References: Nonorientable 3-Genus


For knots of 11 crossings or less, there were three for which Seifert's algorithm did not produce a surface of genus less than 2g+1, where g is the orientable genus. These were: 9_46, 11n_139, 11n_141. There were several more at 12 crossings (all those marked with references other than 12a1166 and 12a1287.).

Torus knots T(2,n) have nonorientable genus 1. There are two other torus knots on the table. T(4,3) = 8_19 and T(3,5) = 10_124. These both have nonorientable genus 2, as described in the reference by Teragaito "Crosscap numbers of torus knots."

Teragaito also proved (in "Creating Klein bottles") that the only genus 1 knots of nonorientable genus 2 are twist knots. All of these in the range of the table are do not have companions and the nonorientable genus is thus 2 for all of these, with the exception of the trefoil, 3_1. (These are 3_1, 4_1, 5_2, 6_1, 7_2, 8_1, 9_2, 10_1, 11a247, and 12a803.)

It followed that the nontwist knots of genus 1 are not nonorientable genus 2. This applies to 8_3, 9_5, 9_35, 9_46, 10_3, 11a343, 11a362, 11a363, 11n139, 11n141, 12a1166 and 12a1287.

Recently Teragaito and Hirasawa presented an algorithm computing the nonorientable genus of an arbitrary 2-bridge knot, and did computation for those with 12 crossings or less.



References:

Murakami, Hitoshi and Yasuhara, Akira. "Crosscap number of a knot," Pacific J. Math. 171 (1995), no. 1, 261--273.

Teragaito, Masakazu. "Crosscap numbers of torus knots," Topology Appl. 138 (2004), no. 1-3, 219--238.

Teragaito, Masakazu. "Creating Klein bottles by surgery on knots", Knots in Hellas '98, Vol. 3 (Delphi), J. Knot Theory Ramifications 10 (2001), no. 5, 781--794.

Teragaito, Masakazu and Hirasawa, Mikami. "Crosscap numbers of 2-bridge knots", Arxiv:math.GT/0504446.