9 43

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9 42.gif

9_42

9 44.gif

9_44

9 43.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

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Visit 9 43 at Knotilus!

[edit 9 43 Quick Notes]
[edit 9 43 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,6,11,5 X8394 X2,9,3,10 X14,8,15,7 X15,1,16,18 X11,17,12,16 X17,13,18,12 X6,14,7,13
Gauss code 1, -4, 3, -1, 2, -9, 5, -3, 4, -2, -7, 8, 9, -5, -6, 7, -8, 6
Dowker-Thistlethwaite code 4 8 10 14 2 -16 6 -18 -12
Conway Notation [211,3,2-]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif

Length is 9, width is 4,

Braid index is 4

9 43 ML.gif 9 43 AP.gif
[{5, 10}, {9, 1}, {10, 8}, {6, 9}, {4, 7}, {3, 6}, {2, 5}, {1, 4}, {7, 2}, {8, 3}]

[edit Notes on presentations of 9 43]


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["9 43"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X10,6,11,5 X8394 X2,9,3,10 X14,8,15,7 X15,1,16,18 X11,17,12,16 X17,13,18,12 X6,14,7,13
In[5]:= GaussCode[K]
Out[5]= 1, -4, 3, -1, 2, -9, 5, -3, 4, -2, -7, 8, 9, -5, -6, 7, -8, 6
In[6]:= DTCode[K]
Out[6]= 4 8 10 14 2 -16 6 -18 -12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [211,3,2-]
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]= [math]\displaystyle{ \textrm{BR}(4,\{1,1,1,2,1,1,-3,2,-3\}) }[/math]
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]= { 4, 9, 4 }
In[11]:= Show[BraidPlot[br]]
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
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."
9 43 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{5, 10}, {9, 1}, {10, 8}, {6, 9}, {4, 7}, {3, 6}, {2, 5}, {1, 4}, {7, 2}, {8, 3}]
In[14]:= Draw[ap]
9 43 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index [math]\displaystyle{ \{4,5\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [1][-10]
Hyperbolic Volume 5.90409
A-Polynomial See Data:9 43/A-polynomial

[edit Notes for 9 43's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 2 }[/math]
Topological 4 genus [math]\displaystyle{ 2 }[/math]
Concordance genus [math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant -4

