Cords and 1-handles attached to surface-knots

Ribbon-moves for 2-knots with 1-handles Attached and Khovanov-jacobsson Numbers

We prove that a crossing change along a double point circle on a 2-knot is realized by ribbon-moves for a knotted torus obtained from the 2-knot by attaching a 1-handle. It follows that any 2-knots for which the crossing change is an unknotting operation, such as ribbon 2-knots and twistspun knots, have trivial Khovanov-Jacobsson number. A surface-knot or -link is a closed surface embedded in 4...

Symplectic 2-handles and Transverse Knots

Surface Distance on Knots

Let K be the set of all oriented knots up to isotopy. There are several distance functions on K, such as the Gordian distance and the ♯-Gordian distance [9]. In general, given an ‘unknotting operation’, that is, a method to untie every knot, such as the ∆-unknotting operation introduced by S. Matveev [8] (see also [10]), one can define the corresponding distance function as the minimal number o...

Widths of surface knots

We study surface knots in 4–space by using generic planar projections. These projections have fold points and cusps as their singularities and the image of the singular point set divides the plane into several regions. The width (or the total width) of a surface knot is a numerical invariant related to the number of points in the inverse image of a point in each of the regions. We determine the...

microleakage evaluation in regard to cavo-surface designs & treatments

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