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Cardinality of Blanchfield isometry-orbit quotients under surface homeomorphisms

Determine the cardinality of the quotient set Aut(Bl_{P_{U,g}(c_+,c_-)})/Homeo_\alpha(Σ_g) for all integers g ≥ 0 and c_+, c_- ≥ 0, verifying the following conjectured values: trivial if and only if c_+ + c_- = 0; equal to 2^c c! when (c_+,c_-) ∈ {(c,0),(0,c)}; equal to 2 when (c_+,c_-) = (1,1); and infinite otherwise.

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Background

The classification theorems reduce uniqueness questions for immersed Z-surfaces to the analysis of orbit sets of Blanchfield-form isometries modulo actions of Aut(λ) and of surface homeomorphisms. Understanding the size of Aut(Bl_{P_{U,g}(c_+,c_-)})/Homeo_\alpha(Σ_g) is therefore central to determining when equivariant intersection forms determine immersed surfaces.

The authors show this set is nontrivial in certain small cases and conjecture a complete description of its cardinality across all (c_+,c_-), delineating scenarios where the set is finite and where it is infinite.

Proving or disproving these conjectures would directly impact the uniqueness theory for closed immersed Z-surfaces and clarify the algebraic-topological obstructions encoded by Blanchfield pairings and mapping class group actions.

References

In fact, we conjecture that this set is trivial if and only if $c_++c_-=0$. We also conjecture that for $(c_+,c_-)={(0,c),(c,0)}$ it contains $2c c!$ elements; for $(c_+,c_-)=(1,1)$ it contains 2 elements; and otherwise it is infinite.

— Immersed surfaces with knot group $\mathbb{Z}$ (2410.04635 - Conway et al., 6 Oct 2024) in Remark \ref{rem:WhyItsHard}