THE WRITHE a signed crossing count — the coiling a knot carries with it
Lay a knot flat as a diagram and look at its crossings: at each, one strand passes over the other, and a right-hand rule gives it a sign, +1 or −1. The writhe is the sum of these signs. On its own the writhe depends on the diagram — but it enters the beautiful Călugăreanu–White–Fuller theorem, Lk = Tw + Wr: the linking number of a closed ribbon (a topological invariant, fixed) equals its twist plus its writhe. So as you deform the ribbon, twist can trade into writhe and back, but their sum cannot change. This is not abstract: it is why a coiled phone cord, or a supercoiled DNA molecule, converts twist into writhing loops — the total linking is conserved.
THE TECHNIQUE writhe = sum of signed crossings; Lk = Tw + Wr conserved
A trefoil diagram with three crossings of the same handedness. The demo sums their signs to the writhe: live demo
HISTORY & CREDIT Călugăreanu–White–Fuller, 1959–71
“Twisting a ribbon and coiling it are unrelated.” — Lk = Tw + Wr says the total is fixed, so twist you put in must come out as writhing coils. A phone cord tangles for exactly this reason. cited
1959–61 · Gheorghe Călugăreanu — the twist–writhe relation for a ribbon. 1969–71 · James White & Brock Fuller — the modern Lk = Tw + Wr theorem. now · DNA supercoiling, polymer physics, fluid vortex helicity.
The linking number is fixed by topology; twist and writhe are free to trade, so long as their sum holds. Coil a twisted band and the twist becomes writhe — the ledger balances. Călugăreanu-White-Fuller
RECOMMEND FOR I-13 trefoil writhe, computed
On the canonical compiler, a trefoil's three like-signed crossings sum to writhe 3:
$ i13 run inv_writhe.i13 # sum of signed crossings
writhe = 3 -- three +1 crossings (a trefoil)
Recommend: the writhe is a signed integer count i13 sums directly, and it earns its place in the batch through Lk = Tw + Wr: the linking number is the true invariant (fixed by topology), and the writhe is the visible half of a conserved ledger. i13 adds the crossing signs to get 3; the deform-and-trade invariance (twist ↔ writhe, total fixed) is the same enacted pattern as Poincaré–Hopf's index bookkeeping (dart 284) — a quantity free to move locally, pinned globally. A knot invariant, computed as a one-line sum of ±1.