Read or Download [Journal] The Mathematical Intelligencer. Vol. 20. No 4 PDF
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Extra resources for [Journal] The Mathematical Intelligencer. Vol. 20. No 4
Sample text
5Kirkman held passionate views on the spelling of certain words. VOLUME 20, NUMBER4. 1998 35 Reidemeister move I Reidemeister move H The flype The 2-pass The Perko move: fix lower left disk, flip the other two disks Reidemeister move III The (3, 2) - pass A double 2-pass: re-route the arc with two overpasses, and also the arc with two underpasses Figure 3. Moves on diagrams. do not appear with fewer than eight crossings, and from Talt's first paper, it is evident that initially he did not believe that nonalternating knots were possible.
In fact, the first proof of the existence of a nonalternating knot did not appear until 1930. Little states that he worked for 6 years, from 1893 to 1899, to produce his list of 43, 10-crossing nonalternating knots [Lit4]. As we shall soon see, his list had no omissions, but it did have one duplication. One obstacle to tabulating nonalternating knots is their sheer quantity. Although nonalternating knots do not predominate until 13 crossings (as mentioned earlier, they do not even appear until 8 crossings), it is plausible that the proportion of knots which are alternating tends exponentially to zero with increasing crossing number.
In particular, the BK n class o f a p p r o x i m a t i o n s converges to e m u c h m o r e rapidly than even the direct Maclaurin series method. Therefore, o u r w o r k m a y have practical application. The impressive n u m e r i c a l a c c u r a c y of these n e w app r o x i m a t i o n s should n o t cloud our eyes to an even m o r e e x t r a o r d i n a r y aspect: the elegance and simplicity of the exp r e s s i o n s for e, particularly MIM. C o m p a r e d to m a n y o t h e r m e t h o d s for computing classical constants, MIM is breathtaking.