Knotty Calculations.

When you learned to tie knots as a child, you probably thought their main use was for making bows on birthday presents or keeping your shoes on your feet. However, if a small band of mathematicians and physicists has its way, knots will form the basis for an entirely new kind of computer, one whose power vastly outstrips that of the machines at our disposal today. In the past decade, however, mathematicians have turned knot theory into a bridge between two seemingly unconnected subjects: computer science and quantum mechanics, the branch of physics that deals with the ultrasmall scale of atoms and subatomic particles. In a paper published in January 2003, researchers propose that this connection between the two fields might finally enable physicists to reach a decades-long goal: to exploit quantum physics to build a computer whose performance would far surpass that of computers based on the classical physics of Isaac Newton. The knots that mathematicians have been studying have a slight quirk: After the theoretical knot is tied, the ends of the string are joined together so the knot can't untie. At the heart of the connection between computer science and quantum physics is a knot invariant called the Jones polynomial, which associates a given knot with an array of numbers. Now, mathematician Michael Freedman of Microsoft Research in Redmond, Wash., and physicist Alexei Kitaev of the California Institute of Technology in Pasadena are pursuing a daring idea: If Witten's physical system somehow does calculations beyond the reach of computers, could this system be harnessed to build a completely new kind of computer?