Knots in the Solid Torus Up to 6 Crossings

Abstract

We classify non-affine, prime knots in the solid torus up to 6 crossings. We establish which of these are amphicheiral: almost all knots with symmetric Jones polynomial are amphicheiral, but in a few cases we use stronger invariants, such as HOMFLYPT and Kauffman skein modules, to show that some such knots are not amphicheiral. Examples of knots with the same Jones polynomial that are different in the HOMFLYPT skein module are presented. It follows from our computations, that the wrapping conjecture is true for all knots up to 6 crossings.

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.…”

Double branched covers of knotoids

“…The knot type of this branched cover is a knotoid invariant; in particular by composing the branched covering construction with any invariant of knots in the solid torus (see e.g. [30], [14] and [19]) we obtain a new knotoid invariant. Note that by definition, the lifts of line-isotopic embedded arcs are ambient isotopic knots, since isotopies of k preserving the branching set lift to equivariant isotopies.…”

Tabulation of Prime Knots in Lens Spaces

“…We will now recall from [5] how to adopt the notation of a Gauss code for T . To extend the Gauss code to a code of a knot in T , we take a punctured disk diagram D and keep track of the 0-and ∞-regions.…”

“…We recall from [5] that unlike the classical case, there is no well defined connected sum operation for knots in T . But for two oriented knots, where at least one of them is affine, the connected sum operation is well defined.…”