# The (4, 27) minimal model — the physics twin of the (4, 27) torus knot

Computed 2026-09-23 from the standard Virasoro minimal-model formulas (Belavin–Polyakov–Zamolodchikov, 1984).
Correspondence with the torus knot T(4, 27): Hikami & Kirillov, *Torus knot and minimal model*, Phys. Lett. B 575 (2003).

## Headline numbers

| Quantity | Formula | Value for (p, q) = (4, 27) |
|---|---|---|
| Central charge | c = 1 − 6 (q − p)² / (p q) | **c = 1 − 6·529/108 = −511/18 ≈ −28.39** |
| Number of primary fields | (p − 1)(q − 1) / 2 | **39** |
| Lowest weight (ground field) | h_min = (1 − (q − p)²) / (4 p q) | **h(1,7) = −11/9** |
| Effective central charge | c_eff = c − 24 h_min = 1 − 6/(p q) | **17/18** |
| Unitary? | only if q = p + 1 | **No** (weights go negative) |
| Genus of the torus knot T(4, 27) | (p − 1)(q − 1) / 2 | **39** — same formula as the field count |
| Crossing number of T(4, 27) | min(p(q − 1), q(p − 1)) | 81 |
| Hikami–Kirillov's field (p − 1, 1) = (3, 1) | h(3,1) | 25/2 |

Every weight has a denominator dividing 4 p q = 432 = 4 × 108; 108 itself is the denominator of the central charge before reduction.

## The Kac table (all 39 fields, sorted by weight)

Fields (r, s) with 1 ≤ r ≤ 3, 1 ≤ s ≤ 26, identified under (r, s) ~ (4 − r, 27 − s). Weight h(r,s) = ((27 r − 4 s)² − 529) / 432.

| (r, s) | h | decimal |
|---|---|---|
| (1,7) | −11/9 | −1.2222 |
| (2,14) | −175/144 | −1.2153 |
| (1,6) | −65/54 | −1.2037 |
| (1,8) | −7/6 | −1.1667 |
| (2,15) | −493/432 | −1.1412 |
| (1,5) | −10/9 | −1.1111 |
| (1,9) | −28/27 | −1.0370 |
| (2,16) | −143/144 | −0.9931 |
| (1,4) | −17/18 | −0.9444 |
| (1,10) | −5/6 | −0.8333 |
| (2,10) | −37/48 | −0.7708 |
| (1,3) | −19/27 | −0.7037 |
| (1,11) | −5/9 | −0.5556 |
| (2,9) | −205/432 | −0.4745 |
| (1,2) | −7/18 | −0.3889 |
| (1,12) | −11/54 | −0.2037 |
| (2,19) | −5/48 | −0.1042 |
| (1,1) | 0 | 0 (the identity) |
| (1,13) | 2/9 | 0.2222 |
| (2,20) | 49/144 | 0.3403 |
| (1,14) | 13/18 | 0.7222 |
| (2,21) | 371/432 | 0.8588 |
| (1,15) | 35/27 | 1.2963 |
| (2,5) | 209/144 | 1.4514 |
| (1,16) | 35/18 | 1.9444 |
| (2,4) | 305/144 | 2.1181 |
| (1,17) | 8/3 | 2.6667 |
| (2,3) | 1235/432 | 2.8588 |
| (1,18) | 187/54 | 3.4630 |
| (2,25) | 529/144 | 3.6736 |
| (1,19) | 13/3 | 4.3333 |
| (2,26) | 73/16 | 4.5625 |
| (1,20) | 95/18 | 5.2778 |
| (1,21) | 170/27 | 6.2963 |
| (1,22) | 133/18 | 7.3889 |
| (1,23) | 77/9 | 8.5556 |
| (1,24) | 529/54 | 9.7963 |
| (1,25) | 100/9 | 11.1111 |
| (1,26) | 25/2 | 12.5 |

## Reading it honestly

- Evidence column: every number above follows from two formulas that any physicist can check, and the field count and the knot genus agree because they are the same expression, (p − 1)(q − 1)/2. That agreement is the structural reason the Hikami–Kirillov correspondence exists at all; it is not special to (4, 27).
- Non-unitary minimal models can describe real statistical systems (the Yang–Lee edge is M(2,5); percolation and polymers use M(2,3) and M(3,2) limits), but (4, 27) has no known physical realization. It is a valid label that nobody has had a reason to use.
- Meaning column: the number picked the model. The model did not pick the number.