Gordian distance and clasper surgery for links

TL;DR

This paper explores the relationship between Gordian distance, clasper surgery, and $C_k$-equivalence in links, revealing limitations of finite type invariants in bounding unlinking numbers and establishing quadratic bounds.

Contribution

It extends known results from knots to links regarding $C_k$-equivalence and crossing changes, providing new bounds and exact calculations for unlinking complexities.

Findings

Any link with zero linking number can be reduced to $C_k$-trivial in at most $n^2$ crossing changes.

Constructed links show crossing change distance to $C_k$-trivial links grows quadratically with the number of components.

Finite type invariants like Milnor's invariants have limited power to bound unlinking numbers.

Abstract

In 2000, Habiro introduced the notion of $C_k$-equivalence of knots and links. This geometric filtration is closely connected to finite type invariants, a class of invariants including Milnor's invariants. Shortly thereafter, Ohyama, Taniyama, and Yamada proved that $C_k$-equivalence, and by extension finite type invariants, say very little about the unknotting number by showing that any knot is at most one crossing change away from being $C_k$-trivial for any $k\in \mathbb{N}$. The same is not true for links, since the pairwise linking number gives a lower bound on unlinking and is an invariant of $C_2$-equivalence. We prove that, aside from the linking number, the result of Ohyama, Taniyama, and Yamada extends to links: any $n$-component link with linking number zero can be reduced to a $C_k$-trivial link in at most $n^2$ crossing changes. As a consequence, Milnor's invariants carry only limited information about the unlinking number. To establish a lower bound, we produce a sequence of $n$-component links for which the crossing change distance to a $C_k$-trivial link grows quadratically in $n$. Notably, these bounds are independent of the choice of $k\in \mathbb{N}$. Finally, we determine the exact number of crossing changes to a $C_k$-trivial link for links with nonzero linking numbers and where no component is $C_k$-trivial.