Getting a handle on the Conway knot

Hom, Jennifer

doi:10.1090/bull/1745

Getting a handle on the Conway knot
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by Jennifer Hom HTML | PDF

Bull. Amer. Math. Soc. 59 (2022), 19-29

Abstract:

A knot is said to be slice if it bounds a smooth disk in the 4-ball. For 50 years, it was unknown whether a certain 11 crossing knot, called the Conway knot, was slice or not, and until recently, this was the only one of the thousands of knots with fewer than 13 crossings whose slice-status remained a mystery. We will describe Lisa Piccirillo’s proof that the Conway knot is not slice. The main idea of her proof is given in the title of this article.

References

Tetsuya Abe, On annulus twists, RIMS Kôkyûroku 2004 (2016), 108–114.
Selman Akbulut, On $2$-dimensional homology classes of $4$-manifolds, Math. Proc. Cambridge Philos. Soc. 82 (1977), no. 1, 99–106. MR 433476, DOI 10.1017/S0305004100053718
Selman Akbulut, Cappell-Shaneson homotopy spheres are standard, Ann. of Math. (2) 171 (2010), no. 3, 2171–2175. MR 2680408, DOI 10.4007/annals.2010.171.2171
Jonathan M. Bloom, Odd Khovanov homology is mutation invariant, Math. Res. Lett. 17 (2010), no. 1, 1–10. MR 2592723, DOI 10.4310/MRL.2010.v17.n1.a1
Sylvain E. Cappell and Julius L. Shaneson, There exist inequivalent knots with the same complement, Ann. of Math. (2) 103 (1976), no. 2, 349–353. MR 413117, DOI 10.2307/1970942
S. K. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), no. 2, 279–315. MR 710056
Michael Freedman, Robert Gompf, Scott Morrison, and Kevin Walker, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, Quantum Topol. 1 (2010), no. 2, 171–208. MR 2657647, DOI 10.4171/QT/5
Ralph H. Fox and John W. Milnor, Singularities of $2$-spheres in $4$-space and cobordism of knots, Osaka Math. J. 3 (1966), 257–267. MR 211392
Ralph H. Fox, Free differential calculus. I. Derivation in the free group ring, Ann. of Math. (2) 57 (1953), 547–560. MR 53938, DOI 10.2307/1969736
Michael H. Freedman, The disk theorem for four-dimensional manifolds, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 647–663. MR 804721
David Gabai, Genera of the arborescent links, Mem. Amer. Math. Soc. 59 (1986), no. 339, i–viii and 1–98. MR 823442, DOI 10.1090/memo/0339
C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), no. 2, 371–415. MR 965210, DOI 10.1090/S0894-0347-1989-0965210-7
Robert E. Gompf, More Cappell-Shaneson spheres are standard, Algebr. Geom. Topol. 10 (2010), no. 3, 1665–1681. MR 2683748, DOI 10.2140/agt.2010.10.1665
Kyle Hayden, Thomas E. Mark, and Lisa Piccirillo, Exotic Mazur manifolds and knot trace invariants, arXiv:1908.05269 (2019).
Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359–426. MR 1740682, DOI 10.1215/S0012-7094-00-10131-7
Artem Kotelskiy, Liam Watson, and Claudius Zibrowius, On symmetries of peculiar modules; or, $\delta$-graded link Floer homology is mutation invariant, arXiv:1910.14584 (2019).
Eun Soo Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005), no. 2, 554–586. MR 2173845, DOI 10.1016/j.aim.2004.10.015
Allison N. Miller and Lisa Piccirillo, Knot traces and concordance, J. Topol. 11 (2018), no. 1, 201–220. MR 3784230, DOI 10.1112/topo.12054
Jeffrey Meier and Alexander Zupan, Generalized square knots and homotopy 4-spheres, arXiv:1904.08527 (2019).
Peter Ozsváth and Zoltán Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003), 615–639. MR 2026543, DOI 10.2140/gt.2003.7.615
Peter Ozsváth and Zoltán Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004), no. 1, 58–116. MR 2065507, DOI 10.1016/j.aim.2003.05.001
Peter S. Ozsváth and Zoltán Szabó, Knot Floer homology and rational surgeries, Algebr. Geom. Topol. 11 (2011), no. 1, 1–68. MR 2764036, DOI 10.2140/agt.2011.11.1
Lisa Piccirillo, The Conway knot is not slice, Ann. of Math. (2) 191 (2020), no. 2, 581–591. MR 4076631, DOI 10.4007/annals.2020.191.2.5
Jacob Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010), no. 2, 419–447. MR 2729272, DOI 10.1007/s00222-010-0275-6
Robert Riley, Homomorphisms of knot groups on finite groups, Math. Comp. 25 (1971). MR 295332, DOI 10.1090/S0025-5718-1971-0295332-4
Horst Schubert, Die eindeutige Zerlegbarkeit eines Knotens in Primknoten, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (1949), no. 3, 57–104 (German). MR 0031733
Stephan M. Wehrli, Mutation invariance of Khovanov homology over $\Bbb F_2$, Quantum Topol. 1 (2010), no. 2, 111–128. MR 2657645, DOI 10.4171/QT/3
Claudius Zibrowius, On symmetries of peculiar modules; or, $\delta$-graded link Floer homology is mutation invariant, arXiv:1909.04267 (2019).