Torus knot signatures, Part 1

I want to tell you about the current mathematics I’m working on, because it’s exciting, surprising, beautiful, deep, and easy to explain!  Everything that a good piece of maths should be.  I’m going to have to explain it in a few parts though, leading you through the different steps I had to solve to get to the solution.

Here’s a problem to get you started.  I’m going to give you two numbers, p and q, which are coprime.  That means they have no common factors.  For example, I can’t give you the numbers 9 and 12, since both of them are divisible by 3.  But I can give you the numbers 9 and 10 since 9=3×3 and 10=2×5, and there is no overlap. Ok.  Now I give you another number n between 1 and pq-1, such that n is neither a multiple of p nor a multiple of q.  Your task: take n and keep minusing multiples of p from it until you get a number that is a multiple of q.  Then tell me whether the multiple of q is positive or negative.

Let’s do an example.  Suppose p=3 and q=5.  I need to give you a number n between 1 and 14, so let’s start with the simple n=1.  First we try 1-3=-2.  Not a multiple of 5.  So we try 1-(2×3) = -5.  Success!  And -5 is negative.  Now let n=8.  Try 8-3=5.  First time lucky! And this time we got a positive number.

For particular numbers p and q, we are going to do this for all possible n.  So if p=3 and q=5, we have to find the answer for n=1,2,4,7,8,11,13,14.  If the answer is positive, put j(n)=1, and if it’s negative we put j(n)=-1.  So, for these particular values of n, the corresponding values for j are -1,-1,-1,-1,1,1,1,1.  Now we keep a running total, and call that number s.  So the values for s are -1,-2,-3,-4,3,2,1,0.  Actually, for reasons that I will explain later, the interesting number to look at is 2 times s.

Let’s draw a graph of the function 2s and see what it looks like.  Does it always go “minus minus minus plus plus plus”?  Well, for p=2, the answer is ‘yes’, no matter what value you choose for q.  Here’s the graph of p=2,q=25, with the values of n running along the bottom and the value of 2s running up the side.

p=2, q=25

p=2, q=25

Notice the lovely ‘V’ shape that it makes.  The values of s decrease until a certain point, and then increase again.

Let’s see what happens when p=3.  Here’s the graph for p=3, q=10:

p=3, q=10

p=3, q=10

Oh no, what happened?  The graph is all wiggly at the bottom!  Minus, plus, minus, plus – what’s going on?  It’s obviously not a proplem with the number 3, since our previous example of p=3, q=5 worked out fine.  Maybe we just picked a funny combination of numbers.  We had better try another pair and see what happens there.  How about p=7, q=16?

p=7, q=16

p=7, q=16

Even more wiggles!  Yikes, it seems that this function is going to be pretty unpredictable.  The pattern seems to be very different for every p and q.  How on earth are we going to get a formula that tells us what is happening?

Tune in next time for more clues…

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