Fibered ribbon pretzels

TL;DR

The article classifies when pretzel knots are both fibered and ribbon up to mutation, proving a complete description for prime non-exceptional knots by combining Gabai’s fibered pretzel knot classification with Donaldson-type lattice obstructions. The authors develop a plumbing-graph framework for the double branched covers, derive embedding obstructions from negative definite lattices, and compute signatures via the Wu class to constrain possibilities. They separate the analysis into no-unitary and unitary parameter cases, explicitly identifying the ribbon examples in each regime and providing ribbon disks where possible. This work advances understanding of sliceness and ribbonness in a rich knot family and illustrates how 4-manifold obstructions can decisively inform knot-theoretic properties.

Abstract

We classify fibered ribbon pretzel knots up to mutation. The classification is complete, except perhaps for members of Lecuona's ``exceptional'' family of [Lec15]. The result is obtained by combining lattice embedding techniques with Gabai's classification of fibered pretzel knots, and exhibiting ribbon disks, some of which lie outside of known patterns for standard pretzel projections.

Figures (8)

• Figure 1: The leftmost figure, without the gray band, is the standard projection of the pretzel knot $P(-5,5,-3,3,7)$. The gray band represents the ribbon move that can always be performed when we have adjacent parameters of opposite sign and same absolute value. The other figures depict the result of performing the ribbon move.
• Figure 2: To the left, a pretzel projection of $K=P([1^4],-3,-3,-3)$ with a band describing a move to two unlinked unknots. To the right, the same band after isotoping $K$ to an alternating projection that allows us to identify it as the knot $10_{75}$.
• Figure 3: Two possible negative definite graphs associated to a Type 2 pretzel knot $K$ with $d$ unitary parameters, depending on the value of $e(Y_K)$. The Wu class in each case is depicted in red.
• Figure 4: Partial embedding of $\Gamma_1$ into the standard negative diagonal lattice with basis $\{e^1_1,\dots,e^r_{|n_r|},f_1,f_2\}$. The embedding of the remaining $-2$-chains uses only basis vectors of the form $e^1_*$.
• Figure 5: Two possible negative definite graphs associated to a Type 3A pretzel knot $K$, depending on the value of $e(Y_K)$. The Wu class in each case is depicted in red, and independent of which is the even parameter.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 2.1: Gabai, Theorem 6.7 in Gabai
• Theorem 2.2
• Remark 3.1
• Proposition 4.1
• proof
• Remark 4.2
• Proposition 4.3
• Proposition 4.4
• proof