The Enumeration of Alternating Pretzel Links

TL;DR

This work delivers a complete enumeration framework for oriented alternating pretzel links by crossing number, partitioning the class into three standard Types and reducing counting to equivalence classes of t-codes under flypes and diagram symmetries. It leverages Polya enumeration with cycle indices of $C_k$ and $D_k$ to derive closed formulas for $\mathcal{P}_1(c)$, $\mathcal{P}_2(c)$, and $\mathcal{P}_3(c)$, and combines them into the total $\mathcal{P}(c)=2(\mathcal{P}_1(c)+\mathcal{P}_2(c)+\mathcal{P}_3(c))$. The key contributions include the necklace-based $N(n,k)$ counts, the dihedral corrections $B(n,k)$, and the two-color bracelet counts $B(n_1,k_1;n_2,k_2)$ with parity refinements for Type-3, all yielding explicit summations over feasible parameter sets. Numerical results up to $c=50$ reveal exponential growth with rate $e^{0.588c}$ and indicate that Type-3 links dominate at large crossing numbers; this work also notes potential extensions to non-alternating pretzel links and braid-index considerations. The methodology provides a solid combinatorial framework for broader tabulations within Montesinos-related link families.

Abstract

In this paper, we tabulate the set of alternating pretzel links. Specifically, for any given crossing number $c$, we derive a closed formula that would allow us to compute $\mathcal{P}(c)$, the total number of alternating pretzel links with crossing number $c$. Numerical computation suggests that $\mathcal{P}(c)\approx 0.155e^{0.588c}$. That is, the number of alternating pretzel links with a given crossing number $c$ grows exponentially in terms of $c$.

Figures (6)

• Figure 1: A diagram depicting a general Montesinos link with $4$ rational tangles and $\delta$ horizontal half-twists (crossings).
• Figure 2: A flype move applied to a single crossing and an adjacent strip in a pretzel link diagram.
• Figure 3: Examples of alternating oriented pretzel links.
• Figure 4: The effect of a flype involving a crossing in $\delta$ on the Seifert circle decomposition of a Type 1 (left) or a Type 3 (right) alternating oriented pretzel link diagram.
• Figure 5: A cyclic view point of the Type 1 standard diagrams after using flypes to place the crossings in $\delta$ between two strips. Shown is the case of $k=4$ with $A_j$ being a vertical strip containing $2\alpha_j+1$ crossings.

Theorems & Definitions (18)

• Theorem 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Example 2.7
• Remark 2.8
• Theorem 3.1
• proof