Pillowcases

As was discussed in an older news post, there exists a double-covering of the boundary of a torus onto the boundary of a ball – essentially a way for us to translate things on the surface of a ball to what they would be on the surface of a torus. As we are already well aware, the rational tangles we have been discussing are considered to exists as non-intersecting line segments contained within an open ball. Suppose then that we take those line segments and push them onto the boundary of our open ball. Due to the fact that the torus is a double-covering of the boundary of our open ball containing our tangle then, we now have a way to picture our tangle as it would exist on a torus.

If we “cut” our torus on the disk where all the endpoints of our tangle lie, we get a “tangle in a tube” (for your convenience!). If we grab the endpoints on each side of the tube and and “pull” we flatten the tube into a “pillowcase” like shape with the tangle wrapping all around it, with the endpoints of the tangle at the corners.

Let’s draw all our tangle pillowcases with just one side facing us. Intuitively, solid lines indicate that the tangle is on the side we are looking at, and dotted lines indicate that it is on the opposite side. We draw tic marks on our diagram wherever our tangle wraps around from one side to the other. Note that these tic marks indicate that our solid lines have a “slope” of one. If we note now many sections the tic marks divide the bottom and side of our pillowcase into, however, we can get the slope of the tangle as where is the number of sections on the bottom of the pillowcase, and is the number of sections on the side.

Aside from being a neat way to diagram tangles and find their slopes, pillowcases offer us a visual, intuitive look into the rational structure on rational tangles, specifically the operations of addition and multiplication of tangles.

ADDITION

We’ve mentioned earlier about adding tangles by attaching them side-by-side or top-to-bottom by their endpoints. On pillowcases the process is remarkably similar – simply attach the pillowcase diagrams at the corners. Note the way solid and dotted lines connect. As on the the pillowcases before adding them together, they are considered on a certain side of the pillowcase. Thus solid lines on different pillowcases are sort of “redrawn” to be the same line on the added pillowcases. Normally this would seem a violation of our vow not to cut the tangles when we add, multiply, or otherwise rearrange them. In reality though, we are merely stretching the line segment to suit our needs.

Ideally the lines on the edges where we add our pillowcases match up, but this is not always the case. Our method of finding slope from a pillowcase diagram fails if we do not have parallel lines travelling from one side all the way to the other, as we will have differing number of sections on parallel sides of our pillowcase. To remedy this, we have to draw more lines in our pillowcase, making sure to keep things from intersecting, and that the ratio of sections on the side to the bottom remains the same (i.e. that we do not change the slope of the tangles we are adding). It would seem that by doing this we are once again violating the rules we had established to manipulate our tangles while preserving their properties. However, all this process does is insert a linked component into the tangle diagram that can easily be removed by a series of type 2 and 3 Reidemeister moves, meaning we haven’t really changed the tangle at all!

The slope of the resulting tangle is equivalent to sum of the slopes of the two tangles we added together. This process is pleasingly similar to converting rational numbers to have common denominators for addition.

MULTIPLICATION

Ozawa’s paper defines the multiplication of rational tangles and by replacing each crossing of with a copy of if the crossing is positive, or if the crossing is negative.

On the pillowcase, the process is rather simple. We divide our pillowcase of into a grid based on our tic marks. In each box on the grid, we replace the existing lines with a copy of .

The slope of the resulting tangle is equivalent to the product of the slopes of and . It’s a rather intuitive fact based on the process by which we construct the product tangle. Suppose we have the slope of equivalent to and the slope of equivalent to $\latex frac{c}{d}$. On our grid of , we have squares along the bottom. Replacing each one of these is a copy of the pillowcase of which each have a bottom divided into sections. Thus in our product, the bottom of the pillowcase is divided into sections. By similar reasoning, we determine that the side is divided into sections. Thus the slope of the product tangle is equivalent to , the product of the slopes of and .

CONCLUSION
As in the past, we let be the map that takes our tangle to the rational number equivalent to its slope. As we’ve shown with our pillowcases, demonstrates properties of a homomorphism, namely:
and
.

This opens up possibilities for future research. Most immediately, consider the set of all rational tangles to be the set , and . If we look at the subgroup of mod , then by the First Isomorphism Theorem, there must exist an isomorphism !

Whiteboard from Friday, April 20th 2012

Demonstration of how adding a trivially linked component into our “pillowcase” diagrams doesn’t actually change any properties of the sum tangle

Success! Multiplication of tangles can be done on a pillowcase diagram.

We’ve been thinking too hard!

Lately I’ve been combing Ozawa’s paper on the rational structure on slopes of rational tangles in an attempt to decode just how to compute the slope of a given rational tangle. What I found out was…enlightening to say the least. It seems that the slope of a rational tangle is the continued fraction we’ve been computing for them in the past! Or at least, very similar. I’ll discuss it with Dr. Errthum on Friday and see what he thinks, but either way it’s progress. Some scans from my notes:

Initially I had thought that computing the slope of a rational tangle would involve looking at the explicit functions in the covering map we had pieced together last week. I don’t deny that that map has the topological explanation behind the slope of a tangle, but this method of computation is a lot more intuitive. If we consider tangle addition as shown above and let be a map that takes a tangle to its slope in the rationals, then it seems that a group structure is indeed possible! As seen in the example above, it would make sense that:

Though I imagine considering the infinity tangle (a tangle of slope ) could make a proof of this tricky. Ozawa looks at tangles under both addition and multiplication, so I imagine proving the rational structure of these is where my project is going next.

