A Fox–Artin Sphere

A 2-sphere $P$ (or arc $Q$) in $S^3$ (or $R^3$) is said to be tame if some homeomorphism of $S^3$ (resp., $R^3$) onto itself takes $P$ to the standard 2-sphere (resp. $Q$ to the standard interval). A 2-sphere (resp. arc) is wild if it is not tame. A very famous example is the Alexander sphere (1924) whose set of wild points is a Cantor set was discussed in the seminar previously. The wildness of the embedding hinges on the fact that the unbounded complementary domain of the Alexander sphere is not simply connected (for a tame sphere its complements must be).
However simple connectivity of the complementary domains is not sufficient to characterise wildness as was shown by the myriad of examples constructed by Fox and Artin (1948). They started constructing wild arcs and simple closed curves in $S^3$ by knotting them an infinite number of times. Closed neighbourhoods of these examples may be chosen to be 3-cells and their boundary spheres are wild spheres with different properties. The focus of the talk will be a sphere wild in one point, whose complementary domains are open 3-cells.

Zoom link: https://mi-ras-ru.zoom.us/j/95004507525
Access code: the Euler characteristic of the wedge of two circles
(the password is not the specified phrase but the number that it determines)