Template:Torus Knot Page

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Contents

File:Torus Knot Page.jpg See other torus knots

Visit Torus Knot Page at Knotilus!

Edit Torus Knot Page Quick Notes


Edit Torus Knot Page Further Notes and Views


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, q\leftrightarrow q^{-1}): {{{{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 t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=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.

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).

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