A graph is intrinsically linked (resp. intrinsically knotted) if it contains, in every spatial embedding, a pair of cycles that form a nonsplit link (resp. a cycle that forms a nontrivial knot). In the early 1980s, Conway-Gordon and Sachs showed that the complete graph on 6 vertices is intrinsically linked, and Conway-Gordon showed that the complete graph on 7 vertices is intrinsically knotted. Sachs’ linkless embedding conjecture is that the Petersen Family of graphs (those obtained from $K_6$ by triangle-Y and Y-triangle exchanges) form the complete set of minor-minimal (in some sense, simplest) intrinsically linked graphs. In the early 1990s, Robertson, Seymour and Thomas proved Sachs’ linkless embedding conjecture; in a formidable work spanning three journal articles covering 99 pages. In the early 2000s, Flapan encouraged researchers to find another proof, leveraging more topology. Since that time, the speaker has been in and out of the rabbit hole of seeking a new proof.

Minor-minimal intrinsically knotted graphs have not yet been fully characterized, and hundreds of such graphs have been found (Foisy, Goldberg-Mattman-Naimi, Kohara-Suzuki Schwartz, etc…). The problem of classifying all such graphs seems elusive at this time.

In this talk, we present a survey of minor-minimal intrinsically knotted and intrinsically linked graphs, as well as discuss an outline of an incomplete alternate proof of Sachs’ linkless embedding conjecture, in the hopes that someone within earshot of this talk will be inspired to complete Flapan’s vision of a new proof.