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Geometric Topology Seminar
August 31, 2017 17:00, Moscow, Math Department of the Higher School of Economics (Usachyova, 6), Room 108

On $\Delta$-link homotopy

Yuka Kotoriia, S. A. Melikhovb

a RIKEN – Institute for Physical and Chemical Research
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: M. Gusarov and K. Habiro proved that two knots are not distinguishable by Vassiliev invariants of order $<k$ if and only if they are related by a sequence of $C_k$-moves. The $C_1$-move is a crossing change, and the $C_2$-move can be achieved by taking a connected sum with a copy of the Borromean rings contained in a ball disjoint from the knot. The $C_2$-move can also be presented in a form visually very similar to the third Reidemeister move, and because of this it is also known as the $\Delta$-move.
Two links (or string links) are called self $C_k$-equivalent if they are related by a sequence of $C_k$-moves such that each of them involves strands only from one component. Not only Vassiliev invariants of order $<k$, but also invariants of order $<k$ in the sense of Kirk and Livingston (whose groups are, conjecturally, infinitely generated for $k=2$) are invariant under self $C_k$-equivalence. Self $C_1$-equivalence is better known as link homotopy, and self $C_2$-equivalence is also known as $\Delta$-link homotopy.
We will discuss two new steps in our project of classification of links and string links up to $\Delta$-link homotopy:
In part 1 of the talk, Yuka Kotorii will speak about a crossing change formula for $\mu$-invariants of string links with at most two occurrences of each index. These are precisely those $\mu$-invariants which are invariant under $\Delta$-link homotopy.
In part 2 of the talk, Sergey Melikhov will speak about classification of 3-component string links up to weak $\Delta$-link homotopy ($C_2^{xxx}$ and $C_3^{xx,yz}$ moves) by \mu-invariants of length at most 4 and the generalized Sato-Levine invariant of the closure of each two-component sublink. We also prove that $\bar\mu$-invariants of length at most 4 classify up to weak $\Delta$-link homotopy those 3-component links that are trivial up to link homotopy. As a byproduct of the proof, we compute the image of the Kirk-Koschorke invariant of link maps of three 2-spheres in $S^4$.