Template:Rolfsen Knot Page

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This is the Rolfsen Knot Page master template. For testing purposes, some of the data in it is filled in with data for 8_17.

8 16.gif


8 18.gif



Rolfsen Knot Page.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit Rolfsen Knot Page's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit Rolfsen Knot Page at Knotilus!

Knot presentations

Planar diagram presentation Data:Rolfsen Knot Page/PD Presentation
Gauss code Data:Rolfsen Knot Page/Gauss Code
Dowker-Thistlethwaite code Data:Rolfsen Knot Page/DT Code
Conway Notation Data:Rolfsen Knot Page/Conway Notation

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 8, width is 3,

Braid index is 3

Rolfsen Knot Page ML.gif Rolfsen Knot Page AP.gif
[{7, 10}, {9, 2}, {10, 4}, {3, 5}, {4, 8}, {6, 9}, {5, 1}, {2, 7}, {1, 6}, {8, 3}]

[edit Notes on presentations of Rolfsen Knot Page]

Three dimensional invariants

Symmetry type Data:Rolfsen Knot Page/Symmetry Type
Unknotting number Data:Rolfsen Knot Page/Unknotting Number
3-genus Data:Rolfsen Knot Page/3-Genus
Bridge index Missing
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:Rolfsen Knot Page/ThurstonBennequinNumber
Hyperbolic Volume Data:Rolfsen Knot Page/HyperbolicVolume
A-Polynomial See Data:Rolfsen Knot Page/A-polynomial

[edit Notes for Rolfsen Knot Page's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus Missing
Rasmussen s-Invariant Missing

[edit Notes for Rolfsen Knot Page's four dimensional invariants]

Polynomial invariants

Alexander polynomial Data:Rolfsen Knot Page/Alexander Polynomial
Conway polynomial Data:Rolfsen Knot Page/Conway Polynomial
2nd Alexander ideal (db, data sources) Data:Rolfsen Knot Page/2nd AlexanderIdeal
Determinant and Signature { Data:Rolfsen Knot Page/Determinant, Data:Rolfsen Knot Page/Signature }
Jones polynomial Data:Rolfsen Knot Page/Jones Polynomial
HOMFLY-PT polynomial (db, data sources) Data:Rolfsen Knot Page/HOMFLYPT Polynomial
Kauffman polynomial (db, data sources) Data:Rolfsen Knot Page/Kauffman Polynomial
The A2 invariant Data:Rolfsen Knot Page/QuantumInvariant/A2/1,0
The G2 invariant Data:Rolfsen 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:Rolfsen Knot Page/V 2, Data:Rolfsen 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:Rolfsen Knot Page/V 2,1 Data:Rolfsen Knot Page/V 3,1 Data:Rolfsen Knot Page/V 4,1 Data:Rolfsen Knot Page/V 4,2 Data:Rolfsen Knot Page/V 4,3 Data:Rolfsen Knot Page/V 5,1 Data:Rolfsen Knot Page/V 5,2 Data:Rolfsen Knot Page/V 5,3 Data:Rolfsen Knot Page/V 5,4 Data:Rolfsen Knot Page/V 6,1 Data:Rolfsen Knot Page/V 6,2 Data:Rolfsen Knot Page/V 6,3 Data:Rolfsen Knot Page/V 6,4 Data:Rolfsen Knot Page/V 6,5 Data:Rolfsen Knot Page/V 6,6 Data:Rolfsen Knot Page/V 6,7 Data:Rolfsen Knot Page/V 6,8 Data:Rolfsen 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:Rolfsen Knot Page/Signature is the signature of Rolfsen Knot Page. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:Rolfsen Knot Page/KhovanovTable
Integral Khovanov Homology

(db, data source)

   Data:Rolfsen Knot Page/Integral Khovanov Homology

The Coloured Jones Polynomials