By　using　and　extending　earlier　results　(Liu　&　Ricca,　J.　Phys.　A,　vol.　45,　2012,　205501),　we　derive　the　skein　relations　of　the　HOMFLYPT　polynomial　for　ideal　fluid　knots　from　helicity,　thus　providing　a　rigorous　proof　that　the　HOMFLYPT　polynomial　is　a　new,　powerful　invariant　of　topological　fluid　mechanics.　Since　this　invariant　is　a　two　variable　polynomial,　the　skein　relations　are　derived　from　two　independent　equations　expressed　in　terms　of　writhe　and　twist　contributions.　Writhe　is　given　by　addition/subtraction　of　imaginary　local　paths,　and　twist　by　Dehn＇s　surgery.　HOMFLYPT　then　becomes　a　function　of　knot　topology　and　field　strength.　For　illustration　we　derive　explicit　expressions　for　some　elementary　cases　and　apply　these　results　to　homogeneous　vortex　tangles.　By　examining　some　particular　examples　we　show　how　numerical　implementation　of　the　HOMFLYPT　polynomial　can　provide　new　insight　into　fluid-mechanical　behaviour　of　real　fluid　flows.