[unnatural-transformations]

Quantum Groups & Knot Invariants [1 of n]

This is the first installment of what I hope will eventually be a series of posts. I'll be making them over the next few months as I try to improve my understanding of the relationship between knot invariants and the representation theory of quantum groups. In particular, my minimal goal for the year is to learn how to calculate the Jones polynomial of a link via the representation theory of the q-deformed analogue of the universal enveloping algebra of sl_2 (where sl_2 is the Lie algebra of 2 by 2 matrices with zero trace).

While I hope these posts will be of interest to others (and I will try to write them with some sort of general reader in mind), I should admit up front that I'm mostly writing them for myself. I know (or used to know) a bit about both knot invariants and quantum groups separately, but it's been a while since I looked at either subject and I never really knew much about how they were connected. I'm challenging myself to try to improve on that state of affairs: first by refreshing my old knowledge and then by moving on to something new [or at least, new to me: I'm not doing anything original here].

My idea is that I'm more likely to achieve this goal if I have at least a notional audience to keep me honest about my progress. Hence, this blog and this series of posts.

(I'm also using this as an excuse to try to teach myself to make better use of LaTeX packages like TikZ and xypic. Since Tumblr really doesn't have any good way of rendering mathematics, you should expect to see a lot of embedded images in the posts in this series.)

The next few posts in this series will be my attempts to summarize the key ideas at work as best as I can. These are mostly ideas I'm already fairly familiar with (or at least, ideas I should be familiar with), so I hope this stage won't take more than a few weeks. The limiting factor might just be how much time I have to write things up properly.

Knots and links; their diagrams and the idea of knot invariants

The Jones polynomial via Kauffman's bracket polynomial

Braids, Hecke algebras of type A and the Temperley-Lieb algebra

[+ the Jones polynomial via a trace on the Temperley-Lieb algebra]

[+ monoidal categories; examples and basic properties]

Braided monoidal categories

Motivating bialgebras and Hopf algebras

Representations of some particular quantum groups

Universal R-matrices and quasitriangular Hopf algebras

Schur-Weyl duality (both classical and quantum)

I won't necessarily be posting about these topics in this order, but this order is at least as likely as any other. [Edit: any additions to the list will be indicated in square brackets like this, and prefaced with a + symbol.]

I'll update the list above as I finish writing the posts (both to add links and perhaps to slightly revise the list itself with new or different topics).

After that, what I post about next will depend on where my reading takes me and how much progress I manage to make. At a minimum, I'll try to post a quick summary of whatever I'm currently reading every couple of weeks.

[mathhombre]

Max Bill's Gelbes Feld

Cover art for this article by Barry Cipra in this month's Notices. Gelbes Feld = Yellow Field, and the original Max Bill piece is a magic square.

The math in the article is about pipifying odd magic squares.

But then my eye was caught by something else: The way the 4 appears in the lower right-hand corner is not the way it’s typically represented by pips on a die or domino. The “normal” way to display a 4 is with dots in the four corners, not at the four midpoints. That artistic choice bothered me. Until a possible explanation jumped out: Bill had arranged things so that there are exactly 5 pips across each row, down each column, and along the two main diagonals of his array. I’ll use the term “pipification” to describe such a representation of numbers in an 𝑛 × 𝑛 magic square using pips in an 𝑛 × 𝑛 grid within each of the 𝑛2 cells in the square, arranged so that there are the same number of pips in each row, column, and diagonal; see Figure 3.

Count the pips!

Made me make this...

[the-least-undefinable-ordinal]

Ragebait ideas for your math friends: tell them that their research area is just a special case of your much cooler research

[mathematikerin-freyja]

Category theorist speedrun social pariah status strategy.

[lilith-hazel-mathematics]

Category theory is just a special case in my much cooler area of research, mathematical logic, in which we analyze the notion of a mathematical "theory" in abstract. Category theory is just a manner in which words and symbols are used to encode meaning, convey information, and draw inferences.

