Dehn fillings of knot manifolds containing essential once-punctured tori

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A slope $\frac pq$ is called a characterizing slope for a given knot $K_0$ in $S^3$ if whenever the $\frac pq$-surgery on a knot $K$ in $S^3$ is homeomorphic to the $\frac pq$-surgery on $K_0$ via an orientation preserving homeomorphism, then $K=K_0$. In this paper we try to find characterizing slopes for torus knots $T_{r,s}$. We show that any slope $\frac pq$ which is larger than the number $\frac{30(r^2-1)(s^2-1)}{67}$ is a characterizing slope for $T_{r,s}$. The proof uses Heegaard Floer homology and Agol--Lackenby's 6--Theorem. In the case of $T_{5,2}$, we obtain more specific information about its set of characterizing slopes by applying more Heegaard Floer homology techniques.Comment: Version 2: 19 pages. This is a major revision. The title of the first version was "Towards a Dehn surgery characterization of $T_{5,2}$". We extended the result in the first version to general torus knots. We also fixed a gap in the first version, so our result for $T_{5,2}$ is slightly weaker than the originally claimed on

Section: Proof Of Theorem 11mentioning

confidence: 99%

Characterizing slopes for torus knots

Zhang

2014

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“…Since many of the pretzel knots studied below are genus one, it will be helpful for us to have the following result, which gives particularly strong bounds on small Seifert fibered surgery slopes [BGZ11] of such knots.…”

Section: Some Exceptional Dehn Surgery Resultsmentioning

confidence: 99%

Small Seifert fibered surgery on hyperbolic pretzel knots

Meier

2014

“…As α = (103, 2) = s + 4 = 11, the point (824/11, 16/11) (which lies in the line segment with end points (0, 0) and (103, 2)) has norm 8. So the line segment with endpoints (50, 1) and (824/11, 16/11) is contained in B (8). The intersection point of this line segment with the line passing through (0, 0) and ( So β is possibly 51 or 52.…”

Section: Proof Of Theorem 14-part (I)mentioning

confidence: 99%

“…(7) If β is an O-type finite surgery slope but is not a boundary slope, then β X1 = s 1 + 3 or s 1 + 2 or s 1 + 1 or s 1 corresponding to whether both of or only the O-type character or only the D-type character or neither of the two irreducible characters of X(M (β)) (given by Lemma 3.1 (2)) are or is contained in X 1 respectively. (8) If β is an I-type finite surgery slope but is not a boundary slope, then β X1 = s 1 + 4 or s 1 + 2 or s 1 corresponding to whether both of or only one of or neither of the two irreducible characters of X(M (β)) (given by Lemma 3.1 (3)) are or is contained in X 1 respectively. (9) The half-integral finite surgery slope α is either of T -type or I-type.…”

Section: Tables Of Finite Surgeriesmentioning

confidence: 99%

Finite Dehn surgeries on knots in S3

Zhang

2018

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We show that on a hyperbolic knot K in S 3 , the distance between any two finite surgery slopes is at most two and consequently there are at most three nontrivial finite surgeries. Moreover in case that K admits three nontrivial finite surgeries, K must be the pretzel knot P (−2, 3, 7). In case that K admits two noncyclic finite surgeries or two finite surgeries at distance two, the two surgery slopes must be one of ten or seventeen specific pairs respectively. For D-type finite surgeries, we improve a finiteness theorem due to Doig by giving an explicit bound on the possible resulting prism manifolds, and also prove that 4m and 4m + 4 are characterizing slopes for the torus knot T (2m + 1, 2) for each m ≥ 1.where the base orbifold has genus 0, n > 1, gcd(n, m) = 1. Every prism manifold can be expressed in this form. As a byproduct of the proof of Theorem 1.4, we have the following Theorem 1.5. If P (n, m) can be obtained by Dehn surgery on a knot K in S 3 , then n < |4m|.Theorem 1.5 improves [19, Theorem 2] where no explicit bound on n was given. (In an earlier preprint of [19], the author claimed a bound of n < |16m| without proof.)Recall that on a torus knot T (2m + 1, 2), 4m and 4m + 4 are D-type finite surgery slopes. Our next main result implies that each of the prism manifolds S 3 T (2m+1,2) (4m) and S 3 T (2m+1,2) (4m + 4) can not be obtained by surgery on any other knot in S 3 besides ±T (2m + 1, 2). Theorem 1.6. Suppose that S 3 K (4n) ∼ = εS 3 T (2m+1,2) (4n) for some ε ∈ {±} and n = m or m + 1, where ε ∈ {±} stands for an orientation. Then ε = + and K = T (2m + 1, 2).In the terminology of [42], the above theorem implies that 4m and 4m + 4 are characterizing slopes for T (2m + 1, 2), that is, whenever S 3 K (4n) ∼ = S 3 T (2m+1,2) (4n) for n = m or m + 1, then K = T (2m + 1, 2). Combining Theorem 1.6 with Known Facts 1.2 (5) and Known Facts 1.1 (4), we have Corollary 1.7. Any D-type finite surgery slope of a hyperbolic knot in S 3 is an integer larger than or equal to 28.The bound 28 can be realized as a D-type slope on two hyperbolic knots in S 3 (see Table 5 in Section 2).The results described above suggest the following updated conjectural picture concerning finite surgeries on hyperbolic knots in S 3 .Conjecture 1.8. Let K be a hyperbolic knot in S 3 .(1) (The Berge conjecture) If K admits a nontrivial cyclic surgery, then K is a primitive/primitive Then we have an affine isomorphism σ : Spin c (S 3 K (4m)) → Z/4mZ which sends t i | S 3 K (4m) to i (mod 4m). Let µ be the meridian of K, then µ is isotopic to K ′ in S 3 K (4m). Using (7.5), we see that σ(s + PD[µ]) − σ(s) = [µ] · [G] = 1.So the action of [K ′ ] on Spin c (S 3 K (4m)) is equivalent to adding 1 in Z/4mZ. There is a similar result when we replace K with T .