Title : A group of links and Vassiliev invariants
Speaker : Haruko Aida MIYAZAWA (Tsuda College)
Abstract :

It is known that two knots are $V_n$-equivalent if and only if they are related by a finite sequence of $C_{n+1}$-moves. This result does not hold for links if $n\geq 2$. To generalize the result to links, we define an $SC_2$-move as a special $C_2$-move which satisfies a condition. Then we can show that two links are $V_2$-equivalent if and only if they are related by a finite sequence of $C_3$-moves and $SC_2$-moves. In this talk, I will give another proof of this result. The key of the proof is the following: Let $\mathcal{L}^\mu$ be the set of ambient isotopy classes of oriented $\mu$-component links in $S^3$ and $\mathcal{L}^\mu /\sim$ the set of equivalence classes of $\mathcal{L}^\mu$ with respect to the relation generated by $C_3$-moves and $SC_2$-moves. Then we can define an operation on $\mathcal{L}^\mu /\sim$ and show that $\mathcal{L}^\mu /\sim$ forms a group with the operation. Furthermore we can see that the natural projection $\varphi :\mathcal{L}^\mu \longrightarrow \mathcal{L}^\mu /\sim$ is a Vassiliev invariant of order $2$.

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