Representations of (1,1)-knots

Abstract

Abstract. We present two different representations of (1, 1)-knots and study some connections between them. The first representation is algebraic: every (1, 1)-knot is represented by an element of the pure mapping class group of the twice punctured torus PMCG 2 (T ). Moreover, there is a surjective map from the kernel of the natural homomorphism Ω : PMCG 2 (T ) → MCG(T ) ∼ = SL(2, Z), which is a free group of rank two, to the class of all (1, 1)-knots in a fixed lens space. The second representation is parametric: every (1, 1)-knot can be represented by a 4-tuple (a, b, c, r) of integer parameters such that a, b, c ≥ 0 and r ∈ Z 2a+b+c . The strict connection of this representation with the class of Dunwoody manifolds is illustrated. The above representations are explicitly obtained in some interesting cases, including two-bridge knots and torus knots.

Seifert manifolds and (1, 1)-knots

“…As proved in [18], K(a, b, c, r) is equivalent to K(a, c, b, 2a + b + c − r) and K(a, 0, c, r) is equivalent to K(a, c, 0, r). As a consequence, K(a, 0, c, r) is equivalent to K(a, 0, c, 2a + c − r).…”

“…An integer 4-parametric representation of (1, 1)-knots have been developed in [16] (also see [18]). Each (1, 1)-knot, with the only exception of the "core" knot {P } × S 1 ⊂ S 2 × S 1 , can be represented by four nonnegative integers a, b, c, r, and the represented knot will be referred as K(a, b, c, r).…”

10D massive type IIA supergravities as the uplift of parabolic M2‐brane torus bundles

“…This fiber bundle should be understood as the decompactification of the M2‐brane on the symplectic torus bundle with parabolic monodromy. The mapping class group of the twice punctured torus bundle correpsonds to a (1, 1) knot . There is a residual monodromy associated to the twice punctured torus bundle which is inherited from the parabolic monodromy, indeed the two punctures that lead to a noncompact distinguished 10‐th dimension, are kept invariant under the parabolic monodromy.…”

“…There is a residual monodromy associated to the twice punctured torus bundle which is inherited from the parabolic monodromy, indeed the two punctures that lead to a noncompact distinguished 10‐th dimension, are kept invariant under the parabolic monodromy. More formally and following let be the homeomorphisms group preserving the orientation, a fiber of genus g and P the punctures, such that . The punctures are mapped into punctures though not necessarily become fixed.…”

Semisimple tunnels

“…A braid description for torus knots was obtained by A. Cattabriga and M. Mulazzani [5,Section 4]. Here we will use a similar description due to A. Seo [18].…”