The Dehn function of $V$ is at most sextic

A remarkable result of Guba shows that the Dehn function of Thompson's group $F$ is quadratic (Inv. Math. 163:313-342). In another paper (Alg. prob. in Grps. and Semigrps, 91-120), Guba showed that Thompson's group $T$ and $V$ have Dehn functions bounded above by $n^7$ and $n^{11}$, respectively. Recently, Migliorini showed that the Dehn function of Thompson's group $T$ is also quadratic (For. of Math. Sigma, 13:e109). In this paper, we improve the upper bound of the Dehn function of Thompson's group $V$ to $n^6$ and explore the intricacies of its relations via explicit van Kampen diagrams.