Torus Knots — Crossing Number & Ambient Isotopy

Torus Knot T(p, q)

Winding p (around tube) 2

Winding q (around hole) 3

Continuous deformation (ambient isotopy)

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Knot Data

Trefoil knot (3₁)

Live diagram crossings

Minimal crossing number

Genus g = (p-1)(q-1)/2

How it works

A (p, q) torus knot is drawn by winding a curve p times around the tube of a torus and q times through its hole, for coprime integers p and q:

x(t) = (R + r·cos qt)·cos pt
y(t) = (R + r·cos qt)·sin pt
z(t) = r·sin qt ,   t ∈ [0, 2π]

Every knot has infinitely many 2D diagrams — one per viewing angle — and most of them are not the simplest possible. What you see in "Live diagram crossings" is the crossing count of the current projection, recomputed from the actual screen-space positions of the tube's centerline as you rotate or deform it.

The minimal crossing number is a true knot invariant — it never changes, no matter how you rotate or continuously stretch the curve (an ambient isotopy), because it is the smallest crossing count achievable over every possible diagram. For torus knots it is known exactly (Murasugi):

c(p,q) = min( p(q−1), q(p−1) )
genus:  g(p,q) = (p−1)(q−1) / 2

Turn on continuous deformation to wobble the embedding smoothly — the live count jumps around, sometimes far above the minimum, but by the theorem it can never drop below it. That gap is the whole point: crossing number is a property of the knot, not of any one picture of it.

p / q sliders — pick a torus knot type (T(2,3) trefoil, T(2,5) cinquefoil, T(3,4), T(3,5), …). Non-coprime pairs describe a multi-strand torus link rather than a single knot, so a slider that would collide is nudged to the nearest coprime value automatically.

Torus knot tube Guide torus (wireframe)

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