Template:Rolfsen Knot Page

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

8_18

(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		[{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]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["Rolfsen Knot Page"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

Data:Rolfsen Knot Page/PD Presentation

In[5]:=

GaussCode[K]

Out[5]=

Data:Rolfsen Knot Page/Gauss Code

In[6]:=

DTCode[K]

Out[6]=

Data:Rolfsen Knot Page/DT Code

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

Data:Rolfsen Knot Page/Conway Notation

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

Data:Rolfsen Knot Page/BraidWord

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 3, 8, 3 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{7, 10}, {9, 2}, {10, 4}, {3, 5}, {4, 8}, {6, 9}, {5, 1}, {2, 7}, {1, 6}, {8, 3}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

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

Further Quantum Invariants

Rolfsen Knot Page Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["Rolfsen Knot Page"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

Data:Rolfsen Knot Page/Alexander Polynomial

In[5]:=

Conway[K][z]

Out[5]=

Data:Rolfsen Knot Page/Conway Polynomial

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

Data:Rolfsen Knot Page/2nd AlexanderIdeal

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ Data:Rolfsen Knot Page/Determinant, Data:Rolfsen Knot Page/Signature }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

Data:Rolfsen Knot Page/Jones Polynomial

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

Data:Rolfsen Knot Page/HOMFLYPT Polynomial

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

Data:Rolfsen Knot Page/Kauffman Polynomial

"Similar" Knots (within the Atlas)

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

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {{{{same_jones}}}}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["Rolfsen Knot Page"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ Data:Rolfsen Knot Page/Alexander Polynomial, Data:Rolfsen Knot Page/Jones Polynomial }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{{{{same_alexander}}}}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{{{{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

V_2,1 through V_6,9 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 $t^{r}q^{j}$ 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

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	{{{coloured_jones_2}}}
3	{{{coloured_jones_3}}}
4	{{{coloured_jones_4}}}
5	{{{coloured_jones_5}}}
6	{{{coloured_jones_6}}}
7	{{{coloured_jones_7}}}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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