In　this　paper　we　determine　the　effects　of　winding　number　on　the　dynamics　of　vortex　torus　knots　and　unknots　in　the　context　of　classical,　ideal　fluid　mechanics.　We　prove　that　the　winding　number　-　a　topological　invariant　of　torus　knots　-　has　a　primary　effect　on　vortex　motion.　This　is　done　by　applying　the　Moore-Saffman　desingularization　technique　to　the　full　Biot-Savart　induction　law,　determining　the　influence　of　winding　number　on　the　3　components　of　the　induced　velocity.　Results　have　been　obtained　for　56　knots　and　unknots　up　to　51　crossings.　In　agreement　with　previous　numerical　results　we　prove　that　in　general　the　propagation　speed　increases　with　the　number　of　toroidal　coils,　but　we　notice　that　the　number　of　poloidal　coils　may　greatly　modify　the　motion.　Indeed　we　prove　that　for　increasing　aspect　ratio　and　number　of　poloidal　coils　vortex　motion　can　be　even　reversed,　in　agreement　with　previous　numerical　observations.　These　results　demonstrate　the　importance　of　three-dimensional　features　in　vortex　dynamics　and　find　useful　applications　to　understand　helicity　and　energy　transfers　across　scales　in　vortical　flows.