Invariants of 4-Dimensional 2-Handlebodies from the Temperley-Lieb Category in Positive Characteristic

Abstract

We investigate invariants of 4-dimensional 2-handlebodies associated to the Temperley-Lieb category in characteristic $p>2$ and at a primitive fourth root of unity. These invariants depend additionally on a height parameter $n$, and we focus on the case $n=2$. Provided that $p>3$, we show that the height $n=2$ invariant associated to the Temperley-Lieb category at a primitive fourth root of unity vanishes on $\mathbb{C}P^2$, $\overline{\mathbb{C}P}^2$, and $S^2\times S^2$. In particular, it has the potential to detect exotic smooth structures.