

Geometric Topology Seminar
August 31, 2017 17:00–19:50, Moscow, Math Department of the Higher School of Economics (Usachyova, 6), Room 108






On $\Delta$link homotopy
Yuka Kotorii^{a}, S. A. Melikhov^{b} ^{a} RIKEN – Institute for Physical and Chemical Research
^{b} Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

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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 3component string links up to weak $\Delta$link homotopy ($C_2^{xxx}$ and $C_3^{xx,yz}$ moves) by \muinvariants of length at most 4 and the generalized SatoLevine invariant of the closure of each twocomponent sublink. We also prove that $\bar\mu$invariants of length at most 4 classify up to weak $\Delta$link homotopy those 3component links that are trivial up to link homotopy. As a byproduct of the proof, we compute the image of the KirkKoschorke invariant of link maps of three 2spheres in $S^4$.

