Abstract:
(Joint work with S. Melikhov) Rolfsen’s Conjecture (1974): Every knot (tamely or wildly embedded $S^1$) in $S^3$ is non-ambiently isotopic to an unknot. Field medalist Mike Freedman has singled out this striking unresolved conjecture about manifolds as the one with the simplest statement. We will discuss the status of this conjecture including the following results. In 1976, Charles Giffen achieved a partial resolution of Rolfsen's Conjecture which has been updated and extended by S. Melikhov and myself. Also, Melikhov has recently found remarkable examples showing that analogue of Rolfsen’s Conjecture for $2$-component links is false. An easily proved folk theorem asserts that every knot in $S^3$ that pierces a tame disk is non-ambiently isotopic to an unknot. We will show that the same conclusion holds for knots in $S^3$ that pierce wild disks. Also, we will exhibit a wild knot in $S^3$ that pierces a wild disk but pierces no tame disks, thereby showing that the previously stated result has non-trivial applications.
Link for connecting to the seminar: https://mian.ktalk.ru/j1xwg956wc7a
"PIN code": The number of homotopy classes of maps from the Russian word 쬄 to the Russian word ¨Æ (where a word is understood as the subset of the plane formed by its letters)
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