square knot, but two grannies: a left-handed one and a right-handed one.

Costello: I know I have two grannies, but what’s that got to do with knots?

Abbott: That’s what I’m trying to tell you. It’s all in this book.

Costello: It’s all in there?

Abbott: Yes.

Costello: Where you learned all these facts.

Abbott: Yeah.

Costello: When were they born?

Abbott: Who?

Costello: My two grandmothers.

Abbott:What are you talking about?  It’s a book about tying a knot.  It’s got nothing to do with babies being born.

Costello: My mother told me that Tying the Knot has everything to do with babies bein’ born!

Abbott: Now listen.  You’re being silly.  Do you want to know what’s in this book or not?

Costello: I been askin’ ya what’s in that book.

Abbott: I’m trying to tell you.

Costello: You tell me it’s about my shoes and it ain’t about my shoes.  You tell me it’s about my grandmother and which hand she used to write with.  You tell me it’s about gettin’ married and havin’ babies.  It’s about Boy Scouts and it ain’t about Boy Scouts.

Abbott: OK, OK.  I was trying to give you some examples.  It’s not about any of that.

Costello: None of it?

Abbott: No.  None of those practical matters.

Costello: Nothin’ practical at all?

Abbott: Nothing practical.  It’s a theory book.

Costello: Purely theoretical?

Abbott: Purely theoretical.  all theory.

Costello: All theory from one cover to the other?

Abbott: Cover-to-cover theory.

Costello: OK.

Abbott: OK?

Costello: OK.

Abbott: OK.

Costello: Then just tell me.  Answer me one question.

Abbott: What’s the question?

Costello: This book of theory here.

Abbott: My book of theory.

Costello: The book that explains the theoretical.  What kind of theory does it explain?

Abbott: It’s knot theory.

Costello: Abbott!!!

Math with Balloons

In demonstrating what a mathematical knot is, balloons are the best medium that I’ve found.  They have enough stiffness to keep the knot from collapsing so that you can’t see its structure (a string or rope would either be floppy and hard to work with, or you would have to pull the knot tight, which would make it hard to see it’s structure).  They have enough give to allow knots to be tied in them.  They are very visisble and the ends can be twisted together to complete the loop.

Here I am with not a knot, but a link: the Borromean rings.  It has beautiful symetries that have been recognized for centuries and is important to low dimensional topology (in fact, an early incarnation of my dissertation featured the Borromean rings).

Knot Tying Schema

I’ve noticed that whenever my two-year-old wants to tie a knot, she takes the two strands that she’s interested in and repeatedly twists them around each other.  Of course this doesn’t produce any kind of knot at all (or I should say it only produces the unknot).  But this twisting appears to be the only basic move that she knows (and she learned it herself) for producing knots.

It made me wonder what the basic moves are that we use to tie knots.  Every knot theorist knows that there are three basic moves for transforming one picture of a mathematical knot into another (possibly quite different looking) picture of the same knot.  These moves are called the Reidemeister moves.  Any two pictures of the same knot can be made to look like each other using just these moves (and “plane isotopies”).  Similarly it seems to me that mathematically you really only need two moves to tie any knot, or what I mean to say is to go from a picture of a straight string to a picture of any knot.  Here is an illustration of those two moves:

Of course after doing several of these moves, you will want to glue the two ends of the string together if you wnat to get a picture of a mathematical knot.

When people tie knots, it does seem like there is some finite set of moves they use, but the catalogue of moves seems to be bigger than these two moves I illustrated above.  For instance there are moves like “wrap one end around a loop” or “put one end through a hole” or “follow one strand through a whole series of moves.”  Of course each of these can be broken up into a sequence of the two basic moves, but it is not always useful to break things down into the simplest possible moves in practice.  It makes me wonder what the catalogue of moves is that people use.