Infinitely many SL2(C)-abelian surgeries characterize torus knots

The full torus direction by its actual group. Let $K=T(a,b)$ with signed coprime nonzero parameters. For every positive integer $n$, the rational slope $(abn+1)\mathrm{/}n$ is primitive since $\gcd (abn+1,n)=1$. Its filling relation is ${\mu }^{abn+1}{\lambda }^n=\mu h^n=1$ by [F1]. Since $h$ is central, the meridian becomes central. It normally generates, so the entire quotient group is cyclic and EVERY representation has abelian image. The slopes $ab+1\mathrm{/}n$ are distinct. This proves the direction for all signed torus knots, including the unit-parameter unknot, without an unproved lens-surgery classification or a restriction to irreducible representations.

Actual closed lifts exclude the supplied hyperbolic case. For a primitive rational slope $p\mathrm{/}q$ with $p\mathrm{\ne }0$, filling abelianization is $\mathbb{Z}\mathrm{/}p\mathbb{Z}$: quotient the meridian-generated exterior abelianization by $p[\mu ]+q[\lambda ]=p[\mu ]$. At each all-but-finitely-many hyperbolic filling in [F3], its closed metric has a faithful SL2 lift. An abelian image would make the whole group abelian and hence that finite cyclic group. Its universal cover ${\mathbb{H}}^3$ would then be a finite-sheet cover of a closed compact space and so compact, a contradiction. Thus every such nonzero hyperbolic filling has a nonabelian representation. Discarding the single zero slope is harmless. This proves that a supplied finite-volume hyperbolic knot has only finitely many SL2-abelian rational fillings; the closed Spin lift, not the exterior's second cohomology, is essential.

Filling transfer and the exact conditional limit argument. Every SL2-abelian filling is SU2-abelian since $SU(2)\subset S{L}_2(\mathbb{C})$. For a nontrivial infinitely-SL2-abelian essential satellite WITH the actual fibered and adjacent-top properties, apply [F4], giving $w\mathrm{\ne }0$ and nontrivial pattern closure. In the final reduction these properties are supplied by choosing the actual knot or mirror furnished by [L1] before this row is invoked. Fact [F2] supplies infinitely many actual SL2-abelian slopes for $P(U)$ and $C$, namely $r_i$ and $r_i\mathrm{/}{w}^2$ from distinct satellite slopes $r_i$. Under [L1], the original SU2-abelian slope set is bounded; every infinite subset therefore has a convergent subsequence, and uniqueness of its accumulation point makes its limit $r(K)$. Indeed infinitely many distinct points outside any fixed neighborhood of that point would have a second accumulation point by compactness of a bounded closed interval. Thus the transferred subsets have limits $r(K)$ and $r(K)\mathrm{/}{w}^2$. Applying [L1] to the nontrivial components identifies $r(P(U))=r(K)=w^{2}r(C)$. No assertion that a mere infinite subset shares a finite accumulation point without boundedness has been used.

Conditional minimal-genus reduction. By [L2] and step1.2, a nontrivial infinitely-SL2-abelian nontorus knot would be an essential satellite. Choose such a satellite of least genus. By [L1], one of this knot and its mirror has the actual fibered and specified Alexander properties. Reflection reverses the positive meridian and preserves the degree-one zero-homology longitude by [F1]; the resulting exterior group isomorphism therefore identifies its filling groups with those at negated slopes. It also carries spanning surfaces in both directions, preserving minimal genus, and sends torus knots to torus knots. Replace the least-genus counterexample by that chosen mirror if needed, and choose its essential satellite decomposition by [L2] and step1.2. We now use its ACTUAL supplied fiber and Alexander properties, without an extra assumption on mirror invariance of those coefficients. Thus [F4] and step1.3 give nontrivial $P(U),C$, nonzero $w$, and $g(K)=g(P(U))+\mathrm{\mid }w\mathrm{\mid }g(C)$. Both positive genera are strictly smaller, even for $\mathrm{\mid }w\mathrm{\mid }=1$. Each component has infinitely many SL2-abelian slopes by [F2], so minimality and [L2], with step1.2, make both torus knots. For a signed torus knot the cyclic filling family in step1.1 accumulates at the parameter product; the uniqueness in [L1] identifies its limit slope with that product. Their signed parameter products have the same nonzero sign by step1.3. Let $a>b\ge 2$ and $c>d\ge 2$ be their respective ABSOLUTE parameter pairs, interchanging parameters when needed. Taking absolute values of that slope relation gives $ab=cd{w}^2$. The symmetric torus polynomial in [F5] depends only on these absolute parameters, so no further mirroring of the actual satellite and no extra instanton property for either component is needed.

