10 124 Quick Notes: Difference between revisions
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[[10_124]] is also known as the torus knot [[T(5,3)]] or the pretzel knot P(5,3,-2). It is one of two knots which are both torus knots and pretzel knots, the other being [[8_19]] = [[T(4,3)]] = P(3,3,-2). |
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Also known as "The Torus Knot [[T(5,3)]]". |
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It seems like the prior statement is incorrect. I suspect what this should say is [[10_124]] and [[8_19]] are the only torus knots which are also almost alternating. See page 108 in the Encyclopedia of Knot Theory. Confirmation of this is that [[3_1]] is the pretzel knot P(1,1,1), i.e., the right-handed trefoil. It looks like [[5_1]] is a pretzel knot also, and so on, i.e. [[7_1]], [[9_1]], and should include the Hopf link and the Solomon link etc. These are torus knots/links also. |
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If one takes the symmetric diagram for 10_123 and makes it doubly alternating one gets a diagram for 10_124. That's the torus knot view. There is then a nice representation of the quandle of 10_124 into the dodecahedral quandle Q_30. See http://www.maths.warwick.ac.uk/~bjs/add233.html |
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Latest revision as of 14:36, 2 December 2024
10_124 is also known as the torus knot T(5,3) or the pretzel knot P(5,3,-2). It is one of two knots which are both torus knots and pretzel knots, the other being 8_19 = T(4,3) = P(3,3,-2).
It seems like the prior statement is incorrect. I suspect what this should say is 10_124 and 8_19 are the only torus knots which are also almost alternating. See page 108 in the Encyclopedia of Knot Theory. Confirmation of this is that 3_1 is the pretzel knot P(1,1,1), i.e., the right-handed trefoil. It looks like 5_1 is a pretzel knot also, and so on, i.e. 7_1, 9_1, and should include the Hopf link and the Solomon link etc. These are torus knots/links also.
