What　makes　a　given　knot＇s　unlinking　pathway　more　likely　to　appear　in　certain　topological　decay　processes-from　DNA　to　vortex　dynamics?　Here,　the　authors　introduce　a　measure　of　topological　complexity　as　distance　in　a　suitably　defined　knot　polynomial　space,　allowing　them　to　quantify　the　probability　of　decay　pathways　as　geodesic　flows　in　this　space.　Physical　knots　observed　in　various　contexts　-　from　DNA　biology　to　vortex　dynamics　and　condensed　matter　physics　-　are　found　to　undergo　topological　simplification　through　iterated　recombination　of　knot　strands　following　a　common,　qualitative　pattern　that　bears　remarkable　similarities　across　fields.　Here,　by　interpreting　evolutionary　processes　as　geodesic　flows　in　a　suitably　defined　knot　polynomial　space,　we　show　that　a　new　measure　of　topological　complexity　allows　accurate　quantification　of　the　probability　of　decay　pathways　by　selecting　the　optimal　unlinking　pathways.　We　also　show　that　these　optimal　pathways　are　captured　by　a　logarithmic　best-fit　curve　related　to　the　distribution　of　minimum　energy　states　of　tight　knots.　This　preliminary　approach　shows　great　potential　for　establishing　new　relations　between　topological　simplification　pathways　and　energy　cascade　processes　in　nature.