[edit Notes for 9 43's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^3+3 t^2-2 t+1-2 t^{-1} +3 t^{-2} - t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -z^6-3 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 13, 4 }
Jones polynomial [math]\displaystyle{ -q^7+2 q^6-2 q^5+2 q^4-2 q^3+2 q^2-q+1 }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^6 a^{-4} +z^4 a^{-2} -5 z^4 a^{-4} +z^4 a^{-6} +4 z^2 a^{-2} -7 z^2 a^{-4} +4 z^2 a^{-6} +3 a^{-2} -4 a^{-4} +3 a^{-6} - a^{-8} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^7 a^{-3} +z^7 a^{-5} +z^6 a^{-2} +3 z^6 a^{-4} +2 z^6 a^{-6} -4 z^5 a^{-3} -3 z^5 a^{-5} +z^5 a^{-7} -5 z^4 a^{-2} -13 z^4 a^{-4} -8 z^4 a^{-6} +3 z^3 a^{-3} +z^3 a^{-5} -2 z^3 a^{-7} +7 z^2 a^{-2} +14 z^2 a^{-4} +9 z^2 a^{-6} +2 z^2 a^{-8} +z a^{-7} +z a^{-9} -3 a^{-2} -4 a^{-4} -3 a^{-6} - a^{-8} }[/math]
The A2 invariant [math]\displaystyle{ 1+ q^{-2} + q^{-4} + q^{-6} -2 q^{-12} + q^{-18} + q^{-20} - q^{-26} }[/math]
The G2 invariant [math]\displaystyle{ q^{-2} +2 q^{-6} - q^{-8} + q^{-10} + q^{-12} - q^{-14} +4 q^{-16} - q^{-18} +2 q^{-20} + q^{-22} - q^{-24} +3 q^{-26} - q^{-30} +2 q^{-32} - q^{-34} +2 q^{-38} -3 q^{-40} +2 q^{-42} -2 q^{-44} - q^{-48} -3 q^{-50} + q^{-52} -3 q^{-54} + q^{-56} -2 q^{-58} - q^{-64} + q^{-66} - q^{-68} +2 q^{-72} +3 q^{-78} -2 q^{-80} +4 q^{-82} +2 q^{-88} -2 q^{-90} +2 q^{-92} - q^{-100} - q^{-102} + q^{-104} - q^{-106} - q^{-108} - q^{-112} - q^{-116} + q^{-120} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_43.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q+ q^{-3} + q^{-13} - q^{-15} }[/math]
2 [math]\displaystyle{ q^6-q^2+1+ q^{-2} - q^{-4} + q^{-8} + q^{-10} + q^{-14} + q^{-16} - q^{-18} + q^{-22} - q^{-24} - q^{-26} - q^{-32} + q^{-36} }[/math]
3 [math]\displaystyle{ q^{15}-q^{11}-q^9+q^7+2 q^5-2 q- q^{-1} + q^{-3} +2 q^{-5} + q^{-7} - q^{-11} + q^{-13} +3 q^{-15} + q^{-17} -2 q^{-19} -2 q^{-21} +2 q^{-23} +2 q^{-25} - q^{-27} -2 q^{-29} + q^{-31} +2 q^{-33} - q^{-35} -2 q^{-37} + q^{-39} + q^{-41} -2 q^{-43} -2 q^{-45} + q^{-49} + q^{-51} -2 q^{-55} - q^{-57} +3 q^{-59} +3 q^{-61} -2 q^{-63} -4 q^{-65} +2 q^{-67} +3 q^{-69} -2 q^{-73} - q^{-75} + q^{-77} }[/math]
