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/
About rationaltangela
I am a mathematics student at Winona State University with interests in music, board games, comic books, and mathematics - in particular topology, number theory, and knot theory.
Rational Tangles and Braids Undergraduate Research 