Cube Diagrams and 3-Dimensional Reidemeister-Like Moves for Knots

Abstract

Abstract. In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under five cube diagram moves. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Floer homology.

Lifting Lagrangian immersions in $\mathbb{C} P^{n-1}$ to Lagrangian cones in $\mathbb{C}^n$

“…To construct a grid, cube, or hypercube diagram, one places markings in a 2, 3, or 4 dimensional Cartesian grid, while ensuring that certain marking conditions and crossing conditions hold (cf. Section 2 and 3 in [3], and Section 2 in [5]). In each case, the markings determine a link (cf.…”

Lifting Lagrangian immersions in ℂPn−1 to Lagrangian cones in ℂn

“…These singular fibers are not well understood. The standard approach is to model them locally as special Lagrangian cones C ⊂ ‫ރ‬ 3 (by cone, we mean a subset C ⊂ ‫ރ‬ 3 such that r • C = C for any real number r > 0). Such a cone can be characterized by its link, C ∩ S 5 , which is a Legendrian surface.…”

On the rotation class of knotted Legendrian tori inR5

“…Compared to Legendrian knots in R 3 , little is known about knotted Legendrian submanifolds L n embedded in R 2n+1 . One reason is that in higher dimensions there are no standard representations of embedded Legendrian submanifolds that enable one to study with the same facility as front projections or Lagrangian projections of Legendrian knots in R 3 . For example, one may easily compute the classical invariants of Thurston-Bennequin and rotation numbers by looking at the front projection of a knot in R 3 .…”