A　new　method　based　on　the　use　of　the　Jones　polynomial,　a　well-known　topological　invariant　of　knot　theory,　is　introduced　to　tackle　and　quantify　topological　aspects　of　structural　complexity　of　vortex　tangles　in　ideal　fluids.　By　re-writing　the　Jones　polynomial　in　terms　of　helicity,　the　resulting　polynomial　becomes　then　function　of　knot　topology　and　vortex　circulation,　providing　thus　a　new　invariant　of　topological　fluid　dynamics.　Explicit　computations　of　the　Jones　polynomial　for　some　standard　configurations,　including　the　Whitehead　link　and　the　Borromean　rings　(whose　linking　numbers　are　zero),　are　presented　for　illustration.　In　the　case　of　a　homogeneous,　isotropic　tangle　of　vortex　filaments　with　same　circulation,　the　new　Jones　polynomial　reduces　to　some　simple　algebraic　expression,　that　can　be　easily　computed　by　numerical　methods.　This　shows　that　this　technique　may　offer　a　new　setting　and　a　powerful　tool　to　detect　and　compute　topological　complexity　and　to　investigate　relations　with　energy,　by　tackling　fundamental　aspects　of　turbulence　research.