Crosscap Number

Every knot K in S3 bounds a nonorientable surface F of the form #gP2. The minimum such g among all nonorientable surfaces is called the crosscap number of K.

Notice that if K bounds an orientable surface of genus g, then it bounds a nonorientable surface of crosscap number 2g+1. If a knot is of crosscap number 1, then it bounds a Mobius band, and thus is either a (2,n)-torus knot, or has a companion, and hence is not hyperbolic.

The upper bounds given in the table were obtained by finding nonorientable surfaces usings Seifert's algorithm, using all possible crossing smoothings (states) except for the one that produces an orientable surface. There were three exceptions to this for knots with 11 or fewer crossings, for which Seifert's algorithm produced a genus greater than 2g+1, where g is the orientable genus. These and their 12 crossing counterparts will be marked with references once added to the table.

In [8], Teragaito determines the crosscap number of torus knots. In [6], Murakami and Yasuhara find the crosscap number of the knot 74 to be 3. In [9], Teragaito shows that if a knot is of orientable genus 1 and of crosscap number 2, then it is a twist knot. In [10], Teragaito and Hirasawa present an algorithm computing the crosscap number of an arbitrary 2-bridge knot and did computations for those with 12 crossings or less.

Burton and Ozlen have used normal surfaces and integer programming to find nonorientable surfaces of small crosscap number. Their work has produced new lower bounds for 778 of the knots in the table. Data from those computations are available at data files.

Major progress has been made by Burton, reported in [2].

Kindred [5] has computed the crosscap numbers of all alternating knots of 13 crossings or less; similar results were achieved by Ito-Takimura [4].

Specific Knots

8{10,15,16,17,18} and 9_{16, 22, 24, 25, 28, 29, 30, 32, 33, 34, 36, 37, 38, 39, 40, 41}
   Adams and Kindred have presented an algorithm that determines the crosscap number of an alternating knot. It has been applied to 9 and fewer crossing knots, yielding the previously unknown values for these knots.
   In the published version of that paper, Adams and Kindred applied the algorithm to determine the crosscap numbers of alternating 10 crossing knots [1].

Kalfagianni and Lee have developed new bounds on the crosscap number based on the Jones polynomial [7]. These improved the bounds on the crosscap number for almost half of the 12 crossing knots and precisely determined the number for 283 of the knots.

References

[1] Adams, C. and Kindred, T., A classification of spanning surfaces for alternating links, Algebraic & Geometric Topology 13 (2013), 2967-3007.

[2] Burton, B., "Enumerating fundamental normal surfaces: Algorithms, experiments and invariants," Arxiv preprint.

[3] Burton, B. and Ozlen, M., "Computing the crosscap number of knot using integer programming and normal surfaces," Arxiv preprint.

[4] Ito, N. and Takimura, Y., "A lower bound of crosscap numbers of alternating knots," to appear, JKTR.

[5] Kindred, T., "Crosscap numbers of alternating knots via unknotting splices," Arxiv reprint (2019).

[6] Murakami, H. and Yasuhara, A., "Crosscap number of a knot," Pacific J. Math. 171 no. 1 (1995), 261-273.

[7] Kalfagianni,] E. and Lee, C., Crosscap numbers and the Jones polynomial (2014).

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

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

[10] Teragaito, M. and Hirasawa, M., "Crosscap numbers of 2-bridge knots," Topology 45 no. 3 (2006), 513-530. Arxiv preprint.