Research

My recent work has been on the relationship between fundamental link invariants and quantum link invariants. Fundamental link invariants include invariants such as crossing number, uncrossing number, knot genus, and crosscap number.  Whereas quantum link invariants include examples such as the Colored Jones Polynomial. 

I have recently posted a paper on ArXiv giving two sided bounds for the crosscap number in terms of coefficients of the Jones polynomial. The crosscap number could be seen as the non orientable genus of a link. And as with the genus of a link this value can be quite difficult to compute as you must look across all spanning surfaces of the link. My advisor Efstratia Kalfagianni and her former student Christine Lee originally found similar bounds for alternating links. My work expands this to the Conway sum of strongly alternating links, it also shows that such bounds do not exist for all links.

In terms of the future of this project, my examples for which the crosscap number and Jones polynomial coefficients grow independently are non-hyperbolic links, so this leads to the question of whether two sided linear bounds could be found for all hyperbolic links. The next steps would be to either relax the types of tangles we use in the Conway, or to even consider adequate links as a whole. The roadblock in this case is that we lose the nice properties from a tangle or a link being alternating. This project also has opportunity for undergraduate involvement, through either using the lower bounds to find links for which we can compute an exact crosscap number. This could be done with coding and using a knot database. There is also opportunity for considering specific families of knots and seeing if we can find bounds or if it the two invariants grow independently. 

Another project I have been working on with my advisor is to compute the crossing number for (p,2)-cable knots, where p satisfies certain conditions. This work recently resulted in a paper which is on Arxiv. This uses the relationship between the Jones diameter and the crossing number of a knot. For adequate knots the Jones diameter is known to be twice the crossing number, for non-adequate knots the Jones diameter is at most twice the crossing number. So, this gives opportunity for computation if one finds a strong enough lower bound, namely 2c(K) -2. As with my other project there is opportunity to mirror this work for other families which could be a great opportunity for undergraduate research. 

I have also been working with a group of peers to learn about bordered Floer homology which a professor here believes could lead to a potential project. Another project I have previously worked on related polynomials on graphs with the Jones polynomial which unfortunately did not produce results but would be a topic that would be very approachable for undergraduates.