Understanding the Trefoil Knot and Its Function on a Torus

[sammycaps]

So in the process of giving us a crude definition of a trefoil knot, our professor talks a bit about a function on a torus.

If we view the torus as the identification of sides of a square, and define a function y=(p/q)x, then we may only go from the bottom left corner (0,0) to the top right corner (1,1) (I guess forming a knot) if (p,q)=1. Two questions...

1) Isn't any function y=(m/n)x the same as a function y=(p/q)x with (p,q)=1?

2) Is there a simple way to understand why the (p,q) must be 1, or is it something not so trivial?

 

[jedishrfu]

some info on trefoil and torus:

http://www.rwgrayprojects.com/Lynn/DoubleTorus/dt01.html

 

[sammycaps]

I should mention, my algebra and algebraic topology isn't great. I've studied a good bit of a semester of introductory undergraduate group theory, but not much else. The treatment of knots in my topology class is purely as an example of Van Kampen's theorem and not as a topic in and of itself, so I know very little about them.

 

[lavinia]

1) Isn't any function y=(m/n)x the same as a function y=(p/q)x with (p,q)=1?

yes.

2) Is there a simple way to understand why the (p,q) must be 1, or is it something not so trivial?

it's just that if p and q are relatively prime then they tell you the actual number of rotations around the torus. p/q is just the slope of a straight line on the flat torus. Any common factors cancel out.

 

[sammycaps]

yes.

it's just that if p and q are relatively prime then they tell you the actual number of rotations around the torus. p/q is just the slope of a straight line on the flat torus. Any common factors cancel out.

So if we take p and q not relatively prime rotations around the torus, what happens?

 

[lavinia]

So if we take p and q not relatively prime rotations around the torus, what happens?

try the case of a (2,2) curve. You can draw a picture on a rectangle with opposite edges identified.