4 [math]\displaystyle{ q^{28}-q^{24}-q^{22}-q^{20}+2 q^{18}+2 q^{16}+q^{14}-q^{12}-4 q^{10}-q^8+q^6+3 q^4+3 q^2- q^{-2} -3 q^{-4} -2 q^{-6} +2 q^{-8} +4 q^{-10} +4 q^{-12} -5 q^{-16} -4 q^{-18} + q^{-20} +6 q^{-22} +6 q^{-24} -2 q^{-26} -5 q^{-28} -4 q^{-30} +2 q^{-32} +8 q^{-34} +3 q^{-36} -3 q^{-38} -6 q^{-40} -3 q^{-42} +5 q^{-44} +4 q^{-46} - q^{-48} -5 q^{-50} -4 q^{-52} +3 q^{-54} +3 q^{-56} - q^{-58} -3 q^{-60} -2 q^{-62} +3 q^{-64} +2 q^{-66} - q^{-68} -3 q^{-70} -2 q^{-72} +3 q^{-74} +3 q^{-76} + q^{-78} - q^{-80} -4 q^{-82} -3 q^{-84} +3 q^{-86} +7 q^{-88} +5 q^{-90} -4 q^{-92} -10 q^{-94} -3 q^{-96} +6 q^{-98} +10 q^{-100} +2 q^{-102} -10 q^{-104} -6 q^{-106} +6 q^{-110} +4 q^{-112} -3 q^{-114} -2 q^{-116} - q^{-118} +2 q^{-120} + q^{-122} - q^{-124} }[/math]
5 [math]\displaystyle{ q^{45}-q^{41}-q^{39}-q^{37}+2 q^{33}+3 q^{31}+q^{29}-q^{27}-3 q^{25}-4 q^{23}-2 q^{21}+2 q^{19}+5 q^{17}+4 q^{15}+2 q^{13}-q^{11}-4 q^9-5 q^7-3 q^5+4 q+7 q^{-1} +6 q^{-3} + q^{-5} -6 q^{-7} -9 q^{-9} -7 q^{-11} +9 q^{-15} +13 q^{-17} +8 q^{-19} -3 q^{-21} -11 q^{-23} -12 q^{-25} -4 q^{-27} +8 q^{-29} +16 q^{-31} +12 q^{-33} -12 q^{-37} -16 q^{-39} -8 q^{-41} +6 q^{-43} +16 q^{-45} +13 q^{-47} -13 q^{-51} -16 q^{-53} -5 q^{-55} +9 q^{-57} +16 q^{-59} +9 q^{-61} -6 q^{-63} -15 q^{-65} -12 q^{-67} +2 q^{-69} +12 q^{-71} +11 q^{-73} -11 q^{-77} -11 q^{-79} - q^{-81} +8 q^{-83} +8 q^{-85} -7 q^{-89} -6 q^{-91} +3 q^{-93} +7 q^{-95} +4 q^{-97} -4 q^{-99} -7 q^{-101} -2 q^{-103} +5 q^{-105} +8 q^{-107} +4 q^{-109} -3 q^{-111} -7 q^{-113} -7 q^{-115} -2 q^{-117} +6 q^{-119} +11 q^{-121} +9 q^{-123} + q^{-125} -10 q^{-127} -17 q^{-129} -10 q^{-131} +6 q^{-133} +18 q^{-135} +17 q^{-137} +2 q^{-139} -17 q^{-141} -24 q^{-143} -11 q^{-145} +12 q^{-147} +22 q^{-149} +16 q^{-151} -2 q^{-153} -16 q^{-155} -15 q^{-157} -2 q^{-159} +9 q^{-161} +9 q^{-163} +3 q^{-165} -3 q^{-167} -4 q^{-169} -2 q^{-171} + q^{-173} + q^{-175} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ 1+ q^{-2} + q^{-4} + q^{-6} -2 q^{-12} + q^{-18} + q^{-20} - q^{-26} }[/math]
1,1 [math]\displaystyle{ q^4+4-2 q^{-2} +6 q^{-4} -4 q^{-6} +4 q^{-8} -4 q^{-10} - q^{-12} +2 q^{-14} -4 q^{-16} +6 q^{-18} -4 q^{-20} +10 q^{-22} -4 q^{-24} +8 q^{-26} -4 q^{-28} +2 q^{-30} -4 q^{-32} -4 q^{-34} -6 q^{-38} +4 q^{-40} -2 q^{-42} +4 q^{-44} +2 q^{-48} - q^{-52} -2 q^{-54} + q^{-60} }[/math]
