A Powerful New 'QR Code' Untangles Math's Knottiest Knots | Quanta Magazine

knot-theoryknot-invarianttopologykontsevich-integral

Abstraction: New computable knot invariant both strong and fast visualized as hexagonal QR code

Key points:

Bar-Natan (Toronto) and van der Veen (Groningen) developed a knot invariant that is both strong and easily computable — a "sweet spot" eluding a century of knot theory
Invariant computable for knots with 300+ crossings (even aspects at 600+); identifies more than 97% of knots with 18 crossings, vs. 42% for Jones polynomial and 11% for Alexander polynomial
Output is a colorful hexagonal "QR code" — heat map of polynomial coefficients; two different QR codes guarantee different knots
Method uses a traffic analogy: cars traversing a knot-as-highway, with probabilistic down-ramps at crossings; extended to two car types that can combine/split (like particles), inspired by quantum physics
Conjectured equivalent to the "two-loop polynomial" (second approximation to the Kontsevich integral), which would instantly confirm its deep topological properties
Hexagon diameter may provide a computable lower bound on knot genus — previously hard to calculate for large knots

Connections: Dror Bar Natan · Quanta Magazine · Knot Theory · Topology · Mathematics