Complexity of equal 0-surgeries

TL;DR

This work investigates when two knots have orientation-preserving diffeomorphic $0$-surgeries, calling such pairs friends, and uses a comprehensive census of low-crossing and census knots to quantify this phenomenon. The authors combine verified volume calculations, Alexander polynomials, Regia/Kirby data, and complementary invariants to classify $0$-surgeries, identify minimal complexity pairs achieving extremal sums, and determine which pairs are also $4$-dimensional friends via trace-diffeomorphism extensions. They establish that the minimum possible sum of crossing numbers for friends is $25$ (uniquely realized by $(K6a1,19nh extunderscore 78)$) and the minimum tetrahedral-sum is $12$ (uniquely realized by $(m224,-v3093)$); among low-crossing knots, six pairs are $4$-dimensional friends, with additional mirror-related pairs explored. Beyond direct enumeration, the paper develops obstructions based on symmetry-exceptional slopes, the Arf invariant, and parity arguments to distinguish trace non-equivalences, and it presents several constructive families (Piccirillo friends, special RGB links, flat annulus presentations) yielding explicit $0$-surgery diffeomorphisms and trace relations. The results contribute to the broader program of understanding how knot complements and traces encode 4-manifold topology, with implications for potential approaches to the smooth $4$-dimensional Poincaré conjecture and for the structure of knot traces in general.

Abstract

We say that two knots are friends if they share the same 0-surgery. Two friends with different sliceness status would provide a counterexample to the 4-dimensional smooth Poincaré conjecture. Here we create a census of all friends with small crossing numbers c and tetrahedral complexities t, and compute their smooth 4-genera. In particular, we compute the minimum of c(K)+c(K') and of t(K)+t(K') among all friends K and K'. Along the way, we classify all 0-surgeries of knots of at most 15 crossings. Moreover, we determine for many friends in our census if their traces are equivalent or not. For that, we develop a new obstruction for two traces being homeomorphic coming from symmetry-exceptional slopes of hyperbolic knots. This is enough to also determine the minimum value of c(K)+c(K') among all friends K and K' whose traces are not homeomorphic.

Figures (4)

• Figure 1: Diagrams of $K15n142188$ (left) and $K15n156076$ (right) showing their satellite structures on $K3a1$.
• Figure 2: A symmetric RBG link yielding knots $K_m$ that are friends with their mirrors
• Figure 3: Three special RBG links. $L_1$ is the RBG link from Figure 13 in Manolescu_Piccirillo_0_surgery. $L_2$ and $L_3$ are obtained from $L_1$ by crossing changes.
• Figure 4: Two flat annulus presentations $L_1$ and $L_2$.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• proof : Proof of Theorem \ref{['thm:classification']}
• Lemma 3.1
• proof
• Proposition 3.2
• proof