Two Bridge Notation for a Knot


If a knot is a two-bridge knot, its 2-fold cover is a a lens space, L(p,q). We have abbreviated this in the table by p/q. Two-bridge knots are equivalent if and only if the associated lens spaces are homeomorphic. Knots can be distinguished from their mirror images if one works in the oriented category.

A two-bridge knot can be reconstructed from the fraction p/q by finding a continued fraction expansion of p/q. Because different pairs of numbers can give the same lens space (eg. L(17,5) = L(17,7)), and these have different continued fraction expansions, there is not a unique such description. We always have 0< q < p. By using the continued fraction expansion with all positive entries, one arrives at an alternating diagram for the knot. (For example, 17/5 = [3,2,2] and 17/7 = [2,2,3].) In the table, our continued fractions correspond to our Conway notation (which gives the continued fraction expansion in the case of 2-bridge knots) except that the order might be reversed.

Two lens spaces L(p,q) and L(r,s) (p, r >0) are orientation preserving homeomorphic if and only if r = p and either q = s mod p or qs = 1 mod p. They are orientation reversing homeomorphic if and only if r = p and q = -s mod p or qs = -1 mod p.