2,0 [math]\displaystyle{ q^4+q^2+1+ q^{-4} + q^{-6} - q^{-10} + q^{-20} + q^{-22} + q^{-24} + q^{-26} +2 q^{-28} +2 q^{-30} + q^{-32} -3 q^{-36} -3 q^{-38} -3 q^{-40} -2 q^{-42} - q^{-44} +2 q^{-48} +3 q^{-50} +2 q^{-52} - q^{-58} - q^{-60} - q^{-62} + q^{-66} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ 1+2 q^{-4} +2 q^{-6} +2 q^{-8} +2 q^{-10} + q^{-12} - q^{-16} -2 q^{-18} -2 q^{-20} - q^{-22} - q^{-24} + q^{-28} + q^{-30} +2 q^{-32} + q^{-34} + q^{-36} - q^{-40} - q^{-42} - q^{-44} - q^{-46} + q^{-50} }[/math]
1,0,0 [math]\displaystyle{ q^{-1} + q^{-3} +2 q^{-5} + q^{-7} +2 q^{-9} - q^{-13} -2 q^{-15} -2 q^{-17} - q^{-19} +2 q^{-23} + q^{-25} +2 q^{-27} - q^{-33} - q^{-35} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{-2} + q^{-4} +2 q^{-6} +3 q^{-8} +4 q^{-10} +4 q^{-12} +3 q^{-14} + q^{-16} -2 q^{-20} -4 q^{-22} -4 q^{-24} -2 q^{-26} - q^{-28} +2 q^{-34} + q^{-36} - q^{-38} + q^{-42} +2 q^{-46} +3 q^{-48} + q^{-50} - q^{-56} -3 q^{-58} -2 q^{-60} - q^{-66} + q^{-68} + q^{-70} }[/math]
1,0,0,0 [math]\displaystyle{ q^{-2} + q^{-4} +2 q^{-6} +2 q^{-8} +2 q^{-10} +2 q^{-12} - q^{-16} -3 q^{-18} -2 q^{-20} -3 q^{-22} - q^{-24} +2 q^{-28} +2 q^{-30} +2 q^{-32} +2 q^{-34} - q^{-40} - q^{-42} - q^{-44} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ 1+2 q^{-4} +2 q^{-8} + q^{-12} - q^{-16} -2 q^{-20} + q^{-22} -3 q^{-24} +2 q^{-26} - q^{-28} + q^{-30} + q^{-34} + q^{-36} + q^{-40} - q^{-42} + q^{-44} - q^{-46} - q^{-50} }[/math]
1,0 [math]\displaystyle{ q^2+2 q^{-6} + q^{-8} +2 q^{-14} + q^{-16} - q^{-20} + q^{-24} - q^{-28} - q^{-30} - q^{-38} + q^{-42} - q^{-46} + q^{-50} + q^{-52} - q^{-56} + q^{-58} + q^{-60} - q^{-64} - q^{-72} - q^{-74} + q^{-80} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{-2} +2 q^{-6} + q^{-8} +4 q^{-10} +2 q^{-12} +3 q^{-14} +2 q^{-16} +2 q^{-18} -2 q^{-22} -2 q^{-24} -4 q^{-26} -2 q^{-28} -4 q^{-30} - q^{-32} -3 q^{-34} +2 q^{-36} +3 q^{-40} +2 q^{-42} +3 q^{-44} +2 q^{-46} +2 q^{-48} + q^{-50} - q^{-52} -2 q^{-56} - q^{-58} -2 q^{-60} - q^{-64} + q^{-70} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-2} +2 q^{-6} - q^{-8} + q^{-10} + q^{-12} - q^{-14} +4 q^{-16} - q^{-18} +2 q^{-20} + q^{-22} - q^{-24} +3 q^{-26} - q^{-30} +2 q^{-32} - q^{-34} +2 q^{-38} -3 q^{-40} +2 q^{-42} -2 q^{-44} - q^{-48} -3 q^{-50} + q^{-52} -3 q^{-54} + q^{-56} -2 q^{-58} - q^{-64} + q^{-66} - q^{-68} +2 q^{-72} +3 q^{-78} -2 q^{-80} +4 q^{-82} +2 q^{-88} -2 q^{-90} +2 q^{-92} - q^{-100} - q^{-102} + q^{-104} - q^{-106} - q^{-108} - q^{-112} - q^{-116} + q^{-120} }[/math]

