Recursive Formulas for Riley Polynomials

• The paper establishes that Riley polynomials factor as ±g(u)g(−u) through explicit recurrences dictated by the knot’s ε-sequence.
• It employs coupled recurrences for f(X) and g(X) with linear computational complexity, enabling efficient invariant calculations for 2-bridge knots.
• The derived recurrences reveal key algebraic properties such as unimodality and irreducibility, offering a combinatorial perspective on knot invariants.

The Riley polynomial $\mathcal{R}_K(\lambda)$ is a distinguished invariant associated with 2-bridge knots $K=S(\alpha,\beta)$. Its structure encodes fundamental properties of the knot group’s $\mathrm{SL}_2(\mathbb{C})$ representations. A central fact is the explicit splitting

$\mathcal{R}_K(-u^2)=\pm g(u)g(-u),$

with $g(u)\in\mathbb{Z}[u]$, where $g(u)$ and $\mathcal{R}_K(X)$ are governed by simple recursive relations dictated by the knot’s canonical “$\epsilon$-sequence.” These recurrences, central to the modern arithmetic and combinatorial understanding of Riley polynomials, enable efficient computation and offer insight into irreducibility and unimodality properties (Jo et al., 2022).

1. Definition of the $\epsilon$-Sequence and $\epsilon$-Chebyshev Polynomials

For a 2-bridge knot $K=S(\alpha,\beta)$ with $\alpha=2n+1$, the $\epsilon$-sequence $$\epsilon=(\epsilon_1,\epsilon_2,\ldots,\epsilon_{2n})$$ is given by

$\epsilon_i = (-1)^{\lfloor i\beta/\alpha\rfloor}$

for $i=1,\ldots,2n$, with the symmetry $\epsilon_i = \epsilon_{2n+1-i}$. Two interrelated families of polynomials, parameterized by this sequence, are central:

• $\epsilon$-Chebyshev polynomials of the second kind: $S_n(t)$, defined as $S_n(t)=T_n(0,1;t)$.
• $\epsilon$-Chebyshev polynomials of the third kind: $V_n(t)=T_{n+1}(1,1;t)$.

Both sequences derive from the unified recurrence

$$T_0(a,b;t)=a;\quad T_1(a,b;t)=b; \quad T_{n+1}(a,b;t) = \epsilon_{n+1} t T_n(a,b;t) - T_{n-1}(a,b;t), \quad (1\leq n \leq m).$$

2. Explicit Recursive Formulas

The recursions for the central polynomials are:

• For $S_n(t)$:

$$S_0(t) = 0, \quad S_1(t) = 1,\quad S_{k+1}(t) = \epsilon_{k+1}t S_k(t) - S_{k-1}(t).$$

• For $V_n(t)$:

$$V_0(t) = 1,\quad V_1(t) = \epsilon_2 t - 1, \quad V_{k+1}(t) = \epsilon_{k+2}t V_k(t) - V_{k-1}(t).$$

These recurrences yield polynomials with integer coefficients, and satisfy

$V_n(t)\,V_n(-t) = (-1)^n S_{2n+1}(t).$

3. Splitting and Recurrences for Riley Polynomials

The Riley polynomial for $K=S(\alpha, \beta)$ with $\alpha=2n+1$ is given in the variable $X=-u^2$ as

$R_K(-u^2) = (-1)^n S_{2n+1}(u).$

By the polynomial identity above,

$R_K(-u^2) = V_n(u) V_n(-u).$

Setting $g(u) := V_n(u)$, this yields the canonical splitting

$R_K(-u^2) = \pm g(u) g(-u).$

Alternatively, $R_K(X)$ may be constructed directly via the following coupled recurrences (with $$f_0(X)=1,\ f_1(X)=1+X,\ g_0(X)=0,\ g_1(X)=\epsilon_1$$):

• $g_k(X) = g_{k-1}(X) + \epsilon_k f_{k-1}(X)$,
• $$f_k(X) = f_{k-1}(X) + f_{k-2}(X) + 2\epsilon_k g_{k-1}(X)$$.

For $K=S(2n+1,\beta)$, $R_K(X)=f_n(X)$. In the parabolic case $X\rightarrow -u^2$,

$f_n(-u^2)=(-1)^n S_{2n+1}(u)=V_n(u) V_n(-u).$

4. Computational Implications and Complexity

The coefficients $\epsilon_k$ enter only linearly in the recurrences for $f_k$ and $g_k$, resulting in $O(n)$ computational complexity once the continued-fraction or Schubert form of $K$—and thus the $\epsilon$-sequence—is determined. Calculation does not require matrix multiplication or full determinant expansions, instead relying solely on recursive updates indexed by the $\epsilon$-sequence (Jo et al., 2022).

5. Algebraic and Combinatorial Properties

The splitting $R_K(-u^2)=g_n(u) g_n(-u)$ enables direct analysis of irreducibility. If $\alpha=2n+1$ is prime and $2$ is a primitive root modulo $\alpha$, then $g_n(u)$ is irreducible over $\mathbb{Q}$, and so is $R_K(X)$. Furthermore, the symmetrized Riley polynomial,

$\Phi_K(x) = x^n S_{2n+1}(x+1/x),$

has coefficients forming a unimodal sequence, established via “Pascal-type" inequalities applied to the signed continuant (Jo et al., 2022).

6. Significance and Applications

Recursive formulas for Riley polynomials, via the $\epsilon$-sequence and their interpretation as $\epsilon$-Chebyshev polynomials, simultaneously provide:

• Efficient computation of knot invariants.
• Immediate extraction of the splitting polynomial $g(u)$.
• Elementary proofs of symmetrized Riley polynomial unimodality.
• Rapid irreducibility checks for broad classes of knots.

These results encapsulate Riley polynomials within a combinatorial and algebraic framework that leverages special recurrences traced entirely to the arithmetic data of the knot. This also reflects broader connections between classical special polynomials and the arithmetic of low-dimensional topology (Jo et al., 2022).