In Chapter 2 of "Model Theory", Wilfred Hodges lists three ways that mathematicians use formulas:

"First, a mathematician writes the equation 'y=4x*x'. By writing this equation one names a set of points in the plane.... As a model theorist would put it, the equation defines a 2-ary relation on the reals....

Or second, a mathematician writes down the laws

• ∀ x, y, z, x ≤ y and y ≤ z imply x ≤ z;
  
• ∀ x,y exactly one of x ≤ y, y ≤ x, x=y holds.
  

By doing this, one names a class of relations, namely those relations ≤ for which the laws are true.

Third, a mathematician defines a homomorphism from a group G to a group H to be a map from G to H such that x=y*z implies f(x) = f(y)*f(z). Here the equation x=y*z defines a class of maps."

From this starting point, how should a model theorist regard the skein relations of knot theory?



These specify a small area of a knot where differences are allowed - outside the displayed area, the three knots must be identical.

- posted by Scott @ 10:28 PM