Difference between revisions of "Lecture 01"
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======Slide 014====== | ======Slide 014====== | ||
[[Image:L01_s014.jpg|frame|none|Lecture 01, Slide 014<br> | [[Image:L01_s014.jpg|frame|none|Lecture 01, Slide 014<br> | ||
− | How many alternatives do we have to consider, if we allow maximally nine moves and require at least three for the finishing moves? We noted previously that there are two possibilities for the three finishing moves (LRC or RLC). Since moves cannot repeat into the same sector, each of the possibilities can only have been preceded by two alternatives i.e.. (CLRC or RLRC) and (CRLC or LRLC). These four possibilities again can have been preceded by two alternatives each ... etc. Since we treat knots of different move-numbers as distinct, the total number of moves up to length | + | How many alternatives do we have to consider, if we allow maximally nine moves and require at least three for the finishing moves? We noted previously that there are two possibilities for the three finishing moves (LRC or RLC). Since moves cannot repeat into the same sector, each of the possibilities can only have been preceded by two alternatives i.e.. (CLRC or RLRC) and (CRLC or LRLC). These four possibilities again can have been preceded by two alternatives each ... etc. Since we treat knots of different move-numbers as distinct, the total number of moves up to length L is the sum of all powers of 2 up to (L-2). (-2 because of the finishing moves!). You should be able to figure out reasoning like this on your own! |
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Revision as of 22:55, 11 September 2007
Update Warning! This page has not been revised yet for the 2007 Fall term. Some of the slides may be reused, but please consider the page as a whole out of date as long as this warning appears here.
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