The search for a relative Bar-Natan invariant

Khovanov theory has shown its strength many times since it was constructed in 1999 – it is now known to detect the first four knots in Rolfsen's table as well as a few links, it has lead to a very useful concordance invariant and it is related to other low-dimensional homology theories via spectral sequences. A factor in its strength, and also the main difficulty in working with it, is the sheer amount of information that it keeps track of. My project has been to uncover structure in the Khovanov chain complex of a knot K with a choice of 4-ended tangle T sitting inside of it.

Working with the version of Khovanov theory known as Bar-Natan homology, I will present partial progress in the construction of a relative invariant of (K, T) in the special case where K is the unknot. This progress comes in a roundabout way, through an analogue of a computational device in Heegaard Floer theory (known as "the mapping cone formula") that the invariant should allow us to construct. Towards the end of the talk, I will tell you what this computational device is currently conjectured to look like, as well as provide some evidence.