Twist polynomial as a weight system for set systems

Recently, Chmutov (2023) proved that the partial-dual polynomial considered as a function on chord diagrams satisfies the four-term relation. Deng et al. (2023) then proved that this function on framed chord diagrams also satisfies the four-term relation, i.e., is a framed weight system. In this paper, we extend their results to the twist polynomial of a set system by proving that the twist polynomial on set systems satisfies the four-term relation and therefore determines a finite type link invariant. We investigate the behavior of the polynomial with respect to the Hopf algebra structure on the space of binary delta-matroids and show that the twist polynomial selects $n$ primitive elements in degree $n$.