DT (Dowker-Thistlethwaite) Codes: Difference between revisions
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<a href=' http://chrykne.info/ '> chrykne </a> <br /> |
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<a href=' http://bambulka.info/ ' > bambulka </a> <br /> |
<a href=' http://bambulka.info/ ' > bambulka </a> <br /> |
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====Links==== |
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[[Image:DTCode4L7n2.gif|frame|center|A DT notation example, for the link [[L7n2]]]] |
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DT Codes for links are defined in a similar way (see {{ref|DollHoste}}). Follow the same numbering process as for knots, except when you finish traversing one component, jump straight to the next. It is not difficult to see that there is always a choice of starting points along the components for which the resulting pairing is a pairing between odd and even numbers. (On the figure above one possible choice is indicated). Again, it is enough to only list the even numbers corresponding to <math>1, 3, 5, \ldots</math>; call the resulting list <math>\lambda</math>. (Above, <math>\lambda=(6,-8,-10,12,-14,2,-4)</math>). Notice that the odd indices are naturally subdivided into sublists according to the component of the link on which they lie, and this induces a subdivision of <math>\lambda</math> into sublists. Thus with the choices made in the figure above, the DT code for the link [[L7n2]] is <math>(6,-8\mid -10,12,-14,2,-4)</math>. |
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<tt>KnotTheory`</tt> knows about DT codes for links: |
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<!--$$DTCode[Link[7, NonAlternating, 2]]$$--> |
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<!--Robot Land, no human edits to "END"--> |
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{{InOut| |
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n = 7 | |
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in = <nowiki>DTCode[Link[7, NonAlternating, 2]]</nowiki> | |
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out= <nowiki>DTCode[{6, -8}, {-10, 12, -14, 2, -4}]</nowiki>}} |
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<!--END--> |
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<!--$$MultivariableAlexander[DTCode[{6, -8}, {-10, 12, -14, 2, -4}]][t]$$--> |
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<!--Robot Land, no human edits to "END"--> |
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{{InOut| |
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n = 8 | |
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in = <nowiki>MultivariableAlexander[DTCode[{6, -8}, {-10, 12, -14, 2, -4}]][t]</nowiki> | |
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out= <nowiki>-1 + t[1] + t[2] - t[1] t[2]</nowiki>}} |
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<!--END--> |
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{{note|DollHoste}} H. Doll and J. Hoste, ''A tabulation of oriented links'', Mathematics of Computation '''57-196''' (1991) 747-761. |
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