Same knot?
Eight sculptures by Werner Alois Stanggassinger. Each one is a single closed tube: a mathematical knot.
The puzzle. Imagine each sculpture made of rubber hose: you may bend, stretch and twist it as much as you like, but never cut it. Which of the eight can be deformed into one another?
Bonus. One of them is the mirror image of two others. Which one? Can it be turned into them?
Solution
Eight sculptures, four knots.
| Sculptures | Knot | How we know |
|---|---|---|
| 1, 4, 7 | 12a_1019 (12 crossings) | Equal invariants and isometric hyperbolic complements, checked by computer. By Gordon–Luecke, knots with homeomorphic complements are the same knot up to mirror image; this knot is amphichiral, so the mirror question does not arise. |
| 2, 6 | 18ah_5090151 in Burton's census (18 crossings) | Equal invariants and isometric hyperbolic complements; this knot is amphichiral as well. |
| 5, 8 | 819, the torus knot T(3,4), signature −6 | Identified from the centerline curve alone, without looking anything up. |
| 3 | 819 again, but with signature +6 | The mirror image of 5 and 8. 819 is chiral: no rubber-hose move turns it into its mirror image. The signature tells them apart; the determinant (3 for both) does not. |
So the bonus answer is 3, and no.
One more thing to see. The knot of 1, 4 and 7 is its own mirror image — yet only one of the three sculptures shows it. Sculpture 7 is carried onto itself by a rotary reflection (a sixth of a turn combined with a reflection) and by a point reflection through its centre. Sculptures 1 and 4 have only the threefold rotation: their mirror symmetry is hidden and comes out only after deforming them. Sculpture 4 is even assembled from a module and its mirror image, three times over — and still the finished shape is not mirror-symmetric. (Measured on the centerline curves: every rotation, reflection and rotary reflection that maps the curve onto itself.)
1 Proxima B · 2 Sundown · 3 Number 5 · 4 Double 3 · 5 Why Not · 6 Trainstation · 7 Always a Pleasure · 8 XYZ Free View. Built in stainless steel: 3, 4, 6, 8; the others are rendered designs. Identification in collaboration with Claude Code (Anthropic), using SnapPy, Regina and SageMath. All nineteen sculptures with their invariants: Sculptures.