The knot-complement theorem

Assume the Axiom of Choice. Let $K\subset S^3$ be a smooth knot with chosen exterior $E_K$ and ordered meridian-longitude basis $(\mu ,\lambda )$. A surgery triad is a cyclically ordered triple of primitive unoriented slopes $s_0,s_1,s_{\mathrm{\infty }}$ on $\mathrm{\partial }{E}_K$ whose pairwise geometric intersection numbers are one, together with the oriented fillings $Y_j=E_K(s_j)$ and the three 2-handle surgery cobordisms in the cyclic order $Y_0\to Y_1\to Y_{\mathrm{\infty }}\to Y_0$. Choose primitive representatives $v_j=(p_j,q_j)$ in the ordered basis $(\mu ,\lambda )$ so that $\det (v_0,v_1)=\det (v_1,v_{\mathrm{\infty }})=\det (v_{\mathrm{\infty }},v_0)=1$; the cyclic order is the one for which this orientation convention holds. The geometric intersection number of $s_j$ and $s_k$ is $\mathrm{\mid }{p}_j{q}_k-p_k{q}_j\mathrm{\mid }$.

For a closed oriented Riemannian 3-manifold $Y$, a principal $SO(3)$-bundle $P\to Y$ is a locally trivial bundle with fiber $SO(3)$ and right structure-group action. A connection $A$ is a smooth right-$SO(3)$-invariant horizontal distribution on $P$ complementary to its vertical tangent spaces. In a local trivialization its curvature is $F_A=dA+\frac{1}{2}[A\wedge A]$; it is flat when $F_A=0$. Let $\mathcal{A}(P)$ be the space of smooth connections and let $\mathcal{G}(P)$ be the group of bundle automorphisms covering the identity of $Y$. The configuration space is the quotient $\mathcal{B}(P)=\mathcal{A}(P)\mathrm{/}\mathcal{G}(P)$. The associated oriented rank-three real vector bundle has second Stiefel–Whitney class $w_2(P)\in H^2(Y;\mathbb{Z}\mathrm{/}2)$.

Fix an $SO(3)$-invariant inner product on $\mathfrak{s}\mathfrak{o}(3)$ and normalize the Chern–Simons functional on its covering of $\mathcal{B}(P)$ so that its differential at $A$ in a connection variation $a$ is $d{\mathrm{C}\mathrm{S}}_A(a)={\int }_Y⟨a\wedge F_A⟩$. A holonomy perturbation is a finite sum of gauge-invariant terms formed by integrating smooth class functions of holonomy around a finite family of embedded loops against compactly supported smooth weights. A critical point of a perturbed Chern–Simons functional is called a perturbed flat connection. At a nondegenerate critical point, the gauge-fixed Hessian is self-adjoint; the spectral flow along a path of such Hessians is the signed count of eigenvalues crossing zero and supplies the relative grading convention. Give $\mathbb{R}\times Y$ the product metric and orientation $dt\wedge \mathrm{o}\mathrm{r}(Y)$. Fix a holonomy perturbation $f_{\pi }$ and write $V_{\pi }(B)={\mathrm{\nabla }}_{L^2}{f}_{\pi }(B)$ for its gradient. A finite-energy perturbed anti-self-dual trajectory is a gauge-equivalence class with a temporal-gauge representative $A=B(t)$ on the pullback bundle satisfying $\frac{dB}{dt}=-({\ast }_Y{F}_B+V_{\pi }(B))=-{\mathrm{\nabla }}_{L^2}(\mathrm{C}\mathrm{S}+f_{\pi })(B),$ with specified limiting critical points ${\beta }_{-},{\beta }_{+}$ of that same perturbed functional at the two ends, and with finite flow energy $E_{\pi }(B):={\int }_{\mathbb{R}}{\parallel \frac{dB}{dt}\parallel }_{L^2(Y)}^{2}dt={\int }_{\mathbb{R}}\mathrm{\parallel }{\ast }_Y{F}_B+V_{\pi }(B){\mathrm{\parallel }}_{L^2(Y)}^{2}dt<\mathrm{\infty }\mathrm{.}$ Equivalently, in this gauge its four-dimensional equation is $F_A^{+}+{\widehat{V}}_{\pi }(A)=0$, where ${\widehat{V}}_{\pi }(A)=\frac{1}{2}(dt\wedge V_{\pi }(B)+{\ast }_Y{V}_{\pi }(B))$. When $\pi =0$ this reduces to the ordinary anti-self-duality equation. On a chosen real lift of the Chern–Simons functional along the trajectory, the chain rule gives $d(\mathrm{C}\mathrm{S}+f_{\pi })(B(t))\mathrm{/}dt=-\mathrm{\parallel }dB\mathrm{/}dt{\mathrm{\parallel }}_{L^2}^2$; whenever the functional has the specified endpoint limits, the energy is their difference. For nonzero perturbations those endpoints need not be flat, so no finiteness of the unperturbed curvature integral ${\int }_{\mathbb{R}\times Y}\mathrm{\mid }{F}_A{\mathrm{\mid }}^2$ is imposed. If the analytic Fredholm construction is established, the expected dimension is the index of the gauge-fixed linearization of the perturbed equation with the stated asymptotic conditions; translation decreases dimension by one for a nonconstant trajectory.

