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 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, only 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 less 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 (once added to the table) will be marked with references.

In the references below the crosscap number of torus knots is determined in the first paper by Teragaito. In the Murakami-Yasuhara paper the crosscap number of the knot 74 is found to be 3. In the paper about Klein bottles by Teragaito it is shown that if a knot is of (orientable) genus 1 and of crosscap number 2, then it is a twist knot. In the paper by Teragaito and Hirasawa, they presented 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. There work has lowered produed new lower bounds for 778 of the knots in the table. Data from those computations are avaiable at data files.

Major progress has been made by Burton, reported in his paper "Enumerating fundamental surfaces, Algorithms, experiments and invariants."

(2012) Adams and Kindred have presented an algorithm that determines the crosscap number of an alternating knot. It has been applied to 9 crossing knots and less, yielding the previously unknown values for:

8_{10,15,16,17,18} and 9_{16,22,24,25,28,29,30,32,33,34,36,37,38,39,40,41}.

(2014) Kalfagianni and Lee have developed new bounds on the crosscap number based on the Jones polynomial. 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; these are linked in the results page.



References

C. Adams and T. Kindred, A classification of spanning surfaces for alternating links.

B. Burton, Enumerating fundamental surfaces, Algorithms, experiments and invariants, preprint, arXiv:1111.7055v2

B. Burton and M. Ozlen, Computing the crosscap number of knot using integer programming and normal surfaces, preprint, arXiv:1107.2382

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

E. Kalfagianni and C. Lee, Crosscap numbers and the Jones polynomial. (Arxiv preprint 1408.4493)

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

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

M. Teragaito and M. Hirasawa, Crosscap numbers of 2-bridge knots, Arxiv:math.GT/0504446, Topology 45 (2006), no. 3, 513--530..


Further information on particular knots.