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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.03062 |
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Table of Contents:
- 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.