The candidate instanton surgery-triangle input data consist of a principal $SO(3)$-bundle on each $Y_j$, compatible bundle identifications on the three cobordisms, the invariant inner product and Chern–Simons normalization, the allowed perturbation class, and the grading convention. After separate proofs of analytic existence, regularity, compactness and gluing, these data may be used to define critical-point complexes and cobordism maps. Chain complexes, exactness of the three-term Floer sequence, and independence of the choices are separate theorem obligations. This definition asserts none of those analytic or Floer-theoretic conclusions.

Slope coordinates, meridian-longitude basis and oriented fillings

Kronheimer–Mrowka, Dehn surgery, the fundamental group and SU(2), §§2–4

Kronheimer–Mrowka, Knots, sutures and excision, §§7.1–7.8

Motegi, Dehn surgery on knots: survey, pp. 7–10

The ordinary closed-manifold $SO(3)$ theory in this definition is distinct from the singular earring complex and sutured closure used for Khovanov detection. No Khovanov or singular-instanton rank detector is a logical input to Property P in the proof route considered here.

Kronheimer–Mrowka, Dehn surgery, the fundamental group and SU(2), §2

Let $T$ be the trefoil represented by the standard oriented three-crossing diagram below. Choose a basepoint $x_0\in S^3\setminus T$ in the diagram's upper collar and use the common upper-collar paths from $x_0$ to base the positive-meridian generators. Its under-arcs are labelled $a,b,c$ so that the ordered triples $(o,L,R)$ of over-arc, upper target under-arc and lower source under-arc are $(b,c,a)$, $(c,a,b)$, and $(a,b,c)$ at the three crossings; read each triple in the orientation-preserving local chart that straightens the over-arc to point rightward. The knot group $G_T={\pi }_1(S^3\setminus T,x_0)$, where ${\pi }_1$ is based loops modulo endpoint-fixed homotopy with multiplication by concatenation, has presentation $⟨a,b\mid aba=bab⟩$. This presentation denotes the free group on $a,b$ modulo the normal closure of the displayed relation. Its abelianization, the quotient by its commutator subgroup, is isomorphic to $\mathbb{Z}$, but $G_T$ is nonabelian.

The drawing shows two oriented strands crossing three times; the two bottom endpoints join to the corresponding top endpoints by the outside arcs. The white gaps mark underpasses, and each redrawn strand is over. The crossing triples in the statement fix the generator labels and local chart convention.

Wirtinger presentation theorem

Local crossing conjugacy convention

Cattabriga and Francaviglia, Knot Theory and Applications: Lecture Notes, §4.2

Charbonneau, Computing the Wirtinger Presentation of a Knot Group from a Knot Diagram, §§3–4.1

Assume the Axiom of Choice. Let $K\subset S^3$ be a nontrivial smooth knot, and let $p,q$ be coprime integers with $\mathrm{\mid }p\mathrm{\mid }=1$ and $q\mathrm{\ne }0$. For the Dehn filling $Y_{p\mathrm{/}q}=E_K(p\mathrm{/}q)$ whose attached solid-torus meridian is identified with $p[\mu ]+q[\lambda ]$, the fundamental group ${\pi }_1(Y_{p\mathrm{/}q})$ admits a homomorphism to $SU(2)$ with noncyclic image.

Chosen knot exterior

Meridian and preferred longitude

Filling slope convention

Ordinary closed instanton surgery data

Kronheimer–Mrowka, Dehn surgery, the fundamental group and SU(2), Theorem 1 and §§2–4

Kronheimer–Mrowka, Witten's conjecture and Property P, §§1 and 3

Kronheimer–Mrowka, Knots, sutures and excision, §§7.5 and 7.8

Motegi, Dehn surgery on knots: survey, pp. 19–21 and 26

Kronheimer–Mrowka, published Theorem 1, p. 741

The published primary PDF was obtained from the International Press journal archive. Theorem 1 on printed p. 741 (PDF p. 1) was read directly, including its exact nontrivial-knot and $\mathrm{\mid }r\mathrm{\mid }\le 2$ hypotheses and its noncyclic-image conclusion. The slope definitions and $[pa+qb]$ filling convention on pp. 743 and 745–746 were inspected, together with the application and nonvanishing discussion on pp. 752–753. These source readings identify the imported theorem and its convention; they do not constitute a local proof audit of its gauge theory or of all its cited prerequisites.

The paper explicitly says on p. 741 that its proof avoids explicit instanton Floer homology and the Floer exact triangle, using holonomy perturbations instead. Accordingly, this item's historical identifier does not assert a general surgery-triangle or rank theorem: its full Statement is the numerator-one representation conclusion proved above from the external input. The retained singular-instanton complex and integral resolution cube keep their existing scope; neither is relabelled as supplying this closed-surgery result. Ordinary KHI nonvanishing and varying-meridian representation paths are separate inputs and are not inferred here.

