L-spaces and knot traces

There has been a great deal of interest in understanding which knots are characterized by which of their Dehn surgeries. We study a 4-dimensional version of this question: which knots are determined by which of their traces? We prove several results that are in stark contrast with what is known about characterizing surgeries, most notably that the 0-trace detects every L-space knot. Our proof combines tools in Heegaard Floer homology with results about surface homeomorphisms and their dynamics. We also consider nonzero traces, proving for instance that each positive torus knot is determined by its $n$-trace for any $n\leq 0$, whereas no non-positive integer is known to be a characterizing slope for any positive torus knot besides the right-handed trefoil.

💡 Research Summary

The paper investigates a four‑dimensional analogue of the well‑studied problem of characterizing knots by their Dehn surgeries. For a knot K⊂S³ and an integer n, the n‑trace Xₙ(K) is the smooth, oriented 4‑manifold obtained by attaching an n‑framed 2‑handle to B⁴ along K. The authors ask for which pairs (K,n) the diffeomorphism type of Xₙ(K) determines K, i.e. Xₙ(J)≅Xₙ(K) forces J=K. While it is known that traces often fail to detect knots (e.g. Piccirillo’s construction of a different knot with the same 0‑trace as the Conway knot), the authors prove striking positive results, especially for L‑space knots.

The main theorem (Theorem 1.2) states that if K is an L‑space knot—i.e. some positive surgery on K yields a Heegaard‑Floer L‑space—then the 0‑trace X₀(K) detects K. The proof splits into two complementary ingredients. First (Theorem 1.3) the authors show that a single trace already detects whether a knot is an L‑space knot and recovers its genus. For n=0 this follows from their earlier work