2-Adjacent Knots: Graph Structure & Obstructions

• 2-adjacent knots are defined by the existence of two crossing circles such that changing any nonempty subset yields a knot isotopic to a fixed target.
• The study organizes these relations into a directed graph (Γ₂) that captures infinite out‐valence features and nesting properties across knot classes.
• Obstruction techniques using determinant formulas and Alexander polynomial constraints provide rigorous criteria, culminating in a full classification for knots up to 12 crossings.

2-adjacent knots arise in two closely related senses. In the general $n$-adjacency framework, a knot $K$ is $2$-adjacent to a knot $K'$, written $K \xrightarrow{2} K'$, if there are two crossing circles for $K$ such that generalized crossing changes along either one individually, or along both together, carry $K$ to a knot isotopic to $K'$. In the unknotting-specialized usage, a knot is called $2$-adjacent if it has two distinguished crossings such that changing any nonempty subset of them yields the unknot. Recent work develops both a global graph-theoretic formalism for these relations and a concrete cataloging and obstruction theory for $2$-adjacency to the unknot (Campisi et al., 9 Mar 2026, Carney et al., 30 Sep 2025).

1. Definitions and notation

The general notion begins with $K$0-adjacency. A knot $K$1 is said to be $K$2-adjacent to a knot $K$3, written

$K$4

if there exist $K$5 disjoint crossing circles

$K$6

so that for every nonempty subset $K$7, a single Dehn surgery, described as a generalized crossing change, along each circle in $K$8 carries $K$9 to a knot isotopic to $2$0. For $2$1, this becomes the condition that there are two crossing circles $2$2 for $2$3 of surgery orders $2$4 such that

$2$5

Here $2$6 denotes the result of performing a Dehn surgery of slope $2$7 on the torus bounding a small tubular neighborhood of $2$8, and similarly for the other surgeries (Campisi et al., 9 Mar 2026).

In the unknotting formulation, a knot $2$9 is $K'$0-adjacent if it admits a diagram with two distinguished crossings $K'$1 so that

$K'$2

all yield the unknot. Equivalently, the $K'$3-adjacency set $K'$4 has the property that changing any nonempty subset of it unknots $K'$5 (Carney et al., 30 Sep 2025).

These formulations encode a strong simultaneous unknotting or retargeting condition. The defining feature is not merely that two local modifications exist, but that every nonempty subcollection of the prescribed two modifications has the same target.

2. The adjacency graph $K'$6

To organize all $K'$7-adjacency relations, Campisi–Doleshal–Staron define a directed graph $K'$8. Its vertex set is the set of all isotopy classes of knots in $K'$9, and there is a directed edge

$K \xrightarrow{2} K'$0

if and only if $K \xrightarrow{2} K'$1. In this language, $K \xrightarrow{2} K'$2 is a $K \xrightarrow{2} K'$3-adjacent neighbor of $K \xrightarrow{2} K'$4. If $K \xrightarrow{2} K'$5 and $K \xrightarrow{2} K'$6 admits a non-trivial $K \xrightarrow{2} K'$7-adjacency to itself, a bi-directed loop is placed at $K \xrightarrow{2} K'$8. Equivalently,

$K \xrightarrow{2} K'$9

An induced subgraph $K$0 is obtained by restricting to those edges realized by an adjacency in which at least one crossing circle is cosmetic, meaning non-nugatory in $K$1 (Campisi et al., 9 Mar 2026).

This graph packages $K$2-adjacency as a global directed relation rather than a collection of isolated constructions. The direction records the knot from which the specified generalized crossing changes begin, and the graph structure makes it possible to ask about valence, paths, loops, and limiting behavior across all knots.

3. Structural results and bounds

Several basic structural results determine the large-scale behavior of $K$3 and its higher-$K$4 analogues. For $K$5, the unknot $K$6 is isolated in $K$7; in particular, $K$8 has no incoming edge in $K$9. If the Generalized Cosmetic Crossing Conjecture holds for all knots, then $K$0 is totally disconnected and contains no loops (Campisi et al., 9 Mar 2026).

The family $K$1 is nested. For every $K$2,

$K$3

Moreover, for every $K$4,

$K$5

Thus any $K$6-adjacency is already seen in $K$7. This is the sense in which $K$8 carries the full complexity of higher adjacencies (Campisi et al., 9 Mar 2026).

Genus bounds impose strong restrictions. If $K$9 is nontrivial of genus $K'$0, then $K'$1 cannot be $K'$2-adjacent to the unknot for $K'$3. If $K'$4 and $K'$5, then

$K'$6

A consequence is that if $K'$7 for all $K'$8, then $K'$9 and $2$0 are isotopic. Equivalently, $2$1 has no infinitely many distinct neighbors in the inverse-limit graph

$2$2

In fact, every vertex of $2$3 is isolated (Campisi et al., 9 Mar 2026).

The unknot occupies a special position. For every $2$4, the unknot has at least one nontrivial neighbor in $2$5, hence in $2$6, and in $2$7 it has infinitely many distinct nontrivial neighbors. For $2$8, every nontrivial neighbor of the unknot in $2$9, hence also in $2$0, is neither fibered nor alternating. More generally, if $2$1 is fibered and $2$2 for some $2$3, then either $2$4 or $2$5. In particular, the unknot cannot be a $2$6-adjacency neighbor of any nontrivial fibered knot (Campisi et al., 9 Mar 2026).

4. Explicit $2$7-bridge constructions

A large explicit family shows that $2$8-adjacency is abundant even within the class of $2$9-bridge knots. Let $K$00 be any $K$01-braid whose word in generators $K$02 has odd length and ends in $K$03. Denote by $K$04 the $K$05-bridge knot obtained by the $K$06-bridge closure of $K$07. For any pair of nonzero integers $K$08, define $K$09 to be the $K$10-bridge closure of the braid

$K$11

Two crossing circles $K$12 and $K$13 are placed so that each encloses precisely the block of $K$14 or $K$15 crossings on a trivial disk disjoint from the rest of the diagram (Campisi et al., 9 Mar 2026).