Assume the Axiom of Choice. Let $K\subset S^3$ be a nontrivial smooth knot, and let $p,q$ be coprime integers with $q\mathrm{\ne }0$. For the Dehn filling $Y_{p\mathrm{/}q}=E_K(p\mathrm{/}q)$ whose attached solid-torus meridian is identified with $p[\mu ]+q[\lambda ]$, the fundamental group ${\pi }_1(Y_{p\mathrm{/}q})$ is nontrivial.

Chosen knot exterior

Meridian and preferred longitude

Filling slope convention

First homology of knot fillings

Noncyclic SU(2) representations for numerator-one slopes

Kronheimer–Mrowka, Dehn surgery, the fundamental group and SU(2), Theorem 1

Kronheimer–Mrowka, Knots, sutures and excision, §7.8

Motegi, Dehn surgery on knots: survey, pp. 19–21 and 26

Kronheimer–Mrowka, published Theorem 1, p. 741

The numerator-one case uses the external Kronheimer–Mrowka theorem through [F3], with its exact hypotheses and reading limits recorded in that supplier. The homology cases and the deduction of nontriviality from a noncyclic image are the local argument above.

Assume the Axiom of Choice. Let $K\subset S^3$ be a nontrivial smooth knot and let $p,q$ be coprime integers, not both zero. The Dehn filling $E_K(p\mathrm{/}q)$ is homeomorphic to $S^3$ if and only if $q=0$, in which case $p=\pm 1$ and the slope is the meridional slope $1\mathrm{/}0$.

Chosen knot exterior

Meridian and preferred longitude

Filling slope convention

First homology of knot fillings

Property P for nontrivial knots

Motegi, Dehn surgery on knots: survey, pp. 7–10 and 19–21

Kronheimer–Mrowka, published Theorem 1, p. 741

The exterior $E_K$ and the inserted solid torus meet along $T_K$. Its meridian bounds a disk in the inserted torus and is attached along $p\mu +q\lambda$. When $q=0$, that curve is the original meridian up to orientation, so the filling restores the original tubular neighborhood. The diagram records this gluing; the homology and Property P steps above prove the uniqueness assertion without relying on the picture.

The nonmeridional exclusion uses Property P [F2], whose numerator-one branch explicitly imports Kronheimer–Mrowka, published Theorem 1, p. 741. Its source and reading limits are recorded in the numerator-one supplier. The homology case split and the meridional reconstruction above are retained local arguments.

Assume the Axiom of Choice. Let $K,K^{\mathrm{\prime }}\subset S^3$ be smooth knots with chosen compact exteriors $E_K,E_{K^{\mathrm{\prime }}}$, and let $h:E_K\to E_{K^{\mathrm{\prime }}}$ be a homeomorphism. Either both $K,K^{\mathrm{\prime }}$ are unknotted or both are nontrivial. If both are unknotted, they are ambiently equivalent, but this particular $h$ need not preserve their knot-meridian slopes. If both are nontrivial, $h({\mu }_K)$ is carried to ${\mu }_{K^{\mathrm{\prime }}}$ by an isotopy through homeomorphisms of the boundary torus, and $h$ extends across their tubular solid tori to a homeomorphism of pairs $(S^3,K)\to (S^3,K^{\mathrm{\prime }})$; that extension preserves ambient orientation when $h$ preserves orientation and reverses it when $h$ reverses orientation.

Compact knot exterior

Meridian slope convention

Ambient equivalence of knots and standard unknot

S3 filling detects the meridian for a nontrivial knot

Smooth Loop theorem for continuous essential-boundary disk maps

Extension over a solid torus

Cattabriga and Francaviglia, Knot Theory and Applications: Lecture Notes, §4.1.1

Motegi, Dehn surgery on knots: survey, pp. 19–21

Kronheimer–Mrowka, Dehn surgery, the fundamental group and SU(2), Theorem 1

Kronheimer–Mrowka, published Theorem 1, p. 741

For nontrivial knots, the exterior map $h$ carries the source meridian filling to the target filling along $\gamma$. The unique-S3-filling lemma forces $\gamma$ to be the target meridian, after which the meridian-preserving boundary map extends over the solid tori. The commutative square records the maps; the proof above justifies each extension without relying on the diagram.

The full homeomorphism statement is retained. The unknot/nontrivial category step uses the declared smooth Loop theorem on the continuous essential-boundary disk map, as written in step 2.1; it does not require a relative smoothing theorem for the given topological disk. This local repair consumes that theorem only at its stated scope and makes no new audit claim about its proof. The nontrivial branch uses Property P and the S3-filling lemma, whose numerator-one input is now explicitly supplied by the external Kronheimer–Mrowka published Theorem 1, p. 741. That imported gauge-theoretic theorem is not claimed as a local reconstruction. The local case split, filling transport, and solid-torus extension retain the full Statement above; no new audit or workflow-gate result is asserted here.