Notes for AKT-090917-2/0:09:48: Difference between revisions

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Def: A knot diagram is '''descending''' if there exists a point on the diagram s.t. if we draw the knot starting at that point, whenever the knot has a crossing, the new segment goes under the existing segment.
''Definition'': A knot diagram is '''descending''' if there exists a point on the diagram such that, starting from that point and going along the knot, each time we reach any crossing for the first time we do so along the upper strand of the crossing.


By flipping crossings one can make any knot descending and any descending knots are unknots. Deduce that the Jones skein relation and value of <math>J</math> for union of circles give <math>J</math> explicitly for any knots.
By flipping crossings one can make any knot descending and any descending knot is the unknot. From this we can deduce that the Jones skein relation and the value of <math>J</math> for the union of any number unknots give <math>J</math> explicitly for any knot.

Latest revision as of 09:05, 8 September 2011

Definition: A knot diagram is descending if there exists a point on the diagram such that, starting from that point and going along the knot, each time we reach any crossing for the first time we do so along the upper strand of the crossing.

By flipping crossings one can make any knot descending and any descending knot is the unknot. From this we can deduce that the Jones skein relation and the value of for the union of any number unknots give explicitly for any knot.

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