We present results on the isovector momentum fraction, $langle x rangle_{u-d}$, helicity moment, $langle x rangle_{Delta u-Delta d}$, and the transversity moment, $langle x rangle_{delta u-delta d}$, of the nucleon obtained using nine ensembles of gauge configurations generated by the MILC collaboration using $2+1+1$-flavors of dynamical highly improved staggered quarks (HISQ). The correlation functions are calculated using the Wilson-Clover action and the renormalization of the three operators is carried out nonperturbatively on the lattice in the RI${}^prime$-MOM scheme. The data have been collected at lattice spacings $a approx 0.15, 0.12, 0.09,$ and 0.06 fm and $M_pi approx 310, 220$ and 135 MeV, which are used to obtain the physical values using a simultaneous chiral-continuum-finite-volume fit. The final results, in the $overline{MS}$ scheme at 2 GeV, are $langle x rangle_{u-d} = 0.173(14)(07)$, $langle x rangle_{Delta u-Delta d} = 0.213(15)(22)$ and $langle x rangle_{delta u-delta d} = 0.208(19)(24)$, where the first error is the overall analysis uncertainty and the second is an additional systematic uncertainty due to possible residual excited-state contributions. These results are consistent with other recent lattice calculations and phenomenological global fit values.
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