Four-Genus

The smooth (or topological) 4-genus of a knot is the minimum genus of a smooth (topological, locally-flat) surface embedded in the 4-ball with boundary the link.

For knots, some bounds are determined by the p-signatures and, to avoid being of 4-genus 0 (slice), the Alexander polynomial. In addition, there are bounds determined by gauge theoretic invariants, which apply only in the smooth category. The knots for which these smooth techniques are required are marked in the table with reference links.

For multi-component links, the relevant lower bounds are determined by the Arf-Robertello invariant, the signature, and in the smooth setting the Slice-Bennequin bound.

All the data about the four-genus presented in LinkInfo was computed by Stepan Orevkov. He computed the upper bounds by performing crossing changes in the minimal diagrams for the links.

No specific links to report at this time.

References (for knots).

[1] A'Campo, N., "Generic immersions of curves, knots, monodromy and gordian numbers," Inst. Hautes Etudes Sci. Bubl. Math. 88 (1998), 151-169.

[2] Boileau, M., Boyer, S., and Gordon, C., "Branched covers of quasipositive links and L-spaces," Arxiv preprint.

[3] Fujino, Y., Miyazawa, Y., and Nakajima, K., "H(n)-unknotting number of a knot," Reports of knots and low-dimensional manifolds (1997), 72-85.

[4] Gibson, W. and Ishikawa, M., "Links and gordian numbers associated with generic immersions of intervals," Topology and its Applications, 123 (2002), 609-636.

[5] Kawamura, T., On unknotting numbers and four-dimensional clasp numbers of links, Ph.D. Thesis, University of Tokyo (2000).

[6] Kawamura, T., "The unknotting numbers of 10139 and 10152 are 4," Osaka J. Math. 35 (1998), 539-546.

[7] Lewark, L. and McCoy, D., "On calculating the slice genera of 11- and 12-crossing knots," Arxiv preprint.

[8] Murakami, H. and Nakanishi, Y., "Triple points and knot cobordism," Kobe J. Math, v 1. (1984), 1-16.

[9] Piccirillo, L., "The Conway knot is not slice," Arxiv preprint.

[10] Shibuya, T., Memoirs of the Osaka Institute of Technology, Vol 45 (2000) 1-10.

[11] Stoimenow, A., "Positive knots, closed braids and the Jones polynomial," Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2(2) (2003), 237-285.

[12] Rasmussen, J., "Khovanov homology and the slice genus," Arxiv preprint.

[13] Tanaka, T., "Unknotting numbers of quasipositive knots," Top. Appl 88 (1998), 239-246.