Research ArticleNo Access

Genera of knots in the complex projective plane

Kansas State University, USA

E-mail Address: jacobpichelmeyer@gmail.com

https://doi.org/10.1142/S0218216520500819Cited by:5 (Source: Crossref)

Abstract

Our goal is to systematically compute the $ℂ P^{2}$ -genus of as many prime knots up to 8-crossings as possible. We obtain upper bounds on the $ℂ P^{2}$ -genus via coherent band surgery. We obtain lower bounds by obstructing homological degrees of potential slice discs. The obstructions are pulled from a variety of sources in low-dimensional topology and adapted to $ℂ P^{2}$ . There are 27 prime knots and distinct mirrors up to 7-crossings. We now know the $ℂ P^{2}$ -genus of all of these knots. There are 64 prime knots and distinct mirrors up to 8-crossings. We now know the $ℂ P^{2}$ -genus of all but 6 of these knots, where the $ℂ P^{2}$ -genus was not determined explicitly, it was narrowed down to 2 possibilities. As a consequence of this work, we show an infinite family of knots such that the $ℂ P^{2}$ -genus of each knot differs from that of its mirror.

Keywords:

AMSC: 57M25, 57R40, 57R65, 57R90

References

1. M. Ait Nouh, Genera and degrees of torus knots in $C P^{2}$ , J. Knot Theory Ramifications 18(9) (2009) 1299–1312. Link, Web of Science, Google Scholar
2. A. Daemi and C. Scaduto, Chern-simons functional, singular instantons, and the four-dimensional clasp number, preprint (2020), arXiv:2007.1316. Google Scholar
3. R. H. Fox, A quick trip through knot theory, in Topology of 3-Manifolds and Related Topics (Prentice-Hall, Englewood Cliffs, NJ, 1962), pp. 120–167. Google Scholar
4. R. H. Fox, Some problems in knot theory, in Topology of 3-Manifolds and Related Topics (Prentice-Hall, Englewood Cliffs, NJ, 1962), pp. 168–176. Google Scholar
5. R. H. Fox and J. Milnor, Singularities of 2-spheres in 4-space and cobrdism of knots, Osaka J. Math. 3(2) (1966) 257–267. Google Scholar
6. R. Friedman, Donaldon and Seiberg–Witten invariants of algebraic surfaces, Proc. Sympos. Pure Math. 62 (1997) 85–100. Crossref, Google Scholar
7. P. Gilmer, Configurations of surfaces in 4-manifolds, Trans. Amer. Math. Soc. 264(2) (1981) 353–380. Web of Science, Google Scholar
8. R. Gompf and A. Stipsicz, 4-Manifolds and Kirby Calculus (American Mathematical Society, Providence, RI, 1999). Crossref, Google Scholar
9. S. Jabuka and T. Kelly, The nonorientable 4-genus for knots with 8 or 9 crossings, Algebr. Geom. Topol. 18(3) (2018) 1823–1856. Crossref, Web of Science, Google Scholar
10. T. Kawamura, The rasmussen invariants and the sharper slice-bennequin inequality of knots, Topology 46(1) (2007) 29–38. Crossref, Web of Science, Google Scholar
11. P. Kronheimer and T. Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1(6) (1994) 797–808. Crossref, Web of Science, Google Scholar
12. T. Lawson, Smooth embeddings of 2-spheres in 4-manifolds, Expo. Math. 10 (1992) 289–309. Google Scholar
13. L. Lewark and D. McCoy, On calculating the slice genera of 11- and 12-crossing knots, Experiment. Math. 28(1) (2019) 81–94. Crossref, Web of Science, Google Scholar
14. P. Lisca, Lens spaces, rational balls and the ribbon conjecture, Geom. Topol. 11(1) (2007) 429–472. Crossref, Web of Science, Google Scholar
15. P. Lisca and G. Matić, Stein 4-manifolds with boundary and contact structures, Topology Appl. 88(1–2) (1998) 55–66. Crossref, Google Scholar
16. C. Livingston and J. C. Cha, KnotInfo: Table of knot invariants, preprint (2019), https://www.indiana.edu/ $\sim$ knotinfo/. Google Scholar
17. C. Livingston and J. C. Cha, LinkInfo: Table of link invariants, preprint (2019), http://www.indiana.edu/ $\sim$ linkinfo/. Google Scholar
18. C. Livingston and S. Naik, Introduction to knot concordance, (2019). Google Scholar
19. D. McCoy, Alternating knots with unknotting number one, Adv. Math. 305 (2017) 757–802. Crossref, Web of Science, Google Scholar
20. A. Moore and M. Vasquez, A note on band surgery and the signature of a knot, preprint (2018), arXiv:1806.02440. Google Scholar
21. L. Nicolaescu, Notes on Seiberg–Witten Theory (American Mathematical Society, Providence, RI, 2000). Crossref, Google Scholar
22. P. Ozvath and Z. Szabo, Knot Floer homology and the four-ball genus, Geom. Topol. 7(2) (2003) 615–639. Crossref, Web of Science, Google Scholar
23. J. Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182(2) (2010) 419–447. Crossref, Web of Science, Google Scholar
24. R. Robertello, An invariant of knot cobordism, Comm. Pure Appl. Math. 18(3) (1965) 543–555. Crossref, Web of Science, Google Scholar
25. A. Scorpan, The WIld World of 4-Manifolds (American Mathematical Society, Providence, RI, 2005). Google Scholar
26. H. Seifert, Über das Geschlect von Knoten, Math. Ann. 110 (1935) 571–592. Crossref, Google Scholar
27. S. Suzuki, Local knots of 2-spheres in 4-manifolds, Proc. Japan Acad. 45(1) (1969) 34–38. Google Scholar
28. O. Y. Viro, Link types in codimension-2 with boundary, Usp. Mat. Nauk 30 (1975) 231–232 (in Russian). Google Scholar
29. L. Williams, Obstructing sliceness in a family of Montesinos knots, preprint (2008), arXiv:0809.1247. Google Scholar
30. A. Yasuhara, (2,15)-torus knot is not Slice in $C P^{2}$ , Proc. Japan Acad. 67(10) (1991) 353–355. Google Scholar
31. A. Yasuhara, On slice knots in the complex projective plane, Rev. Math. 5(2–3) (1992) 255–276. Google Scholar
32. A. Yasuhara, Connecting lemmas and representing homology classes of simply-connected 4-manfiolds, Tokyo J. Math. 19(1) (1996) 245–261. Crossref, Google Scholar