Sutured instanton knot homology detects the unknot

Given: Full AC, a smooth classical knot $K\subset S^3$, its two oppositely oriented meridional sutures, and the chosen admissible closure from the preceding definition. Remaining established gauge-theoretic inputs below are explicitly imported under the run-specific permission in CLAUDE.md §12, not claimed as locally reconstructed proofs.

The local singular analytic construction gives the finite integral earring complex, its square-zero differential and continuation independence with the declared relative grading and overall sign convention. Analytic foundations for the singular instanton knot complex

The local integral unknot calculation gives $I^{\mathrm{♮}}(U;\mathbb{Z})\cong \mathbb{Z}$, hence ${\dim }_{\mathbb{Q}}{I}^{\mathrm{♮}}(U;\mathbb{Q})=1$. Its generator is specified only up to orientation sign. Singular earring instanton homology of the unknot is integrally rank one

External ordinary determinant-$w$ surface input. For a closed oriented three-manifold $Y$ and a Hermitian line bundle $w$ with odd pairing on an integral surface class, use the determinant-one gauge group of a $U(2)$ lift with fixed determinant $w$. Its admissible instanton homology and commuting operators $\mu (R),\mu (y)$ are defined. For $S^1\times R$, $g(R)\ge 1$, with $c_1(w)$ dual to $S^1$, the ordinary-$w$ product calculation has the spectrum in Kronheimer–Mrowka, Knots, sutures and excision, Proposition 7.1; the common extreme eigenspace $I_{\ast }(S^1\times R\mid R{)}_w$ is one-dimensional by Proposition 7.4. On the ordinary point-$+2$ kernel the extreme surface factor is simple: the paragraph immediately after Proposition 7.4 explicitly states, for every $N\ge 1$, $\ker (\mu (y)-2)\cap \ker (\mu (R)-(2g(R)-2){)}^N=\ker (\mu (y)-2)\cap \ker (\mu (R)-(2g(R)-2))\mathrm{.}$ This holds for the admissible $Y,R,w$ of Definition 7.3, not only product manifolds; no general diagonalizability is asserted. Corollary 7.2 and Proposition 7.5 give the surface bounds on a general admissible $Y$, including the case where $w$ pairs evenly with the other surface; Corollary 7.6 gives the simultaneous generalized-eigenspace decomposition. These are imported published all-genus ordinary-$w$ results, not consequences inferred here from an altered $u$ or $uw$ module. Locators: arXiv:0807.4891v2, §§7.1–7.2, PDF pp. 52–56.

External excision and closure input. Theorem 7.7 of the same source gives excision for two oriented embedded positive-equal-genus surfaces, with equal odd $c_1(w)$ evaluations, an orientation-compatible gluing diffeomorphism and its chosen determinant-bundle lift; when $Y$ is connected the two surfaces are required not to be homologous, and for two components one lies in each. The disconnected group is interpreted as a tensor product. Definition 7.10 and the closure-independence argument following it establish that the sutured group is independent, up to isomorphism, of the allowed auxiliary surface and marked gluing choices. The closure has $c_1(w)$ dual to the standard circle through the fixed auxiliary point. Genus one is permitted. In the meridional knot case Definition 7.13 identifies the annular-auxiliary closure with the torus closure; its group is the generalized surface-$0$ subspace of the ordinary point-$+2$ kernel used in the repaired Definition. By the exact extreme-surface identity in [F3], with $g(\bar{R})=1$, it is also precisely the ordinary simultaneous $(\mu (\bar{R}),\mu (y))=(0,2)$ eigenspace. Definition 7.13's following paragraph identifies this group with the ordinary point-$+2$ half, not the doubled full Floer group. Locators: arXiv:0807.4891v2, Theorem 7.7, Definition 7.10 and its following argument, Definition 7.13, PDF pp. 56–61. The source works over $\mathbb{C}$ for convenience; the rational version is explicitly allowed in the detector paper's §1.1, PDF p. 3.

External decomposition, nonvanishing and genus/rank input. For a balanced sutured manifold decomposed along a surface $S$ with no closed components, and with the closed components of $S\cap V$ parallel oriented boundary-coherent simple closed curves for each component $V$ of $R(\gamma )$, Proposition 7.11 gives the decomposed sutured instanton group as a direct summand (the precise hypotheses of Theorem 6.8, PDF p. 48); Theorem 7.12 states that a taut balanced sutured manifold has nonzero sutured instanton homology. For a classical genus-$g$ knot, Proposition 7.16 gives $KHI(K,g)\mathrm{\ne }0$, with the genus decomposition and its opposite extreme as in §7.6. In particular a nontrivial classical knot has $\dim KHI(K)>1$. Locators: arXiv:0807.4891v2, §§7.5–7.6, PDF pp. 59–64, and Kronheimer–Mrowka, Khovanov homology is an unknot-detector, §1.1, PDF p. 3, which explicitly records this rank consequence and permits $\mathbb{Q}$ coefficients. These decomposition/nonvanishing conclusions are imported established results; no local cap/module or taut-hierarchy proof is asserted here.

External rational singular-to-sutured bridge. For a based smooth knot in $S^3$, Proposition 1.4 of Khovanov homology is an unknot-detector states $I^{\mathrm{♮}}(K;\mathbb{Q})\cong KHI(K;\mathbb{Q})$. Its §5.4, Proposition 5.7, uses the actual earring link $K\cup L$ and its connecting $w_2$ arc; the two tubular-boundary tori are glued exchanging meridians and longitudes and matching the arc endpoints. The excision has a Hopf-link factor $\mathbb{Z}$ and the complementary torus closure. The determinant-one group and the larger gauge quotient are distinguished there: the full ordinary knot group has twice the rank, and the point-$+2$ summand is $KHI$. The stated rational bridge includes this normalization. Locators: Proposition 1.4, PDF p. 3, and §5.4, Proposition 5.7 and its following paragraphs, PDF pp. 74–75. No integral bridge or distinguished canonical vector is inferred.