Here is a list of all the prime knots up to 10 crossings. But to be honest, it’s just on Wikipedia, so you probably could have googled it yourself. For more excitement, you’re going to need the Knot Atlas.
List of Prime Knots (on Wikipedia)
The Knot Atlas (website)
As calculated by the researchers at Queen Mary University of London (the university where I’m based!), here are their results on how graph theory can predict the performance of football teams.
2010 Football World Cup Graphs (website)
To stay abreast of the absolute latest developments in the field of colouring in maps with four or fewer colours, these are the links you’ll need:
The Four Colour Theorem (website)
An Update on the Four Colour Theorem (PDF)
Download: Perfect-Herschel polyhedron net
Download and print this net of a Perfect-Herschel Polyhedron so you can have one of your very own. Then challenge a friend to find a path that goes through each corner once only. It’s such a small polyhedron; it must be possible! (Spoiler: it’s not.)
Herschel Polyhedron Net (PDF)
Sorry everyone, we lied. The Four Colour Theorem has been disproven by this map. Unless of course, you can find a way to colour it in with four colours and save mathematics!
Martin Gardner’s 5-colour map (PDF)
Can you tell the difference between the layout of the states of the USA and the counties of England? Here they’re both represented as a graph, and it’s a puzzle to work out which one is which.
Graphs of England and the USA (PDF)
Some great animations of 4D shapes, by the awesome Davide P. Cervone. Second-order credit also goes to Davide’s PhD supervisor Thomas F. Banchoff who created the first computer visualisations of 4D shapes.
Some notes on the Fourth Dimension (website)
If a standard 3D Rubik’s Cube gets you so frustrated you want to throw it away in a direction orthogonal to all three normal spacial dimensions, then maybe this 4D Rubik’s cube is not for you.
4D Rubik’s Cube (website)