Journal of Knot Theory and Its Ramifications

This paper is a survey paper that gives detailed constructions and illustrations of some of the standard examples of non-orientable surfaces that are embedded and immersed in 4-dimensional space. The illustrations depend upon their 3-dimensional projections, and indeed the illustrations here depend upon a further projection into the plane of the page. The concepts used to develop the illustrations will be developed herein.

The Chekanov–Eliashberg differential graded algebra of a Legendrian knot L is a rich source of Legendrian knot invariants, as is the theory of generating families. The set P(L) of homology groups of augmentations of the Chekanov–Eliashberg algebra is an invariant, as is a count of objects from the theory of generating families called graded normal rulings. This paper gives two results demonstrating the usefulness of computing the homology group of an augmentation using a combinatorial interpretation of a generating family called a Morse complex sequence (MCS). First, we show that if the projection of L to the xz-plane has exactly 4 cusps, then |P(L)| ≤ 1. Second, we show that two augmentations associated to the same graded normal ruling by the many-to-one map between augmentations and graded normal rulings defined by Ng and Sabloff [The correspondence between augmentations and rulings for Legendrian knots, Pacific J. Math.224(1) (2006) 141–150] need not have isomorphic homology groups.

It is known that a knot/link can be nullified, i.e. can be made into the trivial knot/link, by smoothing some crossings in a projection diagram of the knot/link. The minimum number of such crossings to be smoothed in order to nullify the knot/link is called the nullification number. In this paper we investigate the nullification numbers of a particular knot family, namely the family of torus knots and links.

We study and compare two factorization systems for surjective homomorphisms in the category of quandles. The first one is induced by the adjunction between quandles and trivial quandles, and a precise description of the two classes of morphisms of this factorization system is given. In doing this we observe that a special class of congruences in the category of quandles always permute in the sense of the composition of relations, a fact that opens the way to some new universal algebraic investigations in the category of quandles. The second factorization system is the one discovered by E. Bunch, P. Lofgren, A. Rapp and D. N. Yetter. We conclude with an example showing a difference between these factorization systems.

In this paper, we show that any 11-colorable knot has a diagram with a non-trivial 11-coloring such that at most six colors among eleven colors are assigned to arcs of the diagram.

For a boundary-reducible 3-manifold M with ∂M a genus-g surface, we show that if M admits a genus-(g + 1) Heegaard surface S, then the disk complex of S is simply connected. Also we consider the connectedness of the complex of reducing spheres. We investigate the intersection of two reducing spheres for a genus-3 Heegaard splitting of (torus) × I.