Cinquefoil Knot

cinquefoil rotate x 001
The cinquefoil knot is a 5-2-torus-knot, 5 times around the torus axis, 2 times around the central circle of the torus (see the anaglyph picture). usually the cinquefoil knot is drawn as the much nicer looking 2-5-torus-knot, which, in its standard position, has only 5 crossings. this is the minimal number of crossings for the cinquefoil knot. in the Clifford Torus page we show how a torus can be turned inside out, with only one point of the torus passing through infinity. if the knot avoids this point, we have a deformation of the 5-2-torus-knot into the 2-5-torus-knot.

The cinquefoil knot is a torus knot. Meaning, it can be drawn as a curve winding around a torus.

cinquefoil oblique rot 001
Cinquefoil knot, rotating around the torus's axis of revolution.

knot2 5to5 2 001
knot2 5to5 2 001

We can deform a torus m_n_knot with the help of the inside-out-deformation of the Clifford torus [see Clifford Torus] into a torus n_m_knot. We only need to place the knot in such a way on the torus that the torus point, which passes through infinity under the inside-out deformation, does not lie on the knot. In the animation the knot is realized as a strip on the torus by restricting a suitable parametrization to a narrow band.


cinquefoil knot cross eyed
The cinquefoil knot in its 5-2 version looks confusing without the torus, we show a cross-eye stereogram of this knot on the torus.
cinquefoil knot ana
the cinquefoil knot in its 5-2 version looks somewhat confusing without the torus, we show two stereo images of this knot on the torus.

Cinquefoil_Knot.pdf