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Stable equivalence for knots and links in arbitrary 3‑manifolds

Develop a version of stable equivalence for bridge positions of knots and links in arbitrary 3‑manifolds, analogous to Ozawa’s stable equivalence for handlebody‑knots in S^3, in order to enable the definition of stable complexity invariants in this broader setting.

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Background

In the paper, stable invariants are defined for handlebody‑knots using dual curve distances and stabilization moves inspired by Ozawa’s notion of stable equivalence in S3.

The authors point out that an analogous theory of stable equivalence does not currently exist for knots and links in arbitrary 3‑manifolds, limiting the extension of their complexity‑based invariants to this more general context.

Establishing such a theory would pave the way for defining stable invariants for knots and links beyond S3, potentially mirroring the constructions used for handlebody‑knots.

References

There is not a version of stable equivalence known for knots and links in arbitrary 3-manifolds, it would be interesting to explore the possibility of developing such a theory and trying to define similar invariants.

— Estimating distances in simplicial complexes with applications to 3-manifolds and handlebody-knots (2505.00815 - Mondal et al., 1 May 2025) in Section “Future directions and questions”