Slice-ribbon problem

Importance: Outstanding ✭✭✭✭
Author(s): Fox, R
Subject: Topology
Recomm. for undergrads: no
Posted by: rybu
on: November 7th, 2009

\begin{conjecture} Given a knot in $S^3$ which is slice, is it a ribbon knot? % Enter your conjecture in LaTeX % You may change "conjecture" to "question" or "problem" if you prefer. \end{conjecture}

The definitions of slice and ribbon: \Def{slice knot} \Def{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.

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{}


% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

*[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.