Distance one lens space fillings and band surgery on the trefoil knot

Abstract

We prove that if the lens space L(n, 1) is obtained by a surgery along a knot in the lens space L(3, 1) that is distance one from the meridional slope, then n is in {−6, ±1, ±2, 3, 4, 7}. This result yields a classification of the coherent and noncoherent band surgeries from the trefoil to T (2, n) torus knots and links. The main result is proved by studying the behavior of the Heegaard Floer d-invariants under integral surgery along knots in L(3, 1). The classification of band surgeries between the trefoil and torus knots and links is motivated by local reconnection processes in nature, which are modeled as band surgeries. Of particular interest is the study of recombination on circular DNA molecules.1991 Mathematics Subject Classification. 57M25, 57M27, 57R58 (primary); 92E10 (secondary).

Recent advances on the non-coherent band surgery model for site-specific recombination

“…In the biological literature there are numerous other studies of site-specific recombinases acting at two sites in direct repeats along a single circle, or at two sites on separate circles. Several instances specific to reactions involving the trefoil knot and other T (2, n) torus knots and links are mentioned in [59,Section 5]. Here, we review several more specific examples of site-specific recombination on DNA knots at sites in inverted repeats.…”

“…Bandings along the trefoil. Motivated by our interest in site-specific recombination and by the biological significance of the class of T (2, n) torus knots and links, and in particular of the trefoil knot T (2, 3), in a joint work with Lidman [59] we proved the following classification theorem. Note that T (2, 3) is the right-handed trefoil; an analogous statement holds for the left-handed trefoil after mirroring.…”

“…Each d-invariant d(Y, t) is defined to be the minimal grading of a non-torsion class in HF + (Y, t) [71], and is in fact a Spin c rational homology cobordism invariant. The most technical aspect of Theorem 4.2 required us to prove series of formulas (see [59]) which relate the d-invariants under surgery along knots in the lens space L(3, 1) to certain integer-valued knot invariants that are related to Rasmussen's local h-invariant [76]. These formulas generalize work of Ni and Wu [70].…”

“…A criterion from determinant and signature. Using a similar strategy as that in [59], that is, by studying the d-invariants under surgery in the branched double cover, we established a new criterion on banding via the determinant and the signature σ(K) of a knot. Before stating our result, we need to recall the following definition of Ozsváth and Szabó [73].…”

“…[59, Theorem 1.1] The lens space L(n, 1) is obtained by a distance one surgery along a knot in the lens space L(3, 1) if and only if n is one of ±1, ±2, 3, 4, −6 or 7.…”