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Within a day, the printer spat out a knotted wing made of rigid plastic.
However, the higher density of the knot, within knotted specimens, skews the SG of the specimens with knots.
Topologically, a knotted string is not a real knot, as long as its ends are free.
Unlike a knotted piece of rope, a mathematical knot has no free ends.
Saitta and his colleagues modeled the physical behavior of a knotted hydrocarbon chain, representing a portion of a polyethylene molecule.
He proposed that different elements would correspond to vortices bent into different types of knotted tubes forming closed loops.
The quantum states of space depend on how these loops are knotted and linked.
Shaped like a bead on a string with knotted ends, compounds called rotaxanes can act as molecular shuttles, sliding back and forth and ferrying electrons from one end of the string to the other.
Nonetheless, the mathematics of knots has proved useful in recent years for describing knotted polymer strands (SN: 11/16/96, p.
These tubular objects, in turn, serve as useful mathematical models of knotted DNA loops and polymer chains.
However, "to synthesize a specific knot with a small molecular weight is a real achievement," says Millett, who has created models that relate the length of a molecule to the different ways it can be knotted. "Being able to move from a simple knot to a composite is--from a mathematical point of view--a real step."
Cross have developed a scheme for viewing a knotted sphere -- the four-dimensional analog of a knotted string.
Birman and colleague Xiao-Song Lin of Columbia have added one more strand to this knotted web by bringing new mathematical techniques to bear on the study of knots.
After four years of laboratory effort, chemist Qun Yi Zheng of the University of Colorado in Boulder has come up with a "hook and ladder" approach for making knotted compounds.
An imaginary ant crawling along a narrow tunnel within the one-dimensional confines of such a curve, even after completing its circuit, would never be able to tell whether the curve is knotted. Instetad, knottedness resides in the way the curve sits in three-dimensional space.