[m---a---x]

Pre-schooler destroying the whole mathematical community:

All mathematics is just specialized counting to 3

[dailypolyhedra]

Polyhedron of the Day #314: Order-6 hexagonal tiling

The order-6 hexagonal tiling is a regular tiling of the hyperbolic plane, with 6 hexagons joining at each vertex. Its Schläfli symbol is {6,6}. It can be created through alternation of the order-6 square tiling. It is self-dual.

Image created using KaleidoTile and obtained from Polytope Wiki.

[edensmidian]

[noelements-setempty]

If I want to be right, then proof by contrapositive is not wrong.

[thirdity]

“There are many things that I do not know because I photocopied a text and then relaxed as if I had read it.”

— Umberto Eco, How to Write a Thesis

[fireemoji1f525]

Are you an aspiring math major, but don't know what your field of study should be?

@algebraic-dumbass had the idea to make a flow chart of math sub-fields, so perhaps this will help any future mathematicians decide what to study!

[algebraic-dumbass]

beautiful, but where's my algebra (besides galois and category theory)?

[algebraic-dumbass]

"when are we going to use this math in real life" idk maybe when you trying pursuing happiness

[unnatural-transformations]

Despite what you might assume from popular media, there really isn't much of a connection between chess and mathematics. There are certainly mathematicians who play chess and vice versa: former Chess World Champion Emmanual Lasker and the English Grandmaster and chess problem specialist John Nunn both received doctorates in mathematics, for example. But chess -- being a finite deterministic game of perfect information -- generally does not lend itself to very interesting mathematics.

One of the few (limited) exceptions to this general rule I know of is the following chess problem. It's a variation on a type of artificial chess problem called a seriesmover (specifically it's a serieshelpmate in 34). This problem appears in a couple of graduate level math textbooks (but I'm not sure I can say who the author of those books is without somewhat giving the solution away: see below the cut).

Black is to play. She must make a series of 34 consecutive legal moves, without reply by White. After Black's 34 moves, White makes a single move which results in Black being checkmated. (Black, just to be clear, is fully cooperating in this goal and is trying to create a position in which White can deliver mate: this is precisely what is meant by a helpmate.) By "legal moves" I mean in particular that Black cannot ever put herself into check and that she cannot put White into check unless White is going to move next (in other words: Black can only do this, if at all, on her 34th and final consecutive move).

The chess question: can you find any valid series of legal moves by Black that solve this problem?

The math question: how many different solutions of this chess problem are there? (From the given position, how many different valid series of 34 legal moves by Black allow White to deliver mate in one?)

I don't think either part of this problem is particularly hard, and it's certainly a little bit contrived, but it's not trivial either: you shouldn't try to solve the second part of the problem by brute force enumeration.

The author of this puzzle below the cut (the identity of whom, as I said above, is something of a clue if you recognize their name).

This chess problem was created by the mathematician Richard P. Stanley and appears in several places, including the Finnish chess publication Suomen Tehtdvdniekat in 2005.

[unnatural-transformations]

As promised (if a little late) the solution below. @locally-normal answered the chess part a while ago, but if anybody enumerated the number of solutions I'm afraid I didn't see it.

Solving the chess part first: note that by the established rules the Knight can't move until (at the earliest) the very end of the sequence (since either Ne2 or Nf3 would be check), which means the Rook and all three pawns on the h file all also unable to move. That leaves Black with only two pawns (and the King) that are able to freely move.

Since White has to deliver mate with just one move, she must be checking with a protected pawn (otherwise the Black King could just capture the pawn to get out of check). That means White's winning move has to be b7+ or c7+. If White plays either move, a Black King anywhere but in the corner would have at least three squares to move to not threatened by any White piece. But we only have two Black pawns that can be promoted to pieces that would block these squares. So the Black can only be mated on the corner square a8. White's winning move must be b7+ and we need to promote both Black pawns to pieces to block the Black King's escape on a7 and b8.

