Local BPS invariants organize the Euler characteristics of punctual Hilbert schemes into a finite genus expansion. This viewpoint is part of the broader relation between curve contributions and BPS counts: Pandharipande–Thomas [5] constructed such contributions in stable pair theory, while Shende [7, Definition 7 and Corollary 10] formulated the local expansion for plane curve singularities. The resulting integers connect local algebraic geometry to knot theory. Oblomkov–Shende [4] conjectured that the refined Hilbert-scheme series recovers the HOMFLY polynomial of the singularity link; Maulik [3] proved this correspondence. Rutherford’s relation between HOMFLY coefficients and oriented rulings [6] then gives a combinatorial way to calculate the relevant specialization.

Su–Xie–Yu [8, Conjecture 1.1] conjectured that the local BPS sequence of every reduced plane curve singularity is log-concave and has no internal zeros. They prove the conjecture for irreducible weighted-homogeneous singularities and for ADE singularities. They also conjecture log-concavity for normalized ruling polynomials of rainbow closures of positive braids [8, Conjecture 2.3]. The restriction to this class of Legendrian links is important: Capovilla-Searle–Pan [1, Theorem 1.4] realize arbitrary polynomials in $z^2$ with nonnegative integer coefficients as graded ruling polynomials of Legendrian links, without imposing the positive-braid or algebraic-link hypotheses. This general realization result does not settle the restricted conjecture, as already noted in [8, Remark 2.5].

We construct counterexamples within the algebraic-link class. An ordinary multiple point already violates log-concavity, although every BPS coefficient is positive. Increasing the common contact order of the branches gives an infinite family. The uniform argument rests on a polynomial dependence on the contact order, not on extrapolation from finitely many examples.

Let $(C,0)$ be a reduced complex plane curve singularity, with $\delta$-invariant $\delta$ and $b$ branches. Let ${\mathrm{Hilb}}^{\mathrm{\ell }}(C,0)$ denote the scheme of length-$\mathrm{\ell }$ subschemes supported at $0$, and let $\chi$ denote its topological Euler characteristic. The local expansion of [7, Definition 7 and Corollary 10], in the indexing convention of [8], defines the integers $n_h(C,0)$ by

In this normalization, log-concavity means $n_h^2\ge n_{h-1}{n}_{h+1}$ for $1\le h\le \delta -1$.

We study the reduced weighted-homogeneous singularities

Their $r$ smooth branches $y=\zeta x^k$, ${\zeta }^r=1$, have pairwise intersection multiplicity $k$. Thus the additivity formula for $\delta$ and the plane-curve Milnor formula give, with $N=\left(\genfrac{}{}{0pt}{}{r}{2}\right)$ and $d=kN$,

Write $B_{r,k}(w)={\sum }_{h=0}^d{n}_h(C_{r,k})w^h$.

Theorem 1.1 disproves Conjecture 1.1 of [8], and Theorem 1.2 gives counterexamples of infinitely many embedded topological types, since $\delta (C_{36,k})=630k$. The link of $C_{r,k}$ is the $r$-component torus link $T(r,rk)$; thus the coprimality hypothesis in the torus-knot result of [8, Theorem 5.1] is essential. Strict positivity throughout these examples leaves the absence-of-internal-zeros question intact. The remaining irreducible cases and the tame character-variety conjecture of [8] are also outside the scope of these counterexamples.

The same construction applies to normalized ruling polynomials. Equation (8) identifies the BPS polynomial of $C_{43,1}$ with the normalized ruling polynomial of the rainbow closure of ${({\sigma }_1\cdots {\sigma }_{42})}^{43}$. Its coefficient sequence therefore also disproves Conjecture 2.3 of [8].

The established Hilbert–HOMFLY correspondence and braid-walk formula identify $B_{r,k}$ with $\tau ({\mathrm{\Omega }}^{k+1})$, where $\mathrm{\Omega }$ is the full twist and $\tau$ is the canonical Hecke trace. The shift from $k$ to $k+1$ comes from the trace dual basis. The standard character expansion then gives a hook-length formula for the Hilbert series. Pairing conjugate partitions shows that each fixed BPS coefficient is polynomial in $u=k(k+1)$; the leading coefficients are normalized return counts for transposition walks in $S_r$.

These formulas serve two different computations. Triangular integer extraction yields the entire BPS sequence for the ordinary point. For the infinite family, an expansion over $\mathbf{Q}[u]$ gives the log-concavity difference as a polynomial of degree at most $42$. Translating its argument by $110$ makes all its coefficients negative, proving the inequality for every $k\ge 10$. A separate braid-walk argument proves positivity throughout both sequences. Appendix A contains a complete verifier, including all inputs and the checks on the displayed certificate.