.

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["9 43"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^3+3 t^2-2 t+1-2 t^{-1} +3 t^{-2} - t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^6-3 z^4+z^2+1 }[/math]
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]= [math]\displaystyle{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 13, 4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^7+2 q^6-2 q^5+2 q^4-2 q^3+2 q^2-q+1 }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^6 a^{-4} +z^4 a^{-2} -5 z^4 a^{-4} +z^4 a^{-6} +4 z^2 a^{-2} -7 z^2 a^{-4} +4 z^2 a^{-6} +3 a^{-2} -4 a^{-4} +3 a^{-6} - a^{-8} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^7 a^{-3} +z^7 a^{-5} +z^6 a^{-2} +3 z^6 a^{-4} +2 z^6 a^{-6} -4 z^5 a^{-3} -3 z^5 a^{-5} +z^5 a^{-7} -5 z^4 a^{-2} -13 z^4 a^{-4} -8 z^4 a^{-6} +3 z^3 a^{-3} +z^3 a^{-5} -2 z^3 a^{-7} +7 z^2 a^{-2} +14 z^2 a^{-4} +9 z^2 a^{-6} +2 z^2 a^{-8} +z a^{-7} +z a^{-9} -3 a^{-2} -4 a^{-4} -3 a^{-6} - a^{-8} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n12,}

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["9 43"];
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]= { [math]\displaystyle{ -t^3+3 t^2-2 t+1-2 t^{-1} +3 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ -q^7+2 q^6-2 q^5+2 q^4-2 q^3+2 q^2-q+1 }[/math] }
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]= {}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {K11n12,}

