On the left is a complicated alternating knot, with some of its tangles highlighted. One effective way to study such knots is to pretend the diagram bounds a right angled polyhedron. On the right is what happens if you really turn it into a right angled polyhedron. #KnotTheory

The knot 8_10 shown with three Seifert disks, from two angles. In 1934 Seifert showed that every knot has a surface bounded along it. Here, the Seifert surface is formed by attaching 8 twisted bands to the three disks, one for each crossing. #KnotTheory

The alternating knot 10_15, with a checkerboard surface. This surface has boundary on the knot, with twisted bands connecting gray regions shown at crossings. Every knot has checkerboard surfaces, but they're only guaranteed to be essential for alternating knots. #KnotTheory

Ben Burton's work has been featured in a recent New Scientist article, which also discusses how sophisticated algorithms can help solve all sorts of seemingly impossible mathematical problems. #KnotTheory UQ Science newscientist.com/article/mg2553…

A weakly generalised alternating (WGA) diagram of the knot 10_161. A WGA diagram lives on a surface -- here, a torus -- and its crossings alternate over and under on the surface. Such a knot has essential checkerboard surfaces, just like classical alternating knots. #KnotTheory

This link comes from dynamical systems. It consists of three periodic orbits of the flow determined by Arnold's cat map. The black curve is the period 1 orbit. The two blue curves are the two period 2 orbits. I thought the period 2 orbits would be more exciting. #KnotTheory

Could you design the new AustMS logo? austms.org.au/could-you-desi…

Like last week's tweet, this link consists of periodic orbits arising from a flow given by Arnold's cat map. The black curve is the period 1 orbit. The blue curve is another, much more interesting, orbit. This appeared in a paper of Birman and Williams in 1983. #KnotTheory

You may not like it, but this is what the perfect body looks like. #rats #pets #science

Today's knot is actually the unknot: you can move it around without cutting to become a simple circle with no crossings. This diagram is in a paper of Lebrecht Goeritz from 1934. #KnotTheory

The knot K6_3. This is an example of a twisted torus knot. Although the diagram has 46 crossings, the knot complement (the space around the knot) can be completely filled with only 6 ideal tetrahedra. #KnotTheory

The knot K4_3. Continuing the theme of high crossing knots whose complements (the space around them) is made up of low numbers of ideal tetrahedra. This one can be built of only 4 tetrahedra. #KnotTheory

Our next (GT)^2 talk is on Thursday 15 September at 10am AEST with Connie Hui talking about ‘Why and how do we study links in the 3-torus? ’ #MathSeminar #GraduateMathematics #Topology #Geometry MATRIX

The (7,2)-torus knot, on the left, has 7 strands coming in from above and 2 running across horizontally. Shown here are a few steps in a process to turn the (7,2)-Torus knot into the (2,7)-Torus knot. Do you see how to go from here? #KnotTheory

This is a definite improvement, and a sign that the government is listening. But there’s still no compelling reason to retain the National Interest Test at all.

Thanks for hosting us. It has been great spending the week doing mathematics at the MATRIX Institute.

The Government’s Science, Research and Innovation Budget tables reveal that Gov't investment in R&D is at the lowest on record at 0.49% (as a percentage of GDP). In February, Australia’s leading scientists urged the government to commit to a ‘bold & ambitious’ structural reform

I'm going to be giving a public lecture on 12 May, for International Women in Mathematics Day. If you're in the Melbourne area, you are welcome to come along!

Propose special sessions and speakers for the joint meeting of the AMS, NZMS, and Australian Mathematical Society, Dec. 9-13, 2024, in Auckland, NZ. Currently, there is no deadline for session proposals, which will be considered on a rolling basis. Speaker nominations due Sep. 1. ow.ly/33oM50OX60G