Let $v=q^{1/2}$, $z=v-v^{-1}$, and $w=z^2$. The generic Hecke algebra ${\mathcal{H}}_r$ over $\mathbf{Q}(v)$ has generators $T_1,\mathrm{\dots },T_{r-1}$, the braid relations, and

Its standard basis is ${(T_{\sigma })}_{\sigma \in S_r}$. If $s_i$ is the $i$th simple transposition and $\mathrm{\ell }(\sigma )$ is the inversion number of $\sigma$, then

Thus a multiplication step always permits a move $\sigma \mapsto s_i\sigma$, and permits a stay, of weight $z$, precisely at a descent.

Let $w_0$ be the longest permutation. For a positive braid word $\beta =\sigma {}_{i_1}\cdots \sigma {}_{i_m}$ read from left to right, put

Here $[T_{\sigma }]$ denotes coefficient extraction in the standard basis. Thus $W_{\beta }$ counts the $w_0$-to-$w_0$ walks of [8, Definition 2.1] by their number of stays. Let ${\mathrm{\Lambda }}_{\beta }$ be the Legendrian rainbow closure of $\beta$, and let $R_{\beta }(z)$ be its $2$-graded ruling polynomial, with a ruling of $j$ switches weighted by $z^{j-r}$. The walk–ruling correspondence [8, (2.1)] gives

If the closure has $b$ components, its normalized ruling polynomial is ${\tilde{R}}_{\beta }(z)=z^b{R}_{\beta }(z)$. These are the conventions used in [8, Conjecture 2.3].

We next fix the normalization relating this polynomial to the local BPS invariants. Use the HOMFLY skein convention $a^{-1}{P}_{+}-a{P}_{-}=z{P}_0$ and $P_{\mathrm{unknot}}=1$. For the link $L$ of $(C,0)$, put $H_C(q)={\sum }_{\mathrm{\ell }}\chi ({\mathrm{Hilb}}^{\mathrm{\ell }}(C,0))q^{\mathrm{\ell }}$ and $\mu =2\delta +1-b$. The Hilbert–HOMFLY correspondence [4, 3], specialized to the lowest $a$-degree, reads

The sign is due to the choice $z=q{}^{1/2}-q{}^{-1/2}$: the skein convention in [4] has the opposite sign of $z$, and $P_L(a,-z)={(-1)}^{b-1}{P}_L(a,z)$. Indeed, smoothing changes the component number by one, so ${(-1)}^{b-1}{P}_L(a,-z)$ satisfies the same skein relation and unknot normalization as $P_L(a,z)$. In particular,

For a positive $r$-strand braid representative of an algebraic link, its fiber surface gives $\mu =m-r+1$. The rainbow closure has Thurston–Bennequin number $\mathrm{tb}({\mathrm{\Lambda }}_{\beta })=m-r$. Rutherford’s identity [6], in the convention above, is therefore $[a^{\mu }]P_L=z{R}_{\beta }$. Combining these identities gives

This recovers the normalized formulas [8, (4.2), (4.3)].

Let $\mathrm{\Delta }$ be the positive half twist. The link of $C_{r,k}$ is the closure of

It is pure and has length $2d$. Hence $b=r$, the prefactor in (8) is $1$, and exactly $2h$ stays contribute to $n_h(C_{r,k})$. A closed walk has an even number of stays, since each move changes permutation parity and the total length is even.

The walk polynomial can now be evaluated through the character theory of the Hecke algebra. The canonical symmetrizing trace is $\tau (T_{\sigma })={\delta }_{\sigma ,e}$, where $e$ is the identity permutation. The standard dual-basis identity is

To verify this identity, equip ${\mathcal{H}}_r$ with the symmetric bilinear form for which the standard basis is orthonormal. Right multiplication by $T_i$ is self-adjoint: on a pair $(T_{\sigma },T_{\sigma s_i})$ with $\mathrm{\ell }(\sigma s_i)=\mathrm{\ell }(\sigma )+1$, its matrix is . Moving the factors of a reduced expression for $T_{\rho }$ across this form gives

This also implies $\tau (ab)=\tau (ba)$ by checking basis elements. Put $\mathrm{\Omega }=T_{w_0}^2$, the full twist.

For a partition $\lambda ⊢r$, let ${\lambda }^{\prime }$ be its conjugate and $S^{\lambda }$ the corresponding generic Specht module. Write

For a box $\mathrm{\square }=(i,j)\in \lambda$, its hook length is $h_{\mathrm{\square }}={\lambda }_i-j+{\lambda }_j^{\prime }-i+1$. The dimension of the Specht module is

We use the standard Schur-element and Murphy-element formulas for the generic type-$A$ Hecke algebra [2]. To compare normalizations, $G_i=v{T}_i$ satisfies $(G_i-q)(G_i+1)=0$, and the canonical trace is unchanged by this rescaling of the standard basis. Its character expansion is

In our normalization, the Murphy elements are

On a seminormal tableau vector, $L_j$ acts by $q^{\mathrm{col}(j)-\mathrm{row}(j)}$, where the row and column refer to the box containing $j$. Consequently $\mathrm{\Omega }$ acts on $S^{\lambda }$ as $q^{c_{\lambda }}$. At $q=1$, (12) gives $s_{\lambda }(1)=r!/f^{\lambda }$, as required for the normalized regular trace of $\mathbf{Q}[S_r]$.

