Graphs and topology
I've been diving down a [graph theory + topology] rabbit hole lately; here's some of the fruits of my labor.
("Rabbit hole? Fruit? Mix metaphors much?" Not at all, this is a peanut.)
As you likely know, K_n is the "complete graph" with n nodes, i.e. an edge between every pair of nodes. K_1 is a dot, K_2 is a line segment, K_3 a triangle; you can get K_4 by putting a dot in the middle of a triangle and adding spokes.
K_5 is famously one of the simplest graphs you can't draw without two edges crossing. ...on a plane, that is! But if you're drawing on a donut, you're in luck:
The obvious next question is, can we draw K_6 on a donut? Yes! K_7? Yes! K_n? ...turns out no, 7 is as high as you can go before you need to get more topologically interesting.
...and of course, given that, it makes perfect sense...
This exhibits a cellular embedding of the complete graph on 7 vertices on a torus. K_7 is the largest complete graph that embeds on the tor
...that people who spend a lot of time thinking about K_7 tend to be into knitting.
