Hom–Wan exotic-trace conjecture for the satellite patterns

Determine whether, for every knot K in S^3 and all integers m and n, the n-traces of the satellite knots P_{n,m}(K) and Q_{n,m}(K) form an exotic pair, with the sole exception that K is the unknot and m=n=0.

Background

The paper studies the satellite patterns P_{n,m} and Q_{n,m} and compares the smooth structures of their n-traces, X_n(P_{n,m}(K)) and X_n(Q_{n,m}(K)). Earlier work established exoticity in several parameter ranges using the knot-trace invariants ν and |ε|, while the main results of this paper add further cases using immersed-curve methods in bordered Heegaard Floer homology.

The cited conjecture proposes a complete classification: every pair of these traces should be exotic except in the trivial case where the companion K is the unknot and both parameters m and n vanish. The present paper explicitly describes its results as only partial progress toward this conjecture.

References

Hom and Wan additionally posit the following conjecture about exactly which knot traces form exotic pairs; they expect that for every non-trivial case, the resulting knot traces are always exotic. Let $K$ be any knot in $S3$, and $m, n$ integers. Then $X_n(P_{n,m}(K)$) and $X_n(Q_{n,m}(K))$ form an exotic pair, except when $K$ is the unknot and $m = n = 0$.

— Exotic knot traces and the epsilon-invariant  (2609.29432 - Sinha et al., 24 Sep 2026) in Introduction, Conjecture 1.5 (cited as Hom and Wan, Conjecture 1.5)