Vassiliev invariants

V2 and V3: (1, 2)
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
[math]\displaystyle{ 4 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{254}{3} }[/math] [math]\displaystyle{ \frac{82}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{1024}{3} }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 80 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{1016}{3} }[/math] [math]\displaystyle{ \frac{328}{3} }[/math] [math]\displaystyle{ \frac{37231}{30} }[/math] [math]\displaystyle{ -\frac{154}{5} }[/math] [math]\displaystyle{ \frac{31502}{45} }[/math] [math]\displaystyle{ \frac{593}{18} }[/math] [math]\displaystyle{ \frac{2671}{30} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]4 is the signature of 9 43. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -2-1012345χ
15       1-1
13      1 1
11     11 0
9    11  0
7   11   0
5  11    0
3 12     1
1        0
-11       1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ i=5 }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

 [math]\displaystyle{ n }[/math]  [math]\displaystyle{ J_n }[/math]
2 [math]\displaystyle{ q^{17}-q^{16}-q^{15}+2 q^{14}-q^{13}-2 q^{12}+2 q^{11}+q^{10}-3 q^9+q^8+3 q^7-3 q^6+4 q^4-3 q^3-q^2+3 q-1- q^{-1} + q^{-2} }[/math]
3 [math]\displaystyle{ q^{37}-2 q^{36}-q^{35}+2 q^{34}+4 q^{33}-3 q^{32}-7 q^{31}+4 q^{30}+9 q^{29}-3 q^{28}-11 q^{27}+3 q^{26}+11 q^{25}-2 q^{24}-11 q^{23}+2 q^{22}+9 q^{21}-2 q^{20}-8 q^{19}+2 q^{18}+6 q^{17}-q^{16}-5 q^{15}+q^{14}+3 q^{13}-2 q^{11}+q^{10}-q^9+q^7+3 q^6-3 q^5-2 q^4+2 q^3+4 q^2-2 q-3+3 q^{-2} - q^{-4} - q^{-5} + q^{-6} }[/math]
4 [math]\displaystyle{ -q^{60}+2 q^{59}+q^{58}-3 q^{57}-q^{56}-2 q^{55}+9 q^{54}+3 q^{53}-9 q^{52}-7 q^{51}-6 q^{50}+21 q^{49}+11 q^{48}-13 q^{47}-16 q^{46}-13 q^{45}+27 q^{44}+20 q^{43}-11 q^{42}-20 q^{41}-19 q^{40}+26 q^{39}+23 q^{38}-9 q^{37}-18 q^{36}-19 q^{35}+21 q^{34}+22 q^{33}-7 q^{32}-15 q^{31}-18 q^{30}+16 q^{29}+21 q^{28}-5 q^{27}-11 q^{26}-18 q^{25}+9 q^{24}+20 q^{23}-q^{22}-6 q^{21}-17 q^{20}+q^{19}+17 q^{18}+2 q^{17}-12 q^{15}-5 q^{14}+11 q^{13}+q^{12}+3 q^{11}-4 q^{10}-5 q^9+6 q^8-4 q^7+2 q^6+q^5-q^4+6 q^3-6 q^2-2 q+ q^{-1} +7 q^{-2} -3 q^{-3} -2 q^{-4} -2 q^{-5} - q^{-6} +4 q^{-7} - q^{-10} - q^{-11} + q^{-12} }[/math]
5 [math]\displaystyle{ q^{85}-3 q^{83}-2 q^{82}+q^{81}+6 q^{80}+7 q^{79}-14 q^{77}-15 q^{76}+20 q^{74}+25 q^{73}+6 q^{72}-24 q^{71}-38 q^{70}-13 q^{69}+27 q^{68}+44 q^{67}+21 q^{66}-23 q^{65}-50 q^{64}-29 q^{63}+20 q^{62}+51 q^{61}+32 q^{60}-15 q^{59}-48 q^{58}-34 q^{57}+12 q^{56}+46 q^{55}+32 q^{54}-11 q^{53}-41 q^{52}-30 q^{51}+9 q^{50}+39 q^{49}+27 q^{48}-8 q^{47}-33 q^{46}-27 q^{45}+5 q^{44}+30 q^{43}+26 q^{42}-q^{41}-25 q^{40}-27 q^{39}-4 q^{38}+20 q^{37}+26 q^{36}+10 q^{35}-14 q^{34}-26 q^{33}-14 q^{32}+6 q^{31}+23 q^{30}+19 q^{29}+q^{28}-19 q^{27}-21 q^{26}-8 q^{25}+12 q^{24}+22 q^{23}+14 q^{22}-6 q^{21}-18 q^{20}-18 q^{19}-2 q^{18}+14 q^{17}+18 q^{16}+6 q^{15}-6 q^{14}-14 q^{13}-10 q^{12}+2 q^{11}+10 q^{10}+7 q^9+2 q^8-3 q^7-5 q^6-2 q^5+q^4+2+3 q^{-1} +2 q^{-2} -3 q^{-3} -4 q^{-4} -3 q^{-5} +4 q^{-7} +5 q^{-8} -2 q^{-10} -2 q^{-11} -3 q^{-12} +3 q^{-14} + q^{-15} - q^{-18} - q^{-19} + q^{-20} }[/math]
