arXiv subjects
Publications and source records attributed to Mohammed Sabak.
We assign a new polynomial to any checkerboard-colorable 4-valent virtual graph in terms of its Euler circuit expansion. This provides a new combinatorial formulation of the Kauffman-Jones polynomial for checkerboard-colorable virtual links.
We give a non-left-orderability criterion for involutory quandles of non-split links. We use this criterion to show that the involutory quandle of any non-trivial alternating link is not left-orderable, thus improving Theorem 8.1. proven by Raundal et al. (Proceedings of the Edinburgh Mathematical Society (2021 64), page 646). We also use the criterion to show that the involutory quandles of augmented alternating links are not left-orderable. We introduce a new family of links containing all non-alternating and quasi-alternating 3-braid closures and show that their involutory quandles are not left-orderable. This leads us to conjecture that the involutory quandle of any quasi-alternating link is not left-orderable.
For each connected alternating tangle, we provide an infinite family of non-left-orderable L-spaces. This gives further support for Conjecture [3] of Boyer, Gordon, and Watson that is a rational homology 3-sphere is an L-space if and only if it is non-left-orderable. These 3-manifolds are obtained as Dehn fillings of the double branched covering of any alternating encircled tangle. We give a presentation of these non-left-orderable L-spaces as double branched coverings of S^3, branched over some specified links that turn out to be hyperbolic. We show that the obtained families include many non-Seifert fibered spaces. We also show that these families include many Seifert fibered spaces and give a surgery description for some of them. In the process we give another way to prove that the torus knots T(2, 2m+1) are L-space-knots as has already been shown by Ozsváth and Szabó in [24].
We construct an infinite family of links which are both almost alternating and quasi-alternating from a given either almost alternating diagram representing a quasi-alternating link, or connected and reduced alternating tangle diagram. To do that we use what we call a dealternator extension which consists in replacing the dealternator by a rational tangle extending it. We note that all not alternating and quasi-alternating Montesinos links can be obtained in that way. We check that all the obtained quasi-alternating links satisfy Conjecture 3.1 of Qazaqzeh et al. (JKTR 22 (06), 2013), that is the crossing number of a quasi-alternating link is less than or equal to its determinant. We also prove that the converse of the Theorem 3.3 of Qazaqzeh et al. (JKTR 24 (01), 2015) is false.