Knots (with METAFONT)

The knot drawings were first shown on the page Knoten - Seminar in 2021/2022.

Another display of essentially the same drawings is on the page Knots (with METAFONT) - Gallery.

Here is a particular example (click on the image for a high resolution version):

knot 5₂

The pictures were created 1995 using METAFONT, see allknots.pdf and Source files (1995).

These were METAFONT experiments inspired by Exercise 13.8 in The METAFONTbook Zbl 0609.68003. I stopped after saving allknots.pdf (actually allknots.dvi) on July 6, 1995 at 4:11 AM.

There are 4 knot diagrams here with 9 crossings. One is a non-standard drawing of knot 8₁₈ (Torusknoten (4,3)), the others are

knot 9₂₂knot 9₂₉knot 9₄₀

Cf. Appendix D in Burde, Zieschang. Knots. 1985 MR 808776, Zbl 0568.57001:

Screenshots from Burde-Zieschang

Originally all over/under-crossings were drawn alternating. This is incorrect for the torus knots (3,4), (4,3), (6,11), albeit one may prefer the alternating version in a non-mathematical context. The corresponding images have now (March 2026) correct crossings (unless marked with *).

On the page Knots (with METAFONT) - Gallery most torus knots are rectified to have a vertical axis of symmetry, cf. Drawing torus knots on that page.

Quite a few torus knots are depicted here: Category:Torus knots and links (Wikimedia Commons)

The images linked in the following list have high resolutions. The numbers correspond to page numbers in allknots.pdf.

The names "knot C_k", "Knoten C/k" refer to the Alexander-Briggs notation, see Appendix C in Rolfsen, Knots and Links. 1976 MR 515288, Zbl 0339.55004.

torus knot (2,3) - knot 3₁
- Torusknoten (2,3) [thin version]
- Common names: Kleeblattschlinge, trefoil knot (Wikipedia)
- (2,3)_torus_knot.svg (Wikimedia)
knot 4₁
- Viererknoten (four-knot) [thin version]
- By some called Achterknoten, figure-eight knot (Wikipedia).
- The names Viererknoten, four-knot appear in
  - Burde, Zieschang. Neuwirthsche Knoten und Flächenabbildungen. 1967 MR 229229, Zbl 0171.22402.
  - Burde, Zieschang. Knots. 1985 MR 808776, Zbl 0568.57001.
  - Burde, Zieschang. Knots (second edition). 2003 MR 1959408, Zbl 1009.57003.
  - Burde, Zieschang, Heusener. Knots (third, fully revised and extended edition). 2014 MR 3156509, Zbl 1283.57002.
torus knot (2,5) - knot 5₁
- Torusknoten (2,5) [thin version]
- Other name: cinquefoil knot (Wikipedia)
- (2,5)_torus_knot.svg (Wikimedia)
knot 5₂
- "Knoten 5/2 (Versuch zu 9/29)" [thin version]
- "Knoten 5/2 (Versuch zu 9/29)" (gespiegelt) [thin version] (mirrored)
- - well, ok for a first try (as for the drawing)
- This is the twist knot with three half twists, see three-twist knot (Wikipedia).
- Apparently this is an unusual way of drawing knot 5₂ (I couldn't find a similar image elsewhere).
torus knot (2,7) - knot 7₁
- Torusknoten (2,7) [thin version]
- (2,7)_torus_knot.svg (Wikimedia)
torus knot (3,4) - knot 8₁₈
- Torusknoten (3,4) [thin version]
- Torusknoten (3,4) [thin version] (mirrored)
- *Torusknoten (3,4) [thin version] (with alternating crossings)
- (3,4)_torus_knot.svg (Wikimedia)
torus knot (4,3) - knot 8₁₈ by a diagram with 9 crossings
- Torusknoten (4,3) [thin version]
- Torusknoten (4,3) [thin version] (mirrored)
- *Torusknoten (4,3) [thin version] (with alternating crossings)
- I couldn't find the drawing elsewhere.
"Knoten 9/22" [thin version] - a strange knot (I wasn't satisfied with this one)
"Knoten 9/29" [thin version] - looks better (my favorite)
"Knoten 9/40" [thin version] - not bad
torus knot (6,11) - 55 crossings
- Torusknoten (6,11) [thin version]
- Torusknoten (6,11) [thin version] (mirrored)
- *Torusknoten (6,11) [thin version] (with alternating crossings) - another favorite
- I couldn't find the drawing elsewhere.
torus knot (6,11) with numbered crossings is a scan from page 12 of allknots.pdf.

Source files (1995)

The file allknots.txz (xz compressed tar archive) contains the 1995 MF files and a script to build allknots.pdf.

In 2009 I created png images with ImageMagick taking data from MF log files. The cutting at crossings was done with ImageMagick. See directory 2009 in allknots.txz.

In 2026 I added the possibility to draw torus knots with correct over/under-crossings (fortunately there was not much to do).