The Conway knot is not slice

You need to know: Euclidean space , unit ball in , unit sphere in , closed curves in , self-intersecting and non-self-intersecting curves, smoothly embedded 2-dimensional disk in .

Background: A knot is a closed, non-self-intersecting curve in . Two knots and are equivalent if can be transformed smoothly, without intersecting itself, to coincide with . There are tables which lists some knots up to equivalence, which are called the Rolfsen tables. The Conway knot is the knot labelled 11n34 in these tables. A knot is called slice if it bounds a smoothly embedded 2-dimensional disk in .

The Theorem: On 8th August 2018, Lisa Piccirillo submitted to arxiv a paper in which she proved that the Conway knot is not slice.

Short context: A knot diagram is a projection of a knot into a plane, which (i) is injective everywhere, except at a finite number of crossing points, which are the projections of only two points of the knot, and (ii) records over/under information at each crossing. We say that the knot K has n crossings if n is the minimal number of crossing a knot diagram of a knot equivalent to K may have. The notion of slice knot is a central notion in 4-dimensional theory of knots, originated by Fox in 1962. By 2005, all knots of under 13 crossings were classified whether they are slice knots or not, with the only exception of the Conway knot (which has 11 crossings). The Theorem completes this classification.

Links: Free arxiv version of the original paper is here, journal version is here.

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