Bisquick and Levi’s

A lot was covered last Friday in terms of pushing the project in a new direction. First – in what was admittedly quite the aside – I learned from Dr. Errthum what exactly a “pair of pants” was in topological context. If I didn’t love topology before, I sure do now.

More importantly, I found a paper on algebraic tangles that I felt was well worth delving into. After reviewing it Dr. Errthum agreed, and we spent today’s meeting discussing what we each did and did not immediately understand from our own readings, and then worked through what we could bit by bit. The paper was Makoto Ozawa’s “Rational Structure On Algebraic Tangles and Closed Incompressible Surfaces in the Complements of Algebraically Alternating Knots and Links”. What interested me in this paper was Ozawa taking the idea of “slope” of an algebraic tangle being a rational number and showing that there exists a map from algebraic tangles onto the rationals that displayed the group structure of the rationals.

The slope of an algebraic tangle is the first major hurdle in our journey to understanding this paper, and it’s proving to be quite the intriguing challenge. As was stated previous, tangles are defined by an open ball around them in a knot or link. We’re going to call this open ball and the tangle itself . In topology there is something known as a Cover, a collection of subsets of a space such that the union over these subsets is equivalent to . A Double Covering is similar, except that it is a two-to-one map from one space to another, and a Universal Covering maps to all coverings and the space those covers cover. See http://en.wikipedia.org/wiki/Covering_space for more information (and a “stack of pancakes”, which I found amusing). Ozawa states that there exists a map that shows is a double covering of $\partial B$, the boundary of our ball that defines our tangle*. Since is the universal covering of , we know that there exists a map from to that is the universal covering of . By the definition of universal covering then, there exists a map showing that is a covering of .

With me so far? If not that’s okay, at this stage I’m still rather confused too.

We define as the boundary of our tangle – the 4 points on where our tangle intersects the open ball. So , thus must map some subset of to . Ozawa states that maps to – the set of pairs of integers divided by 2.

We’re almost to the slope of an algebraic tangle! There exists a loop on the ball called an Essential Loop that divides into two pieces with two intersection points from on each piece. Symbolically we refer to essential loops as . If this is a loop on the meridian we call it a meridian essential loop or -essential loop, whereas if it is on the longitude we call it a longitude-essential loop or -essential loop. maps the essential loop to a bunch of parallel lines of slope where . (For the -essential loop we get and for the -essential loop we get .) And THAT is the slope of our algebraic tangle!

At the moment it’s sort of a vague understanding of what’s going on, but my assignment for the next meeting is to find explicit maps for what I can of , , and , so that we can get a better understanding of how this all fits together, and actually compute the slopes of some algebraic tangles. It sounds pretty tough, but I’m pretty excited about where this is going, and feel up to the challenge

*For more information on double coverings involving torii and tangles see:
http://sketchesoftopology.wordpress.com/2008/01/25/double-branched-covers-of-rational-tangles-and-the-montesinos-trick/

Dead Ends and Fresh Starts

Last Friday Dr. Errthum and I discussed making the “Knee Bone” from a basic tangle with three positive crossings. To make a long story short, it didn’t work out. Turns out that in order to make the knee bone (without attaching an extra tangle) we’d have to hook one of the endpoints of the tangle through an existing loop between the 3 crossing tangle and one extra crossing added on top of it. Because tangles are not supposed to have their strands cross the open ball that defines them, this is an illegal move, since the endpoints define the open ball! This means that we’ve hit quite a major snag in terms of relating the knee bone back to what we’ve already covered concerning rational tangles, so we’ve decided to try some other directions.

The first new direction is to get “back to basics” so to speak, and take a look at continued fractions of rational tangles again. Flipping through our research materials told us that there exist operations of addition and multiplication on rational tangles. This leads to the question of if we have a rational tangle with continued fraction and another with continued fraction , does multiplying them end up in a tangle with continued fraction ? The question goes for addition as well. It’s important to note here that a ring structure does not exist for the operations of addition and multiplication of tangles (which is a bit of a bummer, admittedly). I am to explore why there is no ring structure on my own, if given the chance.

The other direction we’re considering exploring in is taking a look at algebraic and non-algebraic tangles. An Algebraic Tangle is defined as a tangle obtained from the operations of addition and multiplication on rational tangles. A Non-Algebraic Tangle then, must be one that cannot be obtained in such a manner. This week I’m going to look around for literature concerning algebraic tangles (of three components, if possible), as well as convince myself that an example of a non-algebraic tangle from “The Knot Book” is indeed non-algebraic.

It seems like a major setback, but I’m excited to explore some new areas of knot theory while revisiting some old ones. So far my research has been broader than it is deep, but I’m certain that eventually I’ll stumble across something worth diving down into.

Thus Far…

My project began as an exploration of rational tangles, and since then has ballooned out to cover other areas related to them. Since then it’s also began taking a look at braid groups, representations of braid groups (in particular Lawrence-Krammer Representation, see the Wikipedia link on the “References” page) and how braids can act on tangles:

Dr. Errthum has taken an interest in tangles that do not fit the traditional idea of rational tangles that we’ve seen since the project first started. In particular, he’s intrigued by a tangle we refer to as “The Knee Bone”, and how it can be formed from deforming a rational tangle of 3 positive crossings.