Two-Bridge Notation

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, such knots do not have unique continued fraction descriptions. 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, the continued fractions correspond to the 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.

Sashka (Alexandra) Kjuchukova provided us with two-bridge notation, including for 13 crossing knots, based on her joint work with Ryan Blair, Roman Velazquez, and Paul Villanueva [1]. See also [2].


[1] Blair, R.; Kjuchukova, A.; Velazquez, R.; Villanueva, P. "Wirtinger systems of generators of knot groups." Comm. Anal. Geom. 28 (2020), no. 2, 243-262

[2] Blair, Ryan; Kjuchukova, Alexandra; Ozawa, Makoto. "The incompatibility of crossing number and bridge number for knot diagrams." Discrete Math. 342 (2019), no. 7, 1966-1978.

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