Journal of Knot Theory and Its Ramifications

In this paper, we investigate the interactions between the Legendrian satellite construction and the existence of exact, orientable Lagrangian cobordisms between Legendrian knots. Given Lagrangian cobordisms between two Legendrian knots and between two Legendrian tangles, we construct a Lagrangian cobordism between Legendrian satellites of the knots by the closures of the tangles, with extra twists on both the top and the bottom satellite to compensate for the genus of the cobordism. If the original cobordisms were decomposable, then a decomposable cobordism between satellites exists as well, again with extra twists.

We show that Khovanov link homology is trivial in a range of gradings and utilize relations between Khovanov and chromatic graph homology to determine extreme Khovanov groups and corresponding coefficients of the Jones polynomial. The extent to which chromatic homology and the chromatic polynomial can be used to compute integral Khovanov homology of a link depends on the maximal girth of its all-positive graphs. In this paper, we define the girth of a link, discuss relations to other knot invariants, and describe possible values for girth. Analyzing girth leads to a description of possible all-A state graphs of any given link; e.g., if a link has a diagram such that the girth of the corresponding all-A graph is equal to $\ell >2$, then the girth of the link is equal to $\ell .$

A spiral knot or link diagram (introduced in [C. Adams, R. Hudson, Rachel, R. Morrison, W. George, L. Starkston, S. Taylor and O. Turanova, The spiral index of knots, Math. Proc. Cambridge Philos. Soc. 149(2) (2010) 297–315]) is an oriented knot or link diagram where, when traversing through the planar diagram, the curvature does not change sign. An oriented knot or link type is called curly if it admits a spiral diagram with fewer maxima than the braid index of that knot or link type. We prove that all 2-bridge knots and links with a braid index greater than three are curly. We also show that many alternating oriented Montesinos links are curly.

Niebrzydowski and Przytycki defined a Kauffman bracket magma and constructed the invariant $P$ of framed links in $3$-space. The invariant is closely related to the Kauffman bracket polynomial. The normalized bracket polynomial is obtained from the Kauffman bracket polynomial by the multiplication of indeterminate and it is an ambient isotopy invariant for links. In this paper, we reformulate the multiplication by using a map from the set of framed links to a Kauffman bracket magma in order that $P$ is invariant for links in $3$-space. We define a generalization of a Kauffman bracket magma, which is called a marked Kauffman bracket magma. We find the conditions to be invariant under Yoshikawa moves except the first one and use a map from the set of admissible marked graph diagrams to a marked Kauffman bracket magma to obtain the invariant for surface-links in $4$-space.

We construct an Alexander-type invariant for oriented doodles from a deformation of the Tits representation of the twin group and from the Chebyshev polynomials of the second kind. Like the Alexander polynomial, our invariant vanishes on unlinked doodles with more than one component. We also include values of our invariant on several doodles.

The warping degree of an oriented knot diagram is the minimal number of crossings which we meet as an under-crossing first when we travel along the diagram from a fixed point. The warping degree of a knot projection is the minimal value of the warping degree for all oriented alternating diagrams obtained from the knot projection. In this paper, we consider the maximal number of regions which share no crossings for a knot projection with a fixed crossing, and give lower bounds for the warping degree.

In this paper, we observe that the hat version of the Heegaard Floer invariant of Legendrian knots in contact three-manifolds defined by Lisca-Ozsváth-Stipsicz-Szabó can be combinatorially computed. We rely on Plamenevskaya’s combinatorial description of the Heegaard Floer contact invariant.

In this paper, we deal with the notion of singquandles introduced in [I. R. U. Churchill, M. Elhamdadi, M. Hajij and S. Nelson, Singular knots and involutive quandles, J. Knot Theory Ramifications 26(14) (2017) 1750099]. This is an algebraic structure that naturally axiomatizes Reidemeister moves for singular links, similarly to what happens for ordinary links and quandles. We present a new axiomatization that shows different algebraic aspects and simplifies applications. We also reformulate and simplify the axioms for affine singquandles (in particular in the idempotent case).

In this paper, we examine the properties of the Jones polynomial using dimensionality reduction learning techniques combined with ideas from topological data analysis. Our data set consists of more than 10 million knots up to 17 crossings and two other special families up to 2001 crossings. We introduce and describe a method for using filtrations to analyze infinite data sets where representative sampling is impossible or impractical, an essential requirement for working with knots and the data from knot invariants. In particular, this method provides a new approach for analyzing knot invariants using Principal Component Analysis. Using this approach on the Jones polynomial data, we find that it can be viewed as an approximately three-dimensional subspace, that this description is surprisingly stable with respect to the filtration by the crossing number, and that the results suggest further structures to be examined and understood.

A petal projection of a knot $K$ is a projection of a knot which consists of single multi-crossing and non-nested loops. Since a petal projection gives a sequence of natural numbers for a given knot, the petal projection is a useful model to study knot theory. It is known that every knot has a petal projection. A petal number $p(K)$ is the minimum number of loops required to represent the knot $K$ as a petal projection. In this paper, we find the relation between a superbridge index and a petal number of an arbitrary knot. By using this relation, we find the petal number of $T_{r,s}$ as follows: