The $n$-adjacency graph for knots

Abstract: A knot $K$ is called $n$-adjacent to a knot $K'$ if there is a set of $n$ crossing circles $\mathcal C$ in $K$ so that a generalized crossing change at any nonempty subset of crossings in $\mathcal C$ yields $K'$. In this paper, the authors define a new graph $Γ_n$ to represent $n$-adjacency relationships between knots. We prove several results about this new object.

• The paper introduces directed graphs Γₙ encoding n-adjacency between knots and proves Γₙ ⊆ Γ₂ for every n ≥ 2, concentrating all adjacency information at order two.
• The paper proves that the unknot has a nontrivial neighbor for every n ≥ 2 and infinite valence in Γ₂, while every knot is isolated in the infinite-order graph Γ∞.
• The paper constructs infinitely many 2-bridge knots 2-adjacent to any given 2-bridge knot and recursively produces arbitrarily long diagrammatic paths, subject to unresolved distinctness of knot types.

Overview and motivation

A generalized crossing change is a Dehn surgery along a crossing circle — a curve bounding an embedded disk meeting the knot in two points with algebraic intersection number zero. A knot $K$ is $n$-adjacent to $K'$, written $K \xrightarrow{n} K'$, if there exist $n$ crossing circles for $K$ such that performing generalized crossing changes on any nonempty subset of them yields $K'$. This notion, studied previously by Howards–Luecke, Kalfagianni–Lin, Torisu, Askitas–Kalfagianni, Tao, and Carney–Meike, has served primarily as a tool for obstruction results: bounding Seifert genus, obstructing fibered structures, and constraining adjacency to the unknot.

Campisi, Doleshal, and Staron introduce a graph-theoretic formalism for this relation. For $n \ge 2$, the directed graph $\Gamma_n$ has one vertex per knot type, with an edge from $K$ to $n$0 precisely when $n$1; if $n$2 is $n$3-adjacent to itself, this appears as a bi-directed loop. A neighbor of $n$4 is any knot admitting an incoming edge that is not isotopic to $n$5. They also define the subgraph $n$6, whose edges are those $n$7-adjacencies in which at least one crossing circle is cosmetic (non-nugatory and not trivializable).

The central structural insight connects $n$8 to the Generalized Cosmetic Crossing Conjecture: since an $n$9-adjacency forces each crossing change to leave the other crossing circles nugatory or cosmetic, any class of knots satisfying the conjecture admits no cosmetic circles. Consequently, the unknot is isolated in every $K'$0, by Scharlemann–Thompson's result that the unknot admits no cosmetic generalized crossing changes; more generally, if the conjecture holds universally, then $K'$1 is totally disconnected and loopless.

Structural properties of $K'$2

Two nesting results organize the family of graphs. First, $K'$3: an $K'$4-adjacency restricts to an $K'$5-adjacency on any subset of crossing circles. Second, and more significantly, by induction $K'$6 for all $K'$7, so all higher-order adjacency information is captured by 2-adjacency alone. The paper defines $K'$8 as the opposite extreme.

The interplay between these extremes yields sharp dichotomies. Howards and Luecke showed that a nontrivial knot of genus $K'$9 fails to be $K \xrightarrow{n} K'$0-adjacent to the unknot for all $K \xrightarrow{n} K'$1, yet constructed nontrivial knots that are $K \xrightarrow{n} K'$2-adjacent to the unknot for each fixed $K \xrightarrow{n} K'$3. Combining these, the authors prove:

• For every $K \xrightarrow{n} K'$4, the unknot has a nontrivial neighbor in $K \xrightarrow{n} K'$5.
• In $K \xrightarrow{n} K'$6, the unknot has infinite valence, proved by contradiction using the genus bound to rule out finitely many neighbors.
• By Askitas–Kalfagianni's Alexander polynomial obstruction, for $K \xrightarrow{n} K'$7 none of the unknot's nontrivial neighbors are fibered or alternating.

