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 
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.
Rational Tangles and Braids Undergraduate Research 