Findings and severity labels come from OSVerify’s internal review and do not necessarily establish that the paper’s main theorem is false.
Review note. The whole-paper follow-up completed its search without releasing additional findings; the three previously released Minor issues are retained. Direct source verification of Gordon–Luecke’s “Knots are determined by their complements” remains unresolved because the original paper could not be retrieved.
The single-interval case misstates how many parameters disappear
Where the issue occurs
Section 5.4, proof of Lemma 5.8, single-interval case k = 1 · p. 63
For k = 1, both parameters disappear and the two test assignments are u, 1 − u, with u ∈ {0, 1}.
The phrase “both parameters” is the error: this case has one interval parameter, whose removal leaves both pieces with p = 0.
Issue & evidence
In the single-interval k=1 case in the quoted passage, “both parameters disappear” misstates the parameter count: there is only one interval parameter. Its removal leaves both resulting pieces with p=0, consistent with the displayed parameter total and the bounds immediately following it.
Proposed repair by OSVerify
Replace “both parameters disappear” with “the single interval parameter disappears, so both pieces have p=0.” This correction preserves the test assignments and the subsequent single-interval score calculations.
Related context
BackgroundSection 5.4, Lemma 5.8 and parameter-total identity in its proof
A single interval has b = 1, and the displayed total is ∑p = b − k.
Downstream useSection 5.4, immediately following single-interval score calculations
The correction preserves these bounds; they are not additional erroneous assertions.
The index equality counts virtually restored lens parameters
Where the issue occurs
Section 9.7, proof of Proposition 9.12, paragraph beginning “For S retain only the pure tangential equations” · p. 125
The actual pure-line problem has index ds + pi, with the prescribed ideal labels adding at most 4ℓi position directions when tests are ignored.
The equality for the actual index counts virtually restored lens parameters. The following upper-bound argument is preserved by the proposed correction.
Issue & evidence
The actual-index equality in the quoted passage from the proof of Proposition 9.12 counts virtually restored lens parameters as actual parameters. With h_i separated lens ends, the open-face pure-line index before tests and particle-position variables is d_s+p_i-h_i: p_i restores those h_i parameters, while the sharp fillings contribute zero tangential correction. This overcount does not invalidate the subsequent upper bound.
Proposed repair by OSVerify
Replace “has index d_s+p_i” with “has index at most d_s+p_i,” explaining the loss of one parameter per separated lens end. Including particle positions gives index at most d_s+p_i-h_i+4ℓ_i≤d_s+p_i+4ℓ_i. Nonemptiness and tangential transversality therefore still imply the required inequality, preserving its use in the outside-component and fixed-type exclusions.
Related context
BackgroundSection 8.3, definition of p_i before Definition 8.7
Specifies that p_i restores one interval parameter at each lens end, which is why it is not the actual retained parameter count.
BackgroundSection 8.2, Proposition 8.3
Supplies the zero tangential correction for the sharp reference fillings.
Section 9.4, universal-surjectivity argument, unnumbered display after “Green’s formula for the gauge row gives” · p. 116
0 = ∫X ⟨ξ, G*G(χe c)⟩ = −⟨be, c⟩L²(Ye).
The minus sign in the final pairing is the reported error; it should be plus.
Issue & evidence
The boundary-flux evaluation in the quoted display has its sign reversed. For Q=G*G with limiting operator −∂t²+Δγ, Green's boundary expression is ⟨∂tξ,χ_e c⟩−⟨ξ,∂t(χ_e c)⟩. With ξ=a_e+t b_e plus a decaying remainder and t increasing outward, its limit is +⟨b_e,c⟩. The conclusion b_e=0 is unaffected.
Proposed repair by OSVerify
Replace −⟨b_e,c⟩_{L²(Y_e)} by +⟨b_e,c⟩_{L²(Y_e)} in the quoted display. Since the pairing is zero for every independently allowed parallel end constant c, this correction retains b_e=0 and the same starting point for the subsequent integration-by-parts argument.
Related context
BackgroundSection 9.4, preceding limiting operator and expansion of ξ
Fixes the sign convention and the linear end coefficient b_e used in the boundary-flux calculation.
Downstream useSection 9.4, paragraph immediately following the display
The pairing still vanishes for every allowed c, so b_e = 0 and the subsequent integration-by-parts argument are unchanged.