At the opposite end, Kalfagianni–Lin's theorem that knots with $K \xrightarrow{n} K'$8 for all $K \xrightarrow{n} K'$9 must be isotopic implies every knot is an isolated vertex of $n$0. Thus the adjacency structure concentrates entirely at low orders: $n$1 carries all edges, while the infinite-order limit carries none. Kalfagianni's result that an $n$2-adjacency ($n$3) into a fibered knot requires either isotopy or strictly larger genus further shows the unknot can be a neighbor only of non-fibered knots in $n$4.

These results should be read against their hypotheses: they depend on the cited genus bounds and on known cases of the cosmetic crossing conjecture (2-bridge knots via Torisu, fibered knots via Kalfagianni, satellites via Balm–Kalfagianni, genus-one pretzels via Ito, among others), not on the full conjecture.

2-adjacency of 2-bridge knots

The main construction concerns 2-bridge knots viewed as closures of 3-string braids under an unusual "2-bridge closure" convention, with braid words $n$5 of odd length ending in $n$6. Given such a $n$7 corresponding to a 2-bridge knot $n$8, define

$n$9

Crossing circles enclosing the $K$0 and $K$1 twist regions have zero algebraic intersection with the knot (guaranteed by odd length), and deleting either or both twist factors recovers $K$2. Hence $K$3 for all nonzero integers $K$4. The main theorem follows:

For every 2-bridge knot $K$5, there are infinitely many 2-bridge knots $K$6 such that $K$7.

This stands in stark contrast to Torisu's proof that 2-bridge knots satisfy the generalized cosmetic crossing conjecture, hence are isolated in $K$8: the vertices have abundant neighbors in $K$9 but no edges there involving cosmetic circles. Since Torisu also produced infinitely many Montesinos knots 2-adjacent to any 2-bridge knot, infinitely many vertices of $K'$0 have infinite valence.

Moreover, the construction iterates: setting $K'$1 preserves odd length, so one may recursively build $K'$2, producing chains $K'$3. This yields infinitely many arbitrarily long paths in $K'$4 — indeed infinitely many infinite paths, one infinite family terminating at each choice of parameters for each of the infinitely many 2-bridge starting points.

One caveat on interpretation: the construction produces distinct braid words, but the paper does not verify that the resulting knot types along a path are pairwise non-isotopic, so "arbitrarily long paths" should be understood as paths in diagrams/braid representatives rather than certified simple paths through distinct knot types in $K'$5.

Limitations and open questions

The paper's structural conclusions about $K'$6 are conditional on the Generalized Cosmetic Crossing Conjecture, which remains open in general (Problem 1.58 of Kirby's list). Several quantitative aspects remain unresolved. The authors pose two explicit questions:

1. Beyond the unknot and the 2-bridge knots, which knots have infinite valence in $K'$7?
2. Since Kalfagianni–Lin provide a constant $K'$8 bounding the adjacency order between non-isotopic knots, and Lemma (nested graphs) shows the number of neighbors is non-increasing in $K'$9, what is the rate at which the neighbor count decreases as $n \ge 2$0 grows, for specific classes of knots?

Additionally, whether the infinite paths constructed in the 2-bridge section correspond to paths through distinct knot types in $n \ge 2$1 is left unaddressed.

Conclusion

By packaging $n \ge 2$2-adjacency into a directed graph, the paper converts existing genus and polynomial obstructions into structural statements: $n \ge 2$3 contains all higher-order adjacencies, the unknot has infinite valence in $n \ge 2$4, $n \ge 2$5 is totally disconnected, and $n \ge 2$6 would be edgeless if the cosmetic crossing conjecture holds universally. The explicit braid construction showing that every 2-bridge knot has infinitely many 2-bridge preimages, together with the recursive iteration yielding long paths, demonstrates that 2-adjacency is far from sparse even within a tightly constrained knot family — while the conjectural emptiness of $n \ge 2$7 suggests all such adjacency is carried by trivializable crossing changes.