On detection probabilities of link invariants
This addresses a fundamental limitation in knot theory for mathematicians and physicists studying quantum invariants, showing their asymptotic ineffectiveness for alternating links.
The authors proved that link invariants insensitive to oriented mutation detect alternating links with exponentially decaying probability as crossing number increases, showing they detect such links with probability zero, and provided evidence that this applies broadly to quantum invariants like Jones polynomials.
We prove that the detection rate of n-crossing alternating links by link invariants insensitive to oriented mutation decays exponentially in n, implying that they detect alternating links with probability zero. This phenomenon applies broadly, in particular to quantum invariants such as the Jones or HOMFLYPT polynomials. We also use a big data approach to analyze several borderline cases (e.g. integral Khovanov or HOMFLYPT homologies), where our arguments almost, but not quite, apply, and we provide evidence that they too exhibit the same asymptotic behavior.