6 [math]\displaystyle{ q^{122}-2 q^{121}-q^{120}+2 q^{119}+q^{118}+2 q^{117}-q^{116}+2 q^{115}-9 q^{114}-8 q^{113}+6 q^{112}+9 q^{111}+14 q^{110}+6 q^{109}+q^{108}-35 q^{107}-34 q^{106}+q^{105}+25 q^{104}+49 q^{103}+38 q^{102}+16 q^{101}-70 q^{100}-85 q^{99}-31 q^{98}+24 q^{97}+86 q^{96}+87 q^{95}+55 q^{94}-81 q^{93}-124 q^{92}-74 q^{91}-3 q^{90}+92 q^{89}+117 q^{88}+96 q^{87}-64 q^{86}-128 q^{85}-95 q^{84}-28 q^{83}+77 q^{82}+115 q^{81}+110 q^{80}-50 q^{79}-116 q^{78}-91 q^{77}-32 q^{76}+68 q^{75}+103 q^{74}+103 q^{73}-49 q^{72}-106 q^{71}-80 q^{70}-26 q^{69}+65 q^{68}+93 q^{67}+93 q^{66}-48 q^{65}-95 q^{64}-71 q^{63}-27 q^{62}+55 q^{61}+81 q^{60}+89 q^{59}-36 q^{58}-76 q^{57}-64 q^{56}-36 q^{55}+35 q^{54}+64 q^{53}+86 q^{52}-15 q^{51}-48 q^{50}-54 q^{49}-47 q^{48}+8 q^{47}+40 q^{46}+79 q^{45}+8 q^{44}-15 q^{43}-36 q^{42}-51 q^{41}-18 q^{40}+8 q^{39}+59 q^{38}+21 q^{37}+17 q^{36}-7 q^{35}-38 q^{34}-30 q^{33}-23 q^{32}+24 q^{31}+14 q^{30}+32 q^{29}+22 q^{28}-8 q^{27}-17 q^{26}-35 q^{25}-10 q^{24}-11 q^{23}+20 q^{22}+30 q^{21}+17 q^{20}+12 q^{19}-20 q^{18}-19 q^{17}-27 q^{16}-6 q^{15}+12 q^{14}+16 q^{13}+28 q^{12}+2 q^{11}-4 q^{10}-17 q^9-15 q^8-6 q^7-q^6+18 q^5+4 q^4+6 q^3-4 q-4-6 q^{-1} +7 q^{-2} -7 q^{-3} + q^{-5} +2 q^{-6} +3 q^{-7} + q^{-8} +9 q^{-9} -7 q^{-10} -4 q^{-11} -4 q^{-12} -2 q^{-13} +2 q^{-15} +9 q^{-16} - q^{-17} -2 q^{-19} -2 q^{-20} -3 q^{-21} - q^{-22} +4 q^{-23} + q^{-25} - q^{-28} - q^{-29} + q^{-30} }[/math]
7 [math]\displaystyle{ -q^{161}+2 q^{160}+q^{159}-2 q^{158}-2 q^{157}-2 q^{156}+4 q^{155}+2 q^{154}+q^{153}+5 q^{152}-q^{151}-11 q^{150}-13 q^{149}-9 q^{148}+13 q^{147}+24 q^{146}+21 q^{145}+22 q^{144}-12 q^{143}-45 q^{142}-57 q^{141}-46 q^{140}+16 q^{139}+74 q^{138}+96 q^{137}+85 q^{136}-3 q^{135}-97 q^{134}-142 q^{133}-145 q^{132}-30 q^{131}+111 q^{130}+194 q^{129}+206 q^{128}+68 q^{127}-101 q^{126}-220 q^{125}-264 q^{124}-129 q^{123}+77 q^{122}+236 q^{121}+307 q^{120}+172 q^{119}-43 q^{118}-221 q^{117}-325 q^{116}-212 q^{115}+6 q^{114}+204 q^{113}+329 q^{112}+228 q^{111}+18 q^{110}-181 q^{109}-318 q^{108}-232 q^{107}-31 q^{106}+164 q^{105}+305 q^{104}+226 q^{103}+32 q^{102}-156 q^{101}-294 q^{100}-215 q^{99}-27 q^{98}+154 q^{97}+283 q^{96}+206 q^{95}+22 q^{94}-153 q^{93}-277 q^{92}-198 q^{91}-17 q^{90}+150 q^{89}+264 q^{88}+194 q^{87}+20 q^{86}-141 q^{85}-255 q^{84}-189 q^{83}-25 q^{82}+127 q^{81}+238 q^{80}+187 q^{79}+37 q^{78}-109 q^{77}-220 q^{76}-184 q^{75}-52 q^{74}+86 q^{73}+199 q^{72}+182 q^{71}+68 q^{70}-57 q^{69}-174 q^{68}-179 q^{67}-87 q^{66}+29 q^{65}+146 q^{64}+168 q^{63}+105 q^{62}+5 q^{61}-112 q^{60}-158 q^{59}-117 q^{58}-34 q^{57}+71 q^{56}+134 q^{55}+125 q^{54}+67 q^{53}-33 q^{52}-108 q^{51}-118 q^{50}-85 q^{49}-10 q^{48}+66 q^{47}+105 q^{46}+99 q^{45}+43 q^{44}-33 q^{43}-74 q^{42}-88 q^{41}-67 q^{40}-12 q^{39}+40 q^{38}+75 q^{37}+73 q^{36}+32 q^{35}-4 q^{34}-36 q^{33}-62 q^{32}-52 q^{31}-25 q^{30}+7 q^{29}+40 q^{28}+39 q^{27}+40 q^{26}+27 q^{25}-9 q^{24}-26 q^{23}-35 q^{22}-39 q^{21}-17 q^{20}-3 q^{19}+21 q^{18}+41 q^{17}+27 q^{16}+21 q^{15}+6 q^{14}-25 q^{13}-29 q^{12}-33 q^{11}-19 q^{10}+9 q^9+13 q^8+25 q^7+31 q^6+8 q^5-q^4-18 q^3-23 q^2-8 q-10+15 q^{-2} +8 q^{-3} +10 q^{-4} +3 q^{-5} -6 q^{-6} +3 q^{-7} -4 q^{-8} -5 q^{-9} + q^{-10} -6 q^{-11} - q^{-12} -4 q^{-14} +6 q^{-15} +5 q^{-16} +4 q^{-17} +7 q^{-18} -4 q^{-19} -4 q^{-20} -3 q^{-21} -7 q^{-22} -2 q^{-23} +3 q^{-25} +8 q^{-26} + q^{-27} + q^{-29} -3 q^{-30} -2 q^{-31} -3 q^{-32} - q^{-33} +3 q^{-34} + q^{-35} + q^{-37} - q^{-40} - q^{-41} + q^{-42} }[/math]

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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