We supply the classical satellite Alexander formula used in [L2], instead of assuming it from a satellite Definition. Write $X=V\setminus \mathrm{Int}\nu P$ for the pattern exterior and $E_K=X{\cup }_T{E}_C$ for the satellite exterior, with outer meridian ${\mu }_V$ identified with ${\mu }_C$ and preferred outer longitude with ${\lambda }_C$. Let $w\mathrm{\ne }0$ be the winding number, and let $\mathrm{\Lambda }=\mathbb{Z}[t^{\pm 1}]$ act on covers by the satellite's positive-meridian deck transformation. Total linking restricts to $w$ times companion linking on $E_C$, to $w$ on ${\mu }_V$ and zero on the outer longitude. The pulled-back cover of $T$ has $\mathrm{\mid }w\mathrm{\mid }$ cylinder components, so $H_1(\tilde{T})=\mathrm{\Lambda }\mathrm{/}(t^w-1)$ generated by a lift of ${\lambda }_C$; its $H_0$ is the same module. The cover of $E_C$ has $\mathrm{\mid }w\mathrm{\mid }$ components, with module $H_1=\mathrm{\Lambda }{\otimes }_{\mathbb{Z}[s^{\pm 1}]}{A}_C$, where $s$ acts by $t^w$. Its $H_0$ map from $T$ is an isomorphism, because the two linking images are the same subgroup $w\mathbb{Z}$. Its longitude class is zero in $H_1$: a Seifert surface for $C$ lifts, since linking vanishes on all its loops by normal push-off, and its boundary is that lifted longitude. Replace $C$ by the unknot to obtain the pattern closure $P(U)$. Its companion-side cover has $\mathrm{\mid }w\mathrm{\mid }$ contractible components, the same $H_0$ and zero $H_1$. In both lifted Mayer–Vietoris sequences the $H_0$ map is injective, since its companion-side coordinate is an isomorphism. The only $H_1(\tilde{T})$ image is consequently in the SAME pattern-side module. Quotienting that image yields, respectively, $A_K\cong A_{P(U)}\oplus \left(\mathrm{\Lambda }{\otimes }_{\mathbb{Z}[s^{\pm 1}]}{A}_C\right)$. All gluings and the chosen longitude bounds are in the actual peripheral marking; no companion character transfer is used in this module calculation.

For $w\mathrm{\ne }0$, $\mathrm{\Lambda }$ is free of rank $\mathrm{\mid }w\mathrm{\mid }$ over $\mathbb{Z}[s^{\pm 1}]$ via $s=t^w$, with basis $1,t,\dots ,t^{\mathrm{\mid }w\mathrm{\mid }-1}$ after reversing the exponent convention if $w<0$. Extension of scalars is therefore exact, and [F5] supplies a square presentation of its companion summand by the companion Seifert matrix with $s$ replaced by $t^w$. The direct sum in step 3.1 has a block-diagonal square presentation. Taking its determinant gives ${\mathrm{\Delta }}_K(t)\doteq {\mathrm{\Delta }}_{P(U)}(t){\mathrm{\Delta }}_C(t^w)$. With symmetric normalization and value one at $t=1$, the Laurent unit ambiguity disappears. The winding-zero case gives the same polynomial identity with ${\mathrm{\Delta }}_C(1)=1$: its companion-side cover is now a disjoint union of copies of $E_C$, with $H_1=\mathrm{\Lambda }[{\mu }_C]$ and $H_0=\mathrm{\Lambda }$. The torus-side groups are $H_1={\mathrm{\Lambda }}^2$, $H_0=\mathrm{\Lambda }$, and the meridian map onto the companion $H_1$ is an isomorphism while the longitude maps to zero. The same maps hold for an unknot companion. Eliminate the companion meridian generator in both sequences; their identical remaining pattern quotient is $A_K\cong A_{P(U)}$. This proves the Alexander satellite formula also for $w=0$, without claiming that such a pattern can be an instanton L-space satellite.

