Diffeomorphic vs Isotopic Links in Lens Spaces

Abstract

Links in lens spaces may be defined to be equivalent by ambient isotopy or by diffeomorphism of pairs. In the first case, for all the combinatorial representations of links, there is a set of Reidemeistertype moves on diagrams connecting isotopy equivalent links. In this paper we provide a set of moves on disk, band and grid diagrams that connects diffeo equivalent links: there are up to four isotopy equivalent links in each diffeo equivalence class. Moreover, we investigate how the diffeo equivalence relates to the lift of the link in the 3-sphere: in the particular case of oriented primitive-homologous knots, the lift completely determines the knot class in L(p, q) up to diffeo equivalence, and thus only four possible knots up to isotopy equivalence can have the same lift. Mathematics Subject Classification 2010: Primary 57M27, 57M10; Secondary 57M25.the same family, but in a general lens space: using results of [4,27] and the diffeo moves, we completely characterize, up to isotopy, which knots among the family have the same lift, in terms of the parameters of the lens space and of the amphicheirality of the lift (see Theorem 8).Further development In [13] a tabulations of knots, represented via band diagrams and up to isotopy equivalence is given. Diffeo-moves can be used to detect diffeo equivalent knots in the tabulation. For example, the two knots 5 26 and 5 27 , which are non-isotopic in L(p, q) with p ≤ 12 and conjecturally non-isotopic in the other lens spaces, are related by a τ move.Another interesting development could be to study the diffeo moves in more general manifolds such Seifert one, for which isotopy equivalence moves are already known (see [15]).

Decompositions of the 3-sphere and lens spaces with three handlebodies

“…In this case, M also has a decomposition that satisfies F 12 ∼ = F 13 ∼ = F 23 ∼ = A ∪ A and has four branched loci. In fact, if M is not L(4, 1), M admits type-(1, 1, 1) decompositions that satisfy (1), ( 2), (3), (4), and (5) above. In addition, if M ∼ = L(4, 1), M admits type-(1, 1, 1) decompositions that satisfy (1), ( 2), (3), ( 4), (5), and (6) above.…”

The Alexander polynomial for closed braids in lens spaces

“…The braid representative of this knot is represented in Figure 1 det I − ρ(tσ 3 1 )(t q , t p ) = −t 2p+q + t 3p+q − t 4p+q + t 2p − t p + 1. Since [tσ 3 1 ] = 1, Equation (8) yields…”

Knot Invariants in Lens Spaces

“…Example 2. The knots 526 and 527 in Figure 11 differ by exchanging both the orientation of the fixed and mixed sublinks, which can be interpreted as 527 being the image of 526 under the self-homomorphism of the torus T that reverses both the meridian and the longitude (a so-called flip in the language of [13], see also [3]). The question whether 526 = 527 is equivalent to the question whether the links are non-invertible.…”

“…−az 2 − a −1 z 2 + z 3 + t1 (−a − az 2 − a 2 z 2 + z 3 + az 4 ) + t 2 1 a(−z 2 + az 3 + z 4 ) + t2 (za 7 − a 5 z 2 − a 7 z 2 − a 6 z 3 + az 4 ) − t 3 1 az 2 + t1t2(za 2 + a −1 z 2 − a 2 z 2 + 2z 3 + az 4 ) ≥ 6 −az 2 − z 2 /a + z 3 + t1 (−a − az 2 − a 2 z 2 + z 3 + az 4 ) + t 2 1 a(−z 2 + az 3 + z 4 ) + t2az 4 −t 3 1 az 2 + t1t2 (za 2 + a −1 z 2 − a 2 z 2 + 2z 3 + az 4 ) + t3 (za 3 − az 2 − a 3 z 2 − a 2 z 3 ) p S −1 3,∞ (526) = S −1 3,∞ (527) 2 −az 2 + a −1 z 2 − 2z 3 + a 2 z 3 + az 4 + t1 (z + 2az − za −2 + za 2 + a 3 − za 3 + 2az 2 −2a −1 z 2 − z 3 + a 2 z 3 ) + t 2 1 (az 2 + a 2 z 3 − az 4 ) − t 3 1 az 2 3…”

“…−az 2 + a −1 z 2 − z 3 − t 3 1 az 2 + t1 (a − az 2 + a 2 z 2 − z 3 − az 4 ) + t 2 1 a(z 2 + az 3 − z 4 ) +t1t2 (za 2 + a −1 z 2 − a 2 z 2 + 2z 3 + az 4 ) + t2 (za 7 − a 5 z 2 + a 7 z 2 − a 6 z 3 + az 4 ) ≥ 6 −az 2 + a −1 z 2 − z 3 + t1 (a − az 2 + a 2 z 2 − z 3 − az 4 ) + t 2 1 a(z 2 + az 3 − z 4 ) + t2 az 4 −t 3 1 az 2 + t1t2 (za 2 + a −1 z 2 − a 2 z 2 + 2z 3 + az 4 ) + t3 (za 3 − az 2 + a 3 z 2 − a 2 z 3 ) ∆(526) = ∆(527) = (t 2 − t + 1)(t p+1 − t + 1)(t p+1 − t p + 1).…”