Networking Seifert surgeries on knots, III

Abstract

How do Seifert surgeries on hyperbolic knots arise from those on torus knots? We approach this question from a networking viewpoint introduced in [8]. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; two vertices are connected by an edge if one Seifert surgery is obtained from the other by a single twist along a trivial knot called a seiferter or along an annulus cobounded by seiferters. Successive twists along a "hyperbolic seiferter" or a "hyperbolic annular pair" produce infinitely many Seifert surgeries on hyperbolic knots. In this paper, we investigate Seifert surgeries on torus knots which have hyperbolic seiferters or hyperbolic annular pairs, and obtain results suggesting that such surgeries are restricted.2010 Mathematics Subject Classification. Primary 57M25, 57M50 Secondary 57N10

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

“…The table shows that (T −3,2 , m) has a hyperbolic seiferter if m ∈ {−4, −5}. For m = −5, it is shown in [6] that the lens surgery (T −3,2 , −5) has a hyperbolic seiferter.…”

“…Comparing the indices of the exceptional fibers in the 2-fold branched cover along T −3,2 ∪ c m with that along M ( 1 2 , − 1 3 , − 1 6 ), we see that 2m + 4 = ±6 and so m = 1, −5. The 2-fold branched cover along 6 ) and that along M ( 1 2 , − 1 3 , − 1 6 ) are not homeomorphic because the Euler numbers of these Seifert fibrations are 1 2 [19] shows that the exterior of 3.1 below. This table implies assertion (1).…”

Band surgeries and crossing changes between fibered links

“…Since the numbers of components of L and L ′ differ by 1, their Euler characteristics will never be equal. By [28,15,6], there exists a taut Seifert surface F for L or L ′ , say L, such that F contains b, and so χ(L ′ ) > χ(L). Suppose L is a fibered link.…”

“…(See Section 2 for the definition of i total (α).) By [28,15,6], it is known that if a coherent band surgery increases the Euler characteristic of a link, then the band can be isotoped onto a taut Seifert surface. Hence, such a band surgery between fibered links corresponds to cutting the fiber surface.…”