Template:Torus Knot Page: Difference between revisions

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Visit [http://www.math.toronto.edu/~drorbn/KAtlas/TorusKnots/{{{m}}}.{{{n}}}.html {{PAGENAME}}'s page] at the original [http://www.math.toronto.edu/~drorbn/KAtlas/index.html Knot Atlas]!
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Revision as of 17:48, 31 August 2005

[[Image:Data:Torus Knot Page/Previous Knot.jpg|80px|link=Data:Torus Knot Page/Previous Knot]]

[[Data:Torus Knot Page/Previous Knot]]

[[Image:Data:Torus Knot Page/Next Knot.jpg|80px|link=Data:Torus Knot Page/Next Knot]]

[[Data:Torus Knot Page/Next Knot]]

File:Torus Knot Page.jpg Visit [{{{KnotilusURL}}} Torus Knot Page's page] at Knotilus!

Visit Torus Knot Page's page at the original Knot Atlas!

Edit Torus Knot Page Quick Notes

<ifpageexists>Torus Knot Page Quick Notes</ifpageexists>

Edit Torus Knot Page Further Notes and Views <ifpageexists>Torus Knot Page Further Notes and Views</ifpageexists>


Knot presentations

Planar diagram presentation Data:Torus Knot Page/PD Presentation
Gauss code Data:Torus Knot Page/Gauss Code
Dowker-Thistlethwaite code Data:Torus Knot Page/DT Code
Braid presentation {{{braid_table}}}

Polynomial invariants

Alexander polynomial Data:Torus Knot Page/Alexander Polynomial
Conway polynomial Data:Torus Knot Page/Conway Polynomial
2nd Alexander ideal (db, data sources) Data:Torus Knot Page/2nd AlexanderIdeal
Determinant and Signature { Data:Torus Knot Page/Determinant, Data:Torus Knot Page/Signature }
Jones polynomial Data:Torus Knot Page/Jones Polynomial
HOMFLY-PT polynomial (db, data sources) Data:Torus Knot Page/HOMFLYPT Polynomial
Kauffman polynomial (db, data sources) Data:Torus Knot Page/Kauffman Polynomial
The A2 invariant Data:Torus Knot Page/QuantumInvariant/A2/1,0
The G2 invariant Data:Torus Knot Page/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {{{{same_alexander}}}}

Same Jones Polynomial (up to mirroring, ): {{{{same_jones}}}}

Vassiliev invariants

V2 and V3: (Data:Torus Knot Page/V 2, Data:Torus Knot Page/V 3)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
Data:Torus Knot Page/V 2,1 Data:Torus Knot Page/V 3,1 Data:Torus Knot Page/V 4,1 Data:Torus Knot Page/V 4,2 Data:Torus Knot Page/V 4,3 Data:Torus Knot Page/V 5,1 Data:Torus Knot Page/V 5,2 Data:Torus Knot Page/V 5,3 Data:Torus Knot Page/V 5,4 Data:Torus Knot Page/V 6,1 Data:Torus Knot Page/V 6,2 Data:Torus Knot Page/V 6,3 Data:Torus Knot Page/V 6,4 Data:Torus Knot Page/V 6,5 Data:Torus Knot Page/V 6,6 Data:Torus Knot Page/V 6,7 Data:Torus Knot Page/V 6,8 Data:Torus Knot Page/V 6,9

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where Data:Torus Knot Page/Signature is the signature of Torus Knot Page. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:Torus Knot Page/KhovanovTable
Integral Khovanov Homology

(db, data source)

   Data:Torus Knot Page/Integral Khovanov Homology

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session. {{{computer_talk}}}


Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Torus Knot Page master template (intermediate).

See/edit the Torus Knot_Splice_Base (expert).

Back to the top.

[[Image:Data:Torus Knot Page/Previous Knot.jpg|80px|link=Data:Torus Knot Page/Previous Knot]]

[[Data:Torus Knot Page/Previous Knot]]

[[Image:Data:Torus Knot Page/Next Knot.jpg|80px|link=Data:Torus Knot Page/Next Knot]]

[[Data:Torus Knot Page/Next Knot]]