Identifying non-pseudo-alternating knots by using the free factor property of minimal genus Seifert surfaces

Pseudo-alternating knots and links are defined constructively via their Seifert surfaces. By performing Murasugi sums of primitive flat surfaces, such a knot or link is obtained as the boundary of the resulting surface. Conversely, it is hard to determine whether a given knot or link is pseudo-alternating or not. A major difficulty is the lack of criteria to recognize whether a given Seifert surface is decomposable as a Murasugi sum. In this paper, we propose a new idea to identify non-pseudo-alternating knots. Combining with the uniqueness of minimal genus Seifert surface obtained through sutured manifold theory, we demonstrate that two infinite classes of pretzel knots are not pseudo-alternating.

Classical pretzel knots and left orderability

We consider the classical pretzel knots [Formula: see text], where [Formula: see text] are positive odd integers. By using continuous paths of elliptic [Formula: see text]-representations, we show that (i) the 3-manifold obtained by [Formula: see text]-surgery on [Formula: see text] has left orderable fundamental group if [Formula: see text] and (ii) the [Formula: see text]-cyclic branched cover of [Formula: see text] has left orderable fundamental group if [Formula: see text].

Hyperbolic torsion polynomials of pretzel knots

In this paper, we explicitly calculate the highest degree term of the hyperbolic torsion polynomial of an infinite family of pretzel knots. This gives supporting evidence for a conjecture of Dunfield, Friedl and Jackson that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot. The verification of the genus part of the conjecture for this family of knots also follows from the work of Agol and Dunfield [1] or Porti [19].

Character varieties of even classical pretzel knots

For each even classical pretzel knot P(2k1 + 1, 2k2 + 1, 2k3), we determine the character variety of irreducible SL (2, ℂ)-representations, and clarify the steps of computing its A-polynomial.

Nimble evolution for pretzel Khovanov polynomials

We conjecture explicit evolution formulas for Khovanov polynomials, which for any particular knot are Laurent polynomials of complex variables q and T, for pretzel knots of genus g in some regions in the space of winding parameters $$n_0, \dots , n_g$$n0,⋯,ng. Our description is exhaustive for genera 1 and 2. As previously observed Anokhina and Morozov (2018), Dunin-Barkowski et al. (2019), evolution at $$T\ne -1$$T≠-1 is not fully smooth: it switches abruptly at the boundaries between different regions. We reveal that this happens also at the boundary between thin and thick knots, moreover, the thick-knot domain is further stratified. For thin knots the two eigenvalues 1 and $$\lambda = q^2 T$$λ=q2T, governing the evolution, are the standard T-deformation of the eigenvalues of the R-matrix 1 and $$-q^2$$-q2. However, in thick knots’ regions extra eigenvalues emerge, and they are powers of the “naive” $$\lambda $$λ, namely, they are equal to $$\lambda ^2, \dots , \lambda ^g$$λ2,⋯,λg. From point of view of frequencies, i.e. logarithms of eigenvalues, this is frequency doubling (more precisely, frequency multiplication) – a phenomenon typical for non-linear dynamics. Hence, our observation can signal a hidden non-linearity of superpolynomial evolution. To give this newly observed evolution a short name, note that when $$\lambda $$λ is pure phase the contributions of $$\lambda ^2, \dots , \lambda ^g$$λ2,⋯,λg oscillate “faster” than the one of $$\lambda $$λ. Hence, we call this type of evolution “nimble”.

2-Adjacency between some special knots and pretzel knot

We study 2-adjacency between a classical (3-strand) pretzel knot and the trefoil knot or the figure-eight knot by using the early results about classical pretzel knots and their polynomials and elementary number theory. We show that except for the trefoil knot or the figure-eight knot, a nontrivial classical pretzel knot is not 2-adjacent to either of them, and vice versa.