To study a fixed BPS coefficient as the contact order varies, we expand (15) at $w=0$. Conjugate partitions pair the positive and negative powers of $v$, exposing the dependence on $k(k+1)$. Put $s=v+v^{-1}$, so $s^2=w+4$, and set

Define polynomials $D_a(w)$ by

Explicitly,

In particular, $D_a(0)=a$ for odd $a$, and $D_a(0)=a/2$ for even $a$.

Let $e_{\lambda }$ be the number of even hook lengths of $\lambda$. The identity

implies $e_{\lambda }\equiv c_{\lambda }(mod2)$. Define the power series

It has constant coefficient $1$.

Put $u=k(k+1)$. For each partition define $A_{\lambda ,0}(u)=1$, and for $j\ge 1$ define

These are polynomials of degree at most $j$.

The leading coefficients have a symmetric-group interpretation. Let $T_h(r)$ be the number of ordered products of $2h$ transpositions in $S_r$ whose product is the identity. By the ordinary Murphy-element eigenvalue formula [2], the central element acts on the irreducible $S_r$-module of shape $\lambda$ by $c_{\lambda }$. The coefficient of the identity in $J^{2h}$ is $T_h(r)$; taking the normalized regular character therefore gives

Thus, for $h\ge 1$, a necessary condition for $Q_{r,h}{(u)}^2\ge Q_{r,h-1}(u)Q_{r,h+1}(u)$ for all sufficiently large $u$ is

Formula (13) computes the coefficients $H_{\mathrm{\ell }}=[q^{\mathrm{\ell }}]H_{r,k}$ by a finite partition sum. To compute a summand through degree $d$, start with the array $(1,0,\mathrm{\dots },0)$ and, for each hook length $a$, replace its entries successively by

Multiply the resulting array by $f^{\lambda }$, shift it by $E$, and add it to $(H_{\mathrm{\ell }})$. Thus only ordinary nonnegative integer additions and the hook-length dimension formula are required.

Put $F_{\mathrm{\ell }}=[q^{\mathrm{\ell }}]{(1-q)}^r{H}_{r,k}(q)$. Equation (1) is equivalent to

It gives the triangular recurrence

The function bps_coefficients in Appendix A implements these operations with integer arrays. Applied to $(r,k)=(43,1)$, it produces all $904$ coefficients and checks the three integers in Theorem 1.1. No truncation error is involved: terms of degree greater than $d$ cannot affect (25), and each step determines one coefficient from a term with leading coefficient $1$.

Fix $r=36$ and abbreviate $Q_h=Q_{36,h}$. The leading-coefficient test (23) fails at $(r,h)=(36,21)$. Exact integer evaluation of (22) gives

Consequently the polynomial

has a negative leading coefficient. This proves eventual failure of log-concavity. To establish the explicit range $k\ge 10$, we need the full polynomial rather than its leading term alone.

We compute $Q_{20},Q_{21},Q_{22}$ directly from (20), retaining terms through degree $22$ in $w$. Every $D_a$ has nonzero constant term, so the required truncated inverses exist over $\mathbf{Q}$.

The series $R_{\lambda }$ in (18) can be expanded without manipulating square roots. Write $R_{\lambda }={\sum }_{m\ge 0}{r}_m{w}^m$, with $r_0=1$, and

Then

Only ${\gamma }_1,\mathrm{\dots },{\gamma }_{22}$ and $r_0,\mathrm{\dots },r_{22}$ are needed. For each partition, (19) constructs the coefficient arrays of $A_{\lambda ,j}(u)$ exactly; convolution with these truncated series gives (20). The function contact_polynomials in Appendix A carries out this calculation. It groups the weighted series $R_{\lambda }$ by $|c_{\lambda }|$, since the polynomials $A_{\lambda ,j}$ depend only on that value. This reduces repeated polynomial arithmetic without changing the partition sum.

Exact rational expansion and multiplication give

where the positive denominator is

Table 1 displays the signs compactly: its entry $(d_j,e_j)$ means the exact bound

All $d_j$ are positive. The full integers are determined, without rounding, by $b_j=-\mathcal{D}[x^j]G(110+x)$ and the finite formulas (17)–(20). The appendix generates these numbers and verifies their signs directly; the displayed decimal bounds are a compact record of the output.

Explicitly, if $G(u)={\sum }_{i=0}^{42}{g}_i{u}^i$, the coefficient used in the sign test is

The appendix evaluates these sums over $\mathbf{Q}$, checks that all $43$ are strictly negative, computes the least common multiple of their denominators, and checks every bound in Table 1. It also verifies (26) and compares the three moments with the leading coefficients of $Q_{20},Q_{21},Q_{22}$. Thus the displayed bounds record exact integer inequalities, not rounded numerical evidence.