The Kakimizu complex of a connected sum of links

Banks, Jessica

doi:10.1090/S0002-9947-2013-05839-4

The Kakimizu complex of a connected sum of links
HTML articles powered by AMS MathViewer

by Jessica E. Banks PDF

Trans. Amer. Math. Soc. 365 (2013), 6017-6036 Request permission

Abstract:

We show that $|\mathrm {MS}(L_1\# L_2)|=|\mathrm {MS}(L_1)|\times |\mathrm {MS}(L_2)|\times \mathbb {R}$ when $L_1$ and $L_2$ are any non-split and non-fibred links. Here $\mathrm {MS}(L)$ denotes the Kakimizu complex of a link $L$, which records the taut Seifert surfaces for $L$. We also show that the analogous result holds if we study incompressible Seifert surfaces instead of taut ones.

References

Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, N.J., 1952. MR 0050886, DOI 10.1515/9781400877492
Julian R. Eisner, Knots with infinitely many minimal spanning surfaces, Trans. Amer. Math. Soc. 229 (1977), 329–349. MR 440528, DOI 10.1090/S0002-9947-1977-0440528-3
Osamu Kakimizu, Finding disjoint incompressible spanning surfaces for a link, Hiroshima Math. J. 22 (1992), no. 2, 225–236. MR 1177053
Osamu Kakimizu, Classification of the incompressible spanning surfaces for prime knots of 10 or less crossings, Hiroshima Math. J. 35 (2005), no. 1, 47–92. MR 2131376
W. B. Raymond Lickorish, An introduction to knot theory, Graduate Texts in Mathematics, vol. 175, Springer-Verlag, New York, 1997. MR 1472978, DOI 10.1007/978-1-4612-0691-0
Piotr Przytycki and Jennifer Schultens, Contractibility of the Kakimizu complex and symmetric Seifert surfaces, Trans. Amer. Math. Soc. 364 (2012), no. 3, 1489–1508. MR 2869183, DOI 10.1090/S0002-9947-2011-05465-6
Makoto Sakuma, Minimal genus Seifert surfaces for special arborescent links, Osaka J. Math. 31 (1994), no. 4, 861–905. MR 1315011
Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594