A nonalternating knot with sequence 6 -14 16 - 12 2 - 8 10. Draw a projection of

Question

A nonalternating knot with sequence 6 -14 16 - 12 2 - 8 10. Draw a projection of the knot corresponding to the quence 14 12 - 16 2 18 68 10 - 4. How do you recognize from the sequence of number projection has a trivial crossing in it like this? How about a Type II Reidemeister move that will reduce the number of by two? (See Figure 2.13.) Trivial crossing. Type II Reidemeister move. Dowker's notation allows us to feed projections of knots into the puter simply as a sequence of numbers. In particular, suppose we to attempt a classification of 14 crossing of sequ knots. The number of of the 14 numbers 2, 4, 6, 8, 10, 12, 14, 18, 20, 22, 24, 28 is 14! Is about 87 billion. Then we can put a +1 or -1 in front of each of the numbers, giving us another factor of 2^14. Of courses there aren't this different knots with 14 crossings. Lots of the sequences represent the knot. In fact, lots of the sequences represent the same projection of same not.

Explanation / Answer

A regular diagram of knot K (say) has atmost finite crossing points.

Suppose c(K) be the minimum number of crossing points of all regular diagrams.

The one of the left with 4crossing and the one of the right with 5 crossings.

That is c(a)=4, and c(b)=5.

Thus the sequence of numbers that the projection has trivial crossing using this diagrams.

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