The key verification is direct. Surgery of order $K$16 on $K$17 deletes $K$18, yielding precisely the original $K$19-bridge closure of $K$20. Surgery of order $K$21 on $K$22 deletes $K$23, again yielding $K$24. Doing both surgeries yields the same result $K$25. Consequently, for every $K$26-bridge knot $K$27, and for every nonzero pair $K$28,

$K$29

In particular, each $K$30-bridge knot has infinitely many distinct $K$31-adjacent $K$32-bridge neighbors (Campisi et al., 9 Mar 2026).

This construction also produces arbitrarily long directed paths in $K$33. Starting with $K$34, one forms

$K$35

then $K$36 from $K$37 in the same fashion, and so on. The result is a concrete infinite supply of directed configurations inside $K$38, not merely isolated edges.

5. Obstructions from branched covers and Alexander theory

A new obstruction theory for $K$39-adjacency to the unknot is developed using Ozsváth–Szabó correction terms, the Montesinos trick, and Alexander-polynomial restrictions. If $K$40 has unknotting number $K$41 via changing a crossing $K$42, one considers a crossing arc $K$43 whose band-twist effects that single crossing change. After changing $K$44, one obtains an unknot $K$45 together with an embedded arc $K$46. In the branched double-cover $K$47, that arc lifts to a knot $K$48. The signed Montesinos trick gives

$K$49

when $K$50 is unknotted by changing a negative crossing to positive; the other case is analogous (Carney et al., 30 Sep 2025).

If $K$51 is $K$52-adjacent along crossings $K$53, there are two disjoint crossing arcs $K$54. Changing $K$55 or $K$56 individually still unknots $K$57, so each lift $K$58 gives a half-integral surgery to $K$59. In this setting, $K$60-adjacency forces each surgery slope to be $K$61, and the two lifts together form a two-component link $K$62 with linking number $K$63. The linking-matrix calculation gives

$K$64

The sign choice records whether the two crossing changes in the $K$65-adjacency set have the same sign or opposite signs (Carney et al., 30 Sep 2025).

The main obstruction states that if $K$66 is $K$67-adjacent, then

$K$68

for some $K$69. Moreover, if $K$70 is one crossing arc in the $K$71-adjacency set and $K$72 its lift, then for every

$K$73

one has

$K$74

If either of these conditions fails, $K$75 cannot be $K$76-adjacent (Carney et al., 30 Sep 2025).

The Alexander polynomial enters at several levels. The single-variable Alexander polynomial is written

$K$77

where $K$78 is any Seifert matrix, and

$K$79

For a $K$80-component link $K$81 with $K$82, the Torres formula gives

$K$83

In the $K$84-adjacency setting, each component has trivial Alexander polynomial, so

$K$85

Combined with the Baker–Motegi twisting identity, this yields the exact root-of-unity obstruction $K$86 for $K$87 (Carney et al., 30 Sep 2025).

6. Classification through $K$88 crossings

The complete classification through $K$89 crossings is explicit. Exactly the following twenty prime knots are $K$90-adjacent and have crossing number at most $K$91: $K$92

$K$93

No other prime knots with $K$94 crossings are $K$95-adjacent (Carney et al., 30 Sep 2025).

Several patterns are recorded. Among these $K$96, exactly four are alternating knots of crossing number $K$97: $K$98. Beyond that, the list contains both alternating and nonalternating examples, fibered and nonfibered, and displays no obvious uniform “Conway-polynomial-vanishing” pattern, unlike the case $K$99. The paper also states that one can produce all these examples via a finger-move construction on a $2$00-string tangle (Carney et al., 30 Sep 2025).

This classification confirms conjectures of Ito and Kato. Ito conjectured that the complete list of $2$01-adjacent prime knots with crossing number $2$02 is exactly the twenty knots above, and Kato conjectured that no additional prime knots with $2$03 crossings could admit the $2$04-adjacency property. The verification combines basic obstructions, including unknotting number $2$05, signature constraints, and rational-knot monotonicity of Askitas–Kalfagianni, with polynomial-based obstructions of Tao, McCoy’s result for alternating knots, and the new determinant and Alexander-root obstruction (Carney et al., 30 Sep 2025).

7. Open questions and large-scale picture

The case $2$06 is distinguished by the universality statement $2$07 for every $2$08. Because of this, $2$09 contains all higher-order adjacency information while still exhibiting substantial internal structure. Many vertices of $2$10, including the unknot and every $2$11-bridge knot, have infinite out-valence, and the explicit $2$12-bridge construction produces arbitrarily long directed paths. At the same time,

$2$13

is totally discrete, so no two distinct knots can be adjacent for arbitrarily large $2$14 (Campisi et al., 9 Mar 2026).

The paper isolates two specific questions. One asks which other knots have infinite valence in $2$15. The other asks how the number of neighbors of a given knot $2$16 in $2$17 changes as $2$18 increases. By nestedness, this number is nonincreasing in $2$19, and the Howards–Luecke genus bound gives the first step-function bound for $2$20 (Campisi et al., 9 Mar 2026).

Taken together, these results show that $2$21-adjacency is simultaneously a local crossing-change condition, a global directed-graph relation, and an obstruction-sensitive structure controlled by genus, fiberedness, determinants, and Alexander-polynomial behavior. The current theory supplies infinite families, exact small-crossing classifications, and strong nonexistence criteria, while leaving open the finer enumeration of neighbors in $2$22 and the eventual classification of its large-scale topology.