The 4d-crosscap number of a knot K is the minimum first Betti number among all nonorientable surfaces
bounded by K in B4. In particular, for a slice knot K, the 4d-crosscap number is 1.
Every knot bounds a once punctured connected sum of real projective planes, RP2, in B4.
If the minumum number required is postive, then this minimum is the 4d-crosscap number. If the minimum is 0, the
4d-crosscap number is 1.
A tool for ruling out 4d-crosscap number ≤ 1 is a result of Yasuhara:
If K bounds a Mobius band in B4, then 4Arf(K) - signature(K) = 0, 2, or -2 mod 8.
This applies in both the smooth and topological categories.
Slaven Jabuka and Tynan Kelly [4] provided the complete data for 8 and 9 crossing knots. Nakisa Ghabarian [2] provided the data for 10 crossing knots. Megan Fairchild [8] provided the results for non-alternating 11 crossing knots, leaving all but 11n_{17, 40, 159, 166, 177, 178} undetermined, all in the range [1,2].
Daniel Lee and Joshua Sabloff [9] provided extensive new data. In particular, they extended the KnotInfo data to include 11-crossing alternating knots as well as 12-crossing and 13-crossing knots.
Some upper bounds from [9] were reduced with the following observation. Suppose K bounds an immersed disk with N double points. If N is odd, an additional double point can be added to make N even. Double points can be removed pairwise by replacing two disks with a tube. The resulting surface has first Betti number N. If any pair of double points can be chosen to have the same sign, then the surface is non-orientable. As a result, if u is the unknotting number and u does not equal 2, then the nonorientble crosscap number is at most 2*floor( (u+1)/2)). If u = 2 and the signature is \pm 4, then the two double points are of the same sign; this reduced a few more maximum values.
A secondary search provided more reductions. Here are some examples. (c(K) = 4d crosscap number and u(K) = unknotting number)
• A crossing change in 11a_5 yields the slice knot 6_1, and thus c(K)≤2. This isn't attained from the fact that u(11a_5)=2.
• A crossing change in 11a_197 yields the slice knot 8_8, and thus c(K)≤2. This isn't attained from the fact that u(11a_197)=3.
• A crossing change in 12a_561 yields the knot 5_1 with c(5_1) = 1. Thus c(K)≤3. This isn't attained from the fact that u(12a_561)= 3 or u(5_1) = 2.
• A crossing change in 11a_192 yields the knot 9_13 with c(9_13) = 1. Thus c(K)≤3. This isn't attained from the fact that u(9_13)= 3.
• A crossing change in 13a_3132 yields the knot 12n_725 with c(12n_725) = 2. Thus c(K)≤4. This isn't attained from the fact that u(13a_3132)= 5.
• A crossing change in 13a_3308 yields the knot 3_1 # 8_9. We have 8_9 is slice, and c(3_1) = 1. Thus c(K)≤3. This isn't attained from the fact that u(13a_3132)≤ 3.
[1] Baston, J., Nonorientable four-ball genus can be arbitrarily large, Math. Res. Lett. 21 (2014), no. 3, 423-436.
[2] Ghanbarian, N., "The non-orientable 4-genus for knots with 10 crossings," Arxiv preprint.
[3] Gilmer, P. and Livingston, C., The nonorientable four-genus of knots, J. Lond. Math. Soc. (2) 84 (2011), no. 3, 559-577.
[4] Jabuka, S. and Kelly, T., The nonorientable 4-genus for knots with 8 or 9 crossings, Algebraic and Geometric Topology 18 (2018), 1823-1856.
[5] Ozsvath, P., Stipsicz, A., and Szabo, Z., Unoriented knot Floer homology and the unoriented four-ball genus, Algebraic and Geometric Topology 18 (2018), 1823-1856.
[6] Viro, O., "Positioning in codimension 2 and the boundary," Uspehi Mat. Nauk 30 (1975), 231-232.
[7] Yasuhara, A., "Connecting lemmas and representing homology classes of simply connected 4-manifolds," Tokyo J. Math. 19 (1996), no. 1, 245-261.
[8] Fairchild, M., The Non-Orientable 4-Genus of 11 Crossing Non-Alternating Knots. Arxiv preprint.
[9] Lee, D. and Sabloff, J, An Algorithmic Search for Knots Bounding Mobius Bands in B4. Arxiv preprint.