Since I've not gotten any answers after a bit more than a week, I've now cross-posted to MathOverFlow.

EDIT 2023-08-15: Several commenters here and at MO have asked me to sharpen the original question. I've tried to do that there, and I've reproduced the changes (hopefully, improvements) here.

Papers and books in geometry have been using illustrations for ages - for example there are figures in papyrus rolls containing bits of Euclid's Elements. My question, then, is

What are some of the earliest non-trivial illustrations of topological ideas appearing in published mathematical articles?

I am particularly interested in tracing the history of illustration in modern geometric (or "low-dimensional") topology. So I am mostly looking for early pictures of knots and surfaces (and three-manifolds - I give an example due to Poincaré below). But all examples are welcome!

Examples and non-examples:

  1. Non-example: The diagrams in editions of Euclid are always slightly wonky - drawing the Platonic ideal circle is impossible! So they are topological illustrations of geometric ideas.

  2. Example: The Heegaard diagram [Figure 4] in the fifth supplement to Analysis Situs [Poincaré, 1904]. It is not clear how Poincaré found this example and it is not clear how else he might have communicated the information.

  3. Non-examples: The various carved Celtic knots, or braids appearing as the frames of mosaics, or the coat of arms of the House of Borromeo, or metal links made of many unknots, or indeed textile arts (much older than publishing and, indeed, writing). Of course such artefacts are important, mathematical, and ancient. But it is impossible to point at one of them and say "this was the first one". This is why (perhaps wrongly!) I have restricted to "published" papers.

  4. Non-example: Euler's diagram [1735] of the bridges of Konigsberg does not count. This is because (I feel that) (a) it is a bit too close to a literal (as opposed to topological) figure and (b) graph theory is more properly a subfield of combinatorics, rather than of topology. (Of course, some people (such as Euler) disagree with (b).)

  5. Example: Listing's figures of knots [1848] in Vorstudien zur Topologie. The classic example, inspiring Tait's work on knot tabulation.

  • $\begingroup$ Even if graph theory is combinatorics properties of planar graphs, including Königsberg bridges, involve what is now called combinatorial topology. Would polyhedral diagrams accompanying Euler's formula and the like count? Even Euclid uses topological considerations in some demonstrations, e.g. continuity and topology of the plane to assert intersections (later codified in the line-circle and Pasch's axioms). Would Euclidean diagrams count? $\endgroup$
    – Conifold
    Aug 6 at 1:06
  • $\begingroup$ @Conifold - the link is broken? $\endgroup$
    – Sam Nead
    Aug 6 at 1:47
  • 2
    $\begingroup$ Part of the problem with the question is that what you count seems rather vague and based on personal preferences. Here is from May's Historiographic Vices: Priority Chasing:"The hope of finding a 'first' comes to grief because of the historically dynamic character of ideas... If we describe a result with sufficient vagueness, there seems to be an endless sequence of those who had something within the vague specifications." It is more productive to track gradual crystallization of a conception as it develops, without inclusion/exclusion. $\endgroup$
    – Conifold
    Aug 14 at 18:12
  • 1
    $\begingroup$ It would also help to spell out the underlying reason for why illustrations of topological properties of graphs and polyhedra are out but of knots are in, so that people can tell rather than having to show you the illustrations and ask. "Too literal", "too combinatorial" and "too geometrical" are not exactly illuminating. After all, topology is flashed out through geometry, combinatorics, etc., which is what Listing, and Gauss before him, did with knots. I am not sure if it makes sense to look for Leibniz's or Gauss's illustrations, for example, because they might be too something. $\endgroup$
    – Conifold
    Aug 15 at 8:02
  • 1
    $\begingroup$ Just in case, look at Classical Roots of Knot Theory by Przyticki and Gauss’s Linking Number Revisited by Ricca and Nipoti, and references therein. $\endgroup$
    – Conifold
    Aug 15 at 8:05


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