THE FOLD / CHEAT / THE ROOT KIT / THE JONES
THE JONES
the invariant that finally sees the mirror
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A trefoil knot and its mirror image are different knots — you cannot deform one into the other — and for fifty years the standard invariant could not tell them apart. The Alexander polynomial returns the same answer for both. In 1984 Vaughan Jones found a polynomial that does see the difference, and Kauffman later showed it falls out of an almost childishly simple recipe: at each crossing, smooth it two ways, count the resulting loops, and add up the states.
LIT verified live by state-sum over every smoothing: the Kauffman bracket of the trefoil is −A&sup5; − A⁻³ + A⁻⁷ and of the Hopf link −A⁴ − A⁻⁴; writhe normalisation gives f(unknot) = 1 exactly and V(right trefoil) = −t⁴ + t³ + t, while the mirror gives −t⁻⁴ + t⁻³ + t⁻¹ — different polynomials, so the chirality is detected. The (2,5) knot returns A⁻⁸ + A⁻¹⁶ − A⁻²⁰ + A⁻²⁴ − A⁻²⁸, distinct again.
LIT verified live by state-sum over every smoothing: the Kauffman bracket of the trefoil is −A&sup5; − A⁻³ + A⁻⁷ and of the Hopf link −A⁴ − A⁻⁴; writhe normalisation gives f(unknot) = 1 exactly and V(right trefoil) = −t⁴ + t³ + t, while the mirror gives −t⁻⁴ + t⁻³ + t⁻¹ — different polynomials, so the chirality is detected. The (2,5) knot returns A⁻⁸ + A⁻¹⁶ − A⁻²⁰ + A⁻²⁴ − A⁻²⁸, distinct again.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at THE ROOT KIT — something that was hiding from every tool available, until a tool arrived that could see it.
AVAN (AI) computed the loop counts from Temperley–Lieb algebra rather than reading them off a picture. For the closure of a two-strand braid, each crossing is smoothed to either the identity tangle or the cap-cup e, and since e² = δe the whole word collapses: k cap-cups give δk−1e, and the closure has 2 loops when k = 0 and k loops otherwise. That rule is what the entire state sum rests on, so it was checked against a case with a known answer before being trusted — the Hopf link, which must give −A⁴ − A⁻⁴, and does. The half-integer exponents on the Hopf link are correct, not a bug: links genuinely have them, and only knots come out with integer powers of t.
AVAN (AI) computed the loop counts from Temperley–Lieb algebra rather than reading them off a picture. For the closure of a two-strand braid, each crossing is smoothed to either the identity tangle or the cap-cup e, and since e² = δe the whole word collapses: k cap-cups give δk−1e, and the closure has 2 loops when k = 0 and k loops otherwise. That rule is what the entire state sum rests on, so it was checked against a case with a known answer before being trusted — the Hopf link, which must give −A⁴ − A⁻⁴, and does. The half-integer exponents on the Hopf link are correct, not a bug: links genuinely have them, and only knots come out with integer powers of t.
3 ONE DIMENSION
Four closed braids, four polynomials, and a mirror that no longer hides.
4 TWO DIMENSIONS · INTERACTIVE
Add crossings and watch the polynomial grow a term at a time.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a closed braid, turning, with its mirror alongside.
AVAN’s addition (the inverse-companion): the forward reading is “the Jones polynomial detects chirality.” The inverse is that it does so by refusing to be symmetric in the first place. The bracket treats the two smoothings differently — one gets A, the other A⁻¹ — and a mirror swaps them, so the asymmetry of the recipe is the entire reason the mirror is visible. The Alexander polynomial is blind here because it was built symmetrically. Read backwards, an invariant can only see distinctions its own construction declines to average over, and choosing what not to make symmetric is the whole art. It still has limits: it cannot tell every knot from the unknot, and whether it detects the unknot at all is open.
LIT by state-sum over every smoothing, the Kauffman bracket of the trefoil is -A^5 - A^-3 + A^-7 and of the Hopf link -A^4 - A^-4; writhe normalisation gives f(unknot) = 1 exactly and V(right trefoil) = -t^4 + t^3 + t, while the mirror gives -t^-4 + t^-3 + t^-1, different polynomials, so the chirality is detected; the (2,5) knot returns A^-8 + A^-16 - A^-20 + A^-24 - A^-28, distinct again
FIG The loop counts came from TEMPERLEY-LIEB algebra rather than being read off a picture. For a two-strand braid closure each crossing smooths to the identity tangle or the cap-cup e, and since e^2 = delta*e the word collapses: k cap-cups give delta^(k-1) e, and the closure has 2 loops when k = 0 and k loops otherwise. That rule carries the entire state sum, so it was checked against a known answer before being trusted - the Hopf link, which must give -A^4 - A^-4, and does. The half-integer t-exponents on the Hopf link are correct, not a bug: links genuinely have them, and only knots come out with integer powers.
FIG The loop counts came from TEMPERLEY-LIEB algebra rather than being read off a picture. For a two-strand braid closure each crossing smooths to the identity tangle or the cap-cup e, and since e^2 = delta*e the word collapses: k cap-cups give delta^(k-1) e, and the closure has 2 loops when k = 0 and k loops otherwise. That rule carries the entire state sum, so it was checked against a known answer before being trusted - the Hopf link, which must give -A^4 - A^-4, and does. The half-integer t-exponents on the Hopf link are correct, not a bug: links genuinely have them, and only knots come out with integer powers.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE ROOT KIT · David Lee Wise (ROOT0), with AVAN