What can these blocking pieces be? Not Queens or Rooks, as these would be able to capture the pawn on b7. Not Knights, as a Knight on a1 would be trapped for the same reason the Knight on g1 is trapped: both Nb3 and Nc2 would be check, which is forbidden. So both the blocking pieces will be (dark-squared) Bishops.

The final position will therefore look like this:

Can we set this position up in time? Yes, we can. There is exactly one path for each of the two Black pawns to take, as indicated below by the green arrows. It takes exactly 17 moves for each pawn to promote and get into its assigned blocking position.

So one solution to the chess part of the problem is (abusing normal chess notation slightly):

1. a4 2. a3 3. a2 4. a1=B 5. Bb2 6. Bc1 7. Bd2 8. Be1 9. Bg3 10. Bf4 11. Bh6 12. Bf8 13. Be7 14. Bd8 15. Bc7 16. Bb8 17. Ba7 18. a5 19. a4 20. a3 21. a2 22. a1=B 23. Bb2 24. Bc1 25. Bd2 26. Be1 27. Bg3 28. Bf4 29. Bh6 30. Bf8 31. Be7 32. Bd8 33. Bc7 34. Bb8

and then White plays 35. g7#.

For the math part of the problem, note that Black only has (at most) two legal moves at any point from the initial to the final position. She can move the pawn (later the Bishop) that started on a5, or she can move the pawn (Bishop) that started on a6. And because there's only one path to follow, the a6 pawn can never overtake the a5 pawn. If we color moves by the a5 pawn/Bishop red and moves by the a6 pawn/Bishop blue, that means a move sequence like

1. a4 2. a3 3. a2 4. a1=B 5. Bb2 6. Bc1 7. Bd2 8. Be1 9. Bg3 10. Bf4 11. Bh6 12. Bf8 13. Be7 14. Bd8 15. Bc7 16. Bb8 17. Ba7 18. a5 19. a4 20. a3 21. a2 22. a1=B 23. Bb2 24. Bc1 25. Bd2 26. Be1 27. Bg3 28. Bf4 29. Bh6 30. Bf8 31. Be7 32. Bd8 33. Bc7 34. Bb8

could be represented (with no loss of information) as simply a sequence of 17 up and 17 down arrows:

↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓

And the restriction that the a6 pawn can never overtake the a5 pawn is captured by the fact that (reading left-to-right) the number of down arrows can never exceed the number of up arrows. Moreover every such sequence of up and down arrows corresponds to a (different) series of moves that solve the helpmate problem.

But such a sequence of up and down arrows is just a ballot sequence, and (as shown in the link) such sequences are known to be counted by the Catalan numbers [this is what the Richard Stanley authorship was hinting at: I first saw this problem in his book Catalan Numbers]. So the solution to the math part of our problem is the 17th Catalan number, which is $C_{17} = \frac{1}{18} \begin{pmatrix} 34 \\ 17 \end{pmatrix} = 129644790$.

That means there are 129,644,790 possible solutions to the original chess problem.

[larkandkatydid]

“We should teach kids about personal finance and cooking and old school home economics”

Fuck you. We should teach kids algebra.

[larkandkatydid]

“We should teach kids not to stigmatize non-college career paths.”

We should teach you to kill yourself.

[larkandkatydid]

To make sure this post has real citations:

High-Achieving Black and Latino Students Are Often Shut Out of Algebra 1

Middle schoolers' access to the course is stratified along racial, socioeconomic, and regional lines, new research finds.

And to share an anecdotal citation I once ran an after school program at a “medical careers” based charter high school that was 95% black and, to its credit, set up all kids to graduate with a CNA certification. However, it did not offer AP Bio, AP Chemistry or Algebra II, all essential preparation for any decent pre-med program. So this medical careers school for black kids was preparing them to have only the lowest paying, most physically grueling medical careers and making it almost impossible for them to become doctors.

Donors loved it! But I always wondered how hard-working parents would react to really knowing that they sent their kid to a school that came very close to cutting them off from a career as a doctor at 14.

