The Benard-Conway invariant of two-component links

The Benard-Conway invariant of links in the 3-sphere is a Casson-Lin type invariant defined by counting irreducible SU(2) representations of the link group with fixed meridional traces. For two-component links with linking number one, the invariant has been shown to equal a symmetrized multivariable link signature. We extend this result to all two-component links with non-zero linking number. A key ingredient in the proof is an explicit calculation of the Benard-Conway invariant for (2, 2n)-torus links with the help of the Chebyshev polynomials.

💡 Research Summary

The paper studies the Benard‑Conway invariant h_L, a Casson‑Lin type invariant defined by counting irreducible SU(2) representations of the link group with prescribed meridional traces. While Benard and Conway previously proved that for two‑component links with linking number 1 the invariant coincides with a symmetrized multivariable signature σ_L of Cimasoni‑Florens, the present work extends this equality to all two‑component links whose linking number is non‑zero.

The main theorem (Theorem 1.1) states that for an oriented ordered two‑component link L = L₁ ∪ L₂ with lk(L₁,L₂) ≠ 0, and for any angles (α₁,α₂) ∈ (0,π)² such that the multivariable Alexander polynomial Δ_L does not vanish at any of the four points (e^{±2iα₁}, e^{±2iα₂}), the Benard‑Conway invariant satisfies
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