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We present results on the isovector momentum fraction, $\langle x \rangle_{u-d}$, helicity moment, $\langle x \rangle_{Δu-Δd}$, and the transversity moment, $\langle x \rangle_{δu-δd}$, 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 $M_π\approx 310,\ 220$ 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 $\langle x \rangle_{u-d} = 0.173(14)(07)$, $\langle x \rangle_{Δu-Δd} = 0.213(15)(22)$ and $\langle x \rangle_{δu-δd} = 0.208(19)(24)$, 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.