[frances-kafka]

I want to point out that we used to teach kids all kinds of things (voc tech stuff, shop, the arts, home ec) *and* we used to also teach them Algebra.

What happened?

[larkandkatydid]

We did not! This has always been shittier education that we forced on children that we didn’t believe deserved to go to college.

For some examples: a primary difference between white and black high schools during segregation was almost always the lack of college prep courses offered. This was also a complaint of Chicano students in East LA during the 1970s youth movements, especially young women who were being set up to work in garment factories when they wanted to go to college. Home Ec was attacked by young feminists across all sorts of backgrounds because it was where young women were sent when shut out of science classes. In Donna Gaines’ Teenage Wasteland about suburban teen suicides in the 1980s, she talks about Vo Tech programs as being something that kids who appeared too working class got forced into regardless of what they wanted. This is also a major plot point in Rob Reiner’s Stand by Me.

This has always been a bad argument made in behalf of other people’s children.

[unnatural-transformations]

Mathematicians and linguists and philosophers talk a big game about the importance of their favorite subjects when it comes to understanding what makes us all truly human. But ask any biologist and they'll tell you that taxonomy is literally what separates us from the animals.

[unnatural-transformations]

Despite what you might assume from popular media, there really isn't much of a connection between chess and mathematics. There are certainly mathematicians who play chess and vice versa: former Chess World Champion Emmanual Lasker and the English Grandmaster and chess problem specialist John Nunn both received doctorates in mathematics, for example. But chess -- being a finite deterministic game of perfect information -- generally does not lend itself to very interesting mathematics.

One of the few (limited) exceptions to this general rule I know of is the following chess problem. It's a variation on a type of artificial chess problem called a seriesmover (specifically it's a serieshelpmate in 34). This problem appears in a couple of graduate level math textbooks (but I'm not sure I can say who the author of those books is without somewhat giving the solution away: see below the cut).

Black is to play. She must make a series of 34 consecutive legal moves, without reply by White. After Black's 34 moves, White makes a single move which results in Black being checkmated. (Black, just to be clear, is fully cooperating in this goal and is trying to create a position in which White can deliver mate: this is precisely what is meant by a helpmate.) By "legal moves" I mean in particular that Black cannot ever put herself into check and that she cannot put White into check unless White is going to move next (in other words: Black can only do this, if at all, on her 34th and final consecutive move).

The chess question: can you find any valid series of legal moves by Black that solve this problem?

The math question: how many different solutions of this chess problem are there? (From the given position, how many different valid series of 34 legal moves by Black allow White to deliver mate in one?)

I don't think either part of this problem is particularly hard, and it's certainly a little bit contrived, but it's not trivial either: you shouldn't try to solve the second part of the problem by brute force enumeration.

The author of this puzzle below the cut (the identity of whom, as I said above, is something of a clue if you recognize their name).

This chess problem was created by the mathematician Richard P. Stanley and appears in several places, including the Finnish chess publication Suomen Tehtdvdniekat in 2005.

[locally-normal]

Not familiar with that author. My attempt at a solution lies ahead.

I think you can promote an a-pawn to a bishop and then get it to a7, and then repeat this with the second a pawn getting it to b8. This takes 17 moves per bishop.

This is probably the unique solution, because no other piece can move aside from the king, and if you promote to a knight you can't move it. If you promote to a rook or queen it wouldn't be checkmate. Moving the king out of the corner makes it harder to be checkmated but this is harder to show rigorously. That would be my approach though, anything else would surely take more than 34 moves if it were possible at all. Oh and there's only one fastest route the bishops could take, anything else is slower.

[unnatural-transformations]

Yes, promoting both a-pawns to Bishops and trapping the Black King on a8 is the idea.

I think the simplest argument for why the Black King has to stay in the corner is that

White must be checking with a protected pawn (otherwise Black's King could just capture it), so White's move must be b7+ or c7+

This means Black's King is on one of a8, b8, c8 or d8. By inspection, if the King is on any of b8, c8 or d8 it would be able to move to at least three squares White can't attack (and, because Black can only promote two pawns, Black wouldn't be able to block all those squares with her own pieces).

