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.