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Slice-ribbon problem

Importance: Outstanding ✭✭✭✭
Author(s): Fox, R
Subject: Topology
Keywords: cobordism
knot
ribbon
slice
Recomm. for undergrads: no
Posted by: rybu
on: November 7th, 2009
Conjecture   Given a knot in $ S^3 $ which is slice, is it a ribbon knot?

The definitions of slice and ribbon: slice knot ribbon knot

There is a fairly vast literature on this problem. It is closely related to the problem of determining which homology $ 3 $-spheres bound homology $ 4 $-balls, as both are in essence a type of $ 4 $-dimensional cobordism problem.

Related problems

Which homology 3-spheres bound homology 4-balls?

Bibliography

*[F] Fox, R. H. Some problems in knot theory. 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 168--176

[G] Gilmer, Patrick M. On the slice genus of knots. Invent. Math. 66 (1982), no. 2, 191--197.

[H] Hass, Joel. The geometry of the slice-ribbon problem. Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 1, 101--108.

Ana G. Lecuona. [arXiv:0910.4601] On the Slice-Ribbon Conjecture for Montesinos knots

Brendan Owens. [arXiv:0802.2109] On slicing invariants of knots.


* indicates original appearance(s) of problem.
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