So that forces the King to be on a8 (and it turns out the King doesn't have time to move away from the square and then move back)

And yes, in one sense this is the unique solution (it takes each pawn/Bishop at least 17 moves to get into position and there's only path for each of them to take). But there's more than one sequence of moves that will work! As you say, you could promote one pawn, move it into position and then repeat with the other pawn:

1. a4 2. a3 3. a2 4. a1=B 5. Bb2 6. Bc1 7. Bd2 8. Be1 9. Bg3 10. Bf4 11. Bh6 12. Bf8 13. Be7 14. Bd8 15. Bc7 16. Bb8 17. Ba7 18. a5 19. a4 20. a3 21. a2 22. a1=B 23. Bb2 24. Bc1 25. Bd2 26. Be1 27. Bg3 28. Bf4 29. Bh6 30. Bf8 31. Be7 32. Bd8 33. Bc7 34. Bb8

Or you could shuffle both pawns (and then Bishops) along in tandem, always moving one after the other:

1. a4 2. a5 3. a3 4. a4 5. a2 6. a3 7. a1=B 8. a2 9. Bb2 10. a1=B 11. Bc1 12. Bb2 13. Bd2 14. Bc1 15. Be1 16. Bd2 17. Bg3 18. Be1 19. Bf4 20. Bg3 21. Bh6 22. Bf4 23. Bf8 24. Bh6 25. Be7 26. Bf8 27. Bd8 28. Be7 29. Bc7 30. Bd8 31. Bb8 32. Bc7 33. Ba7 34. Bb8

Or do something between these two extremes: moving one piece for a while and then the other one (although of course you can never have the pawn that started on a6 overtake the one that started on a5).

For the math part of the problem, I'm counting all of those as different solutions (although they are, of course, all executions of the same plan).

[unnatural-transformations]

Despite what you might assume from popular media, there really isn't much of a connection between chess and mathematics. There are certainly mathematicians who play chess and vice versa: former Chess World Champion Emmanual Lasker and the English Grandmaster and chess problem specialist John Nunn both received doctorates in mathematics, for example. But chess -- being a finite deterministic game of perfect information -- generally does not lend itself to very interesting mathematics.

One of the few (limited) exceptions to this general rule I know of is the following chess problem. It's a variation on a type of artificial chess problem called a seriesmover (specifically it's a serieshelpmate in 34). This problem appears in a couple of graduate level math textbooks (but I'm not sure I can say who the author of those books is without somewhat giving the solution away: see below the cut).

Black is to play. She must make a series of 34 consecutive legal moves, without reply by White. After Black's 34 moves, White makes a single move which results in Black being checkmated. (Black, just to be clear, is fully cooperating in this goal and is trying to create a position in which White can deliver mate: this is precisely what is meant by a helpmate.) By "legal moves" I mean in particular that Black cannot ever put herself into check and that she cannot put White into check unless White is going to move next (in other words: Black can only do this, if at all, on her 34th and final consecutive move).

The chess question: can you find any valid series of legal moves by Black that solve this problem?

The math question: how many different solutions of this chess problem are there? (From the given position, how many different valid series of 34 legal moves by Black allow White to deliver mate in one?)

I don't think either part of this problem is particularly hard, and it's certainly a little bit contrived, but it's not trivial either: you shouldn't try to solve the second part of the problem by brute force enumeration.

The author of this puzzle below the cut (the identity of whom, as I said above, is something of a clue if you recognize their name).

This chess problem was created by the mathematician Richard P. Stanley and appears in several places, including the Finnish chess publication Suomen Tehtdvdniekat in 2005.

[mathsuggestions]

Put presents under a connected acyclic graph.

[im-the-antonymph-of-your-mom]

The new xkcd just made me cry

[im-the-antonymph-of-your-mom]

https://xkcd.com/3184

[mathsuggestions]

Play the Real Analysis Game.