Distance One Surgeries in 3-Manifolds

• Distance one surgeries are Dehn surgeries along knots where the surgery slope meets the meridian exactly once, defining a key transformation in 3-manifold theory.
• The classification theorems reveal which lens spaces or Seifert fibered spaces can result from a single distance one surgery, with explicit constructions and arithmetic constraints playing pivotal roles.
• Methodologies such as Heegaard Floer theory and d-invariant computations provide practical obstructions and models, linking the surgery process to biological reconnection via band moves.

A distance one surgery is a type of Dehn surgery along a knot in a 3-manifold where the surgery slope has geometric intersection number one with the meridian. Such surgeries are fundamental in both the theory of 3-manifold transformations and in topological models of natural reconnection processes. The detailed classification of which 3-manifolds can be obtained from others by a single distance one surgery leads to deep constraints, particularly among lens spaces and Seifert fibered spaces.

1. Fundamental Definitions and Concepts

Given a knot $K$ in an oriented 3-manifold $Y$ with tubular neighborhood $\nu(K)$, each slope on $\partial\nu(K)$ corresponds to a primitive element $p\mu + q\lambda$ ($\mu$: meridian, $\lambda$: longitude, $\gcd(p, q)=1$) and is often encoded by $p/q$. The geometric intersection number between two slopes $p/q, r/s$ is $Y$0. A Dehn surgery with slope $Y$1 is called a distance one surgery if $Y$2, i.e., $Y$3 meets the meridian in a single point.

Distance one surgeries are equivalent to integral surgeries when considered in the appropriate basis. If $Y$4 is a knot in a lens space and $Y$5, $Y$6 is called homologically essential if $Y$7, with winding number $Y$8 measuring its class.

2. Classification Theorems for Distance One Surgeries on Lens Spaces

The recent classification results center on lens spaces of the form $Y$9 and which lens spaces $\nu(K)$0 can be obtained from a single distance one surgery. The central results are as follows.

Table: Allowable Distance One Surgeries Between $\nu(K)$1 Spaces

Initial Lens Space	Target After Distance One Surgery	Realization Status
$\nu(K)$2	$\nu(K)$3	Realized
$\nu(K)$4	$\nu(K)$5	Trivial (identity)
$\nu(K)$6	$\nu(K)$7	Realized
$\nu(K)$8	$\nu(K)$9	Realized
$\partial\nu(K)$0	$\partial\nu(K)$1	Realized
$\partial\nu(K)$2	$\partial\nu(K)$3	Open/Exceptional
$\partial\nu(K)$4	$\partial\nu(K)$5	Open/Exceptional
$\partial\nu(K)$6	$\partial\nu(K)$7	Open/Exceptional

This table collates the combinatorial possibilities as described in (Wu et al., 3 Apr 2025, Yang, 2021), and (Wu et al., 2019). Most cases are realized by explicit constructions; four remain open with no known band-move realization ((Wu et al., 3 Apr 2025), Theorem 1.1).

For lens spaces of type $\partial\nu(K)$8, the situation is more restrictive. Aside from three infinite explicit families and 21 sporadic pairs, no other cases arise for $\partial\nu(K)$9 (Wang, 15 Jan 2026).

3. Surgery Formulas and Floer Theoretic Obstructions

The key obstructions and computational tools are provided by Heegaard Floer theory and associated $p\mu + q\lambda$0-invariant formulas.

• For null-homologous knots in $p\mu + q\lambda$1: The Ni–Wu formula states

$p\mu + q\lambda$2

with $p\mu + q\lambda$3 derived from knot Floer local invariants.

• For homologically essential knots: The mapping cone formula applies, encoding the Heegaard Floer complex as a filtered mapping cone with $p\mu + q\lambda$4 controlling the tower structures and grading shifts (Yang, 2021, Wu et al., 2019, Wu et al., 3 Apr 2025).
• For $p\mu + q\lambda$5-space knots, the $p\mu + q\lambda$6-invariant surgery formula (Wu et al., 3 Apr 2025) gives

$p\mu + q\lambda$7

where $p\mu + q\lambda$8 is Floer-simple with the same homology class.