Put $v=\mathrm{\mid }w\mathrm{\mid }$; the symmetric companion polynomial satisfies ${\mathrm{\Delta }}_C(t^w)={\mathrm{\Delta }}_C(t^v)$. The finite torus-pattern obstruction is also elementary once the stated coefficient restrictions in [L1] have been supplied. Write $a>b\ge 2$ coprime and $g=(a-1)(b-1)\mathrm{/}2$. The formula in [F5], expanded at infinity, is ${\mathrm{\Delta }}_{T_{a,b}}(t)=t^g(1-t^{-ab})(1-t^{-1})\mathrm{/}((1-t^{-a})(1-t^{-b}))$. Modulo terms of degree at most $g-a$, its expansion is ${\sum }_{i=0}^{\lfloor a\mathrm{/}b\rfloor }(t^{g-ib}-t^{g-ib-1})$. Thus, for $0\le v<a$, its coefficient $c_{g-v}$ is $1$ if $v=0(\mathrm{m}\mathrm{o}\mathrm{d}b)$, $-1$ if $v=1(\mathrm{m}\mathrm{o}\mathrm{d}b)$, and zero otherwise. Suppose the companion's normalized polynomial has leading terms $t^h-t^{h-1}+O(t^{h-2})$ and the satellite's nonzero coefficients alternate with absolute value one, exactly the two externally supplied instanton conclusions in [L1]. By step 4.1 its coefficient in degree $g+hv-v$ is $c_{g-v}-1$. If $v=1(\mathrm{m}\mathrm{o}\mathrm{d}b)$ this is $-2$, impossible. If $v=qb+r$ with $2\le r<b$, it is $-1$, while the preceding nonzero coefficient is the $-1$ in degree $g+hv-qb-1$: all intervening torus coefficients vanish and no lower companion term reaches those degrees. These consecutive negative coefficients contradict alternation. Consequently $b$ divides $v$ whenever $0<v<a$. This argument is a proved finite polynomial implication and uses the externally supplied instanton properties, without proving their gauge-theoretic foundations.

In particular the torus-pattern configuration required by the minimal-satellite argument is incompatible with those coefficient restrictions. If $ab=k{v}^2$ for an integer $k\ge 1$, then $v<a$, since otherwise $ab\ge a^2>ab$. Step 5.1 makes $b\mid v$, whence $a=kb(v\mathrm{/}b{)}^2$ contradicts $\gcd (a,b)=1$ and $b\ge 2$. Mirroring a torus pattern does not change its symmetric Alexander polynomial, so the same contradiction applies after the prescribed mirror reduction. Also $v=1$ is already excluded as soon as both nontrivial pattern and companion have leading terms $t^g-t^{g-1}$ and $t^h-t^{h-1}$: their product has second leading coefficient $-2$. This is the finite polynomial contradiction needed for the conditional configuration in step2.1; it does not supply the externally supplied instanton conclusions.

The companion torus in step2.1 has the adjacent leading terms required in step5.1 by its explicit torus formula [F5], without another instanton hypothesis on that companion. With $v=\mathrm{\mid }w\mathrm{\mid }$, the configuration of step2.1 satisfies $ab=cd{v}^2$; applying step6.1 to this positive $v$ gives a contradiction. Steps1.1–1.2 unconditionally prove the full signed-torus direction and the supplied-metric hyperbolic exclusion; [F2] proves all actual pattern/companion filling cases. Steps1.3–6.1 supply the remaining converse assembly with the exact externally supplied or explicitly conditional premises [L1, L2]. The exact full iff is now supplied independently by [F6], rather than claimed as a newly local gauge or hyperbolization proof. The original full Statement and representation scope remain unchanged.

Apply [F6] to the given smooth knot. Its rational filling and all-representations abelian conventions coincide with the Statement, and its torus convention includes the unknot. It gives the converse to the locally proved signed-torus direction and hence the full original equivalence.