The Volume of the Knot Potential: Decay and Direction on the Vacuum Disk

VOTKP began as the volume of the knot potential and became the vector of traits kept under peeling. This paper joins the two meanings through the two books of the intrinsic quantum geometric tensor, Q = g - (i/2)Ω: a climb (metric, size) and a turn (curvature, phase). Everything is read on one disk whose centre is the vacuum, the regular ideal tetrahedron, and whose rim is ground. The vacuum disk. Every ideal tetrahedron of a hyperbolic knot complement is placed on a disk centred on the regular tetrahedron. The three ways of reading a tetrahedron are exact 120-degree turns about the centre: the axis switches, the volume does not, like a tennis racket spun about its middle axis. Flat tetrahedra sit on the rim; mirrors sit outside it. Decay. A tetrahedron's volume is a bowl around the vacuum, given by an exact series. Summed over the pieces, with nothing fitted, it gives the hyperbolic volume of 12,954 knots to a median error of 0.076%. The true geometry of a knot is always the solution nearest the vacuum (31 of 31 knots). Direction. The Chern-Simons twist is carried by integer windings. Read in a branching order, it is reproduced exactly in 454 of 454 runs. Our polynomial. The knot potential, read straight off a knot drawing with four dilogarithm terms per crossing, gives one polynomial whose roots each carry decay and direction together. On 79 of 79 hyperbolic knots its top root is the complex volume; its spectrum is the same from different drawings (13 of 13 knots); torus knots give pure-direction roots, zero decay and a rational fraction of a turn. Five of the six knots known to tie themselves in chemistry are of this torus kind. Supporting lines. Jones and HOMFLY strings along the sl(N) lines, the colour of the elements on the same disk (all 35 metals tested sit above the half line, no non-metal does), and the electron's group SL(2,C), whose quotient is exactly the motions of hyperbolic space, complete the chain. Files: the paper as PDF (with the full code appendix) and HTML; the interactive 3D mixer (open in any browser); a narrated film recorded live from the mixer; and a zip with every script, the figures, the source, the research notes and the mixer and film sources. AI-assisted: computations, drafting and literature checks were carried out with Claude (Anthropic) and helper agents, from the author's ideas and earlier papers.