Plumbing techniques and Casson–Walker invariants are also employed to bound possibilities and eliminate spurious candidates (Wu et al., 2019, Wang, 15 Jan 2026).

4. Explicit Constructions, Band Surgeries, and Biological Relevance

Distance one surgeries correspond closely to band surgeries between links in $p\mu + q\lambda$9, via the Montesinos trick: a band move $\mu$0 lifts to a distance one surgery $\mu$1. The enumeration of single band-move transitions is thus governed by the same classification as lens space surgeries (Lidman et al., 2017, Yang, 2021, Wu et al., 3 Apr 2025).

This mechanism models local DNA reconnection (site-specific recombination) in biology, with the knotting type of circular DNA or substrates predicting which transformations are feasible in a single recombination. Only finitely many transitions from, e.g., the trefoil to $\mu$2 torus knots are possible by a single band, reflecting experimental findings (Lidman et al., 2017).

5. Seifert Fibered Surgeries and the Seifert Surgery Network

A parallel paradigm is the Seifert Surgery Network, a 1-complex whose vertices are Seifert surgeries on knots in $\mu$3, with edges corresponding to single twistings (i.e., distance one moves) along "seiferters" (unknotted curves that become fibers in the surgered manifold) or annular pairs (Deruelle et al., 2012). The importance of single-twist moves is highlighted by:

• The algorithmic construction of seiferters using branched covers of tangles and leading-arc detection.
• The connectivity of all three infinite EM-families of Seifert surgeries via explicit finite sequences of distance one moves to torus-knot surgeries.

This framework systematizes the approach to networking, reducing complex surgeries to basic combinatorial paths in the network (Deruelle et al., 2012).

6. Distance One Surgeries to $\mu$4 and Arithmetic Constraints

For transformations of the form $\mu$5, distance one surgery is severely constrained:

• The only infinite families realized correspond to $\mu$6, $\mu$7, $\mu$8.
• Besides these, only 21 exceptional parameter pairs $\mu$9 satisfy all diophantine and Floer-theoretic constraints (Wang, 15 Jan 2026).
• The arithmetic is controlled by difference equations among $\lambda$0-invariants for Seifert fibered intermediates, strictly bounding possible transitions.

The computational method combines explicit computations of $\lambda$1-invariants in both lens spaces (using Ozsváth–Szabó formulas) and small Seifert fibered spaces, along with difference inequalities and Casson–Walker sign tests.

7. Open Problems, Future Directions, and Impact

Despite the completeness of the obstruction theory, two key areas remain open:

• Realizability: For some exceptional or arithmetically allowed $\lambda$2 pairs, it is unknown whether a distance one surgery (or band-move) construction exists (Wu et al., 3 Apr 2025, Wang, 15 Jan 2026).
• Chirally cosmetic bandings: Only for $\lambda$3 with $\lambda$4 are such transformations possible, confirming and extending classic results (Wu et al., 3 Apr 2025).

In the context of DNA topology, these results tightly constrain single-event recombination pathways, lending predictive power to models of recombinase action. In 3-manifold topology, the classification of distance one surgeries via Floer-theoretic invariants provides a template for more general surgery and band-move problems in knot theory.

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References

• (Wu et al., 3 Apr 2025): "Surgeries between lens spaces of type $\lambda$5 and the Heegaard Floer $\lambda$6-invariant"
• (Wang, 15 Jan 2026): "On surgeries from lens space $\lambda$7 to $\lambda$8"
• (Wu et al., 2019): "Studies of distance one surgeries on the lens space $\lambda$9"
• (Yang, 2021): "Distance one surgeries on the lens space $\gcd(p, q)=1$0"
• (Lidman et al., 2017): "Distance one lens space fillings and band surgery on the trefoil knot"
• (Deruelle et al., 2012): "Networking Seifert Surgeries on Knots IV: Seiferters and branched coverings"