Neighbors of Seifert surgeries on a trefoil knot in the Seifert Surgery Network

Abstract

A Seifert surgery is a pair (K , m) of a knot K in S 3 and an integer m such that m-Dehn surgery on K results in a Seifert fiber space allowed to contain fibers of index zero. Twisting K along a trivial knot called a seiferter for (K , m) yields Seifert surgeries. We study Seifert surgeries obtained from those on a trefoil knot by twisting along their seiferters. Although Seifert surgeries on a trefoil knot are the most basic ones, this family is rich in variety. For any m = −2 it contains a successive triple of Seifert surgeries (K , m), (K , m + 1), (K , m + 2) on a hyperbolic knot K , e.g. 17-, 18-, 19-surgeries on the (−2, 3, 7) pretzel knot. It contains infinitely many Seifert surgeries on strongly invertible hyperbolic knots none of which arises from the primitive/Seifertfibered construction, e.g. (−1)-surgery on the (3, −3, −3) pretzel knot.Keywords Dehn surgery · Hyperbolic knot · Seifert fiber space · Trefoil knot · seiferter · Seifert surgery network Dedicated to Fico González-Acuña on his 70th birthday.A. Deruelle

Networking Seifert surgeries on knots, III

“…The following example motivates us to consider the Seifert surgery network. ; see the authors [7]. Since the linking number between c 0 and T 3;2 is 5, the surgery slope changes from 7 to 7 C 5 2 D 18.…”

Networking Seifert surgeries on knots, III

“…Hence, we call (T p,q , m) with a hyperbolic seiferter or a hyperbolic annular pair a spreader. Previously known examples of spreaders [7,8,6,9] have specific patterns and lead us to the following conjecture.…”

“…(1) The meridian c µ for T −3,2 is a seiferter for all (T −3,2 , m) (m ∈ Z). Twisting along c µ yields the horizontal line in Figure 1 pretzel knot P (−2, 3, 7) ( [9]). Since the linking number between c ′ and T −3,2 is 5, the surgery slope changes from −7 to −7 + 5 2 = 18.…”

“…[9]). Take the knot c m in S 3 − T 3,2 illustrated inFigure 4.5; then c m is a hyperbolic seiferter for (T 3,2 , m), (T 3,2 , m+1), and (T 3,2 , m+2) except when m = In particular, (T 3,2 , m) has a hyperbolic seiferter if m = 4, 5.…”

Seifert fibered surgeries on strongly invertible knots without primitive/Seifert positions

“…Since P (−3, 3, 3) is a strongly invertible knot of tunnel number 2, that surgery gives the negative answer to Question 1.1. In [5] we construct a oneparameter family of Seifert fibered surgeries which answer Question 1.1 in the negative and contain Song's example by using the Seifert Surgery Network introduced in [4]. In this paper, we construct a large family of Seifert fibered surgeries giving the negative answer to Question 1.1 by taking 2-fold branched covers of tangles.…”