Benjamin Thompson's 2016 honours thesis in mathematics, on "The Khovanov
homology of rational tangles", supervised by
Scott Morrison, is available here.
Outline
The aim of this thesis is to describe the Khovanov homology of rational tangles. To this
extent we describe rational tangles, followed by Khovanov homology, then combine the two
at the end.
In Chapter 1 we review the main points of the theory of rational tangles. In particular,
we show that rational tangles are classified by a function known as the tangle fraction,
which associates to each rational tangle a rational number. This classification theorem
implies that each rational tangle can be constructed by adding and multiplying together
multiple copies of certain types of tangle.
In Chapter 2 we review the Khovanov homology theory we will use to study rational
tangles. After discussing link invariants, the Kauffman bracket, and categorification in
general, we develop a generalization of Khovanov homology to tangles due to
Bar-Natan
[Bar04]. We then specialize this
to a 'dotted' theory, which is equivalent to Khovanov's
original theory [Kho99] on links
and is significantly easier to work with.
In Chapter 3, we combine the previous topics to develop a theory of the Khovanov
homology of rational tangles. We determine the Khovanov complexes of integer tangles,
before presenting a helpful isomorphism and discussing its implications. We then combine
these to prove Theorem 3.3.1, the main result of this thesis.
Finally, in Chapter 4 we briefly discuss some of the application of Theorem 3.3.1.
Unexpectedly, we find that the bigradings of the subobjects in the Khovanov complexes
can be described by matrix actions, and that one can recover from this action the reduced
Burau representation of $B_3$, the three-strand braid group. The full
ramifications of this
observation are yet to be determined!
