10 156

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10 155.gif

10_155

10 157.gif

10_157

10 156.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 156's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 156 at Knotilus!

[edit 10 156 Quick Notes]
[edit 10 156 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X7,14,8,15 X18,9,19,10 X6,19,7,20 X16,5,17,6 X10,17,11,18 X13,8,14,9 X20,15,1,16 X2,12,3,11
Gauss code 1, -10, 2, -1, 6, -5, -3, 8, 4, -7, 10, -2, -8, 3, 9, -6, 7, -4, 5, -9
Dowker-Thistlethwaite code 4 12 16 -14 18 2 -8 20 10 6
Conway Notation [-3:2:20]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif

Length is 11, width is 4,

Braid index is 4

10 156 ML.gif 10 156 AP.gif
[{3, 10}, {2, 4}, {1, 3}, {8, 11}, {9, 5}, {10, 7}, {4, 8}, {6, 9}, {7, 2}, {11, 6}, {5, 1}]

[edit Notes on presentations of 10 156]


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["10 156"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X12,4,13,3 X7,14,8,15 X18,9,19,10 X6,19,7,20 X16,5,17,6 X10,17,11,18 X13,8,14,9 X20,15,1,16 X2,12,3,11
In[5]:= GaussCode[K]
Out[5]= 1, -10, 2, -1, 6, -5, -3, 8, 4, -7, 10, -2, -8, 3, 9, -6, 7, -4, 5, -9
In[6]:= DTCode[K]
Out[6]= 4 12 16 -14 18 2 -8 20 10 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [-3:2:20]
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,-2,1,-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, 11, 4 }
In[11]:= Show[BraidPlot[br]]
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.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."
10 156 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 10}, {2, 4}, {1, 3}, {8, 11}, {9, 5}, {10, 7}, {4, 8}, {6, 9}, {7, 2}, {11, 6}, {5, 1}]
In[14]:= Draw[ap]
10 156 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-8][-2]
Hyperbolic Volume 11.1634
A-Polynomial See Data:10 156/A-polynomial

[edit Notes for 10 156's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math]
Topological 4 genus [math]\displaystyle{ 1 }[/math]
Concordance genus [math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant 2

[edit Notes for 10 156's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ z^6+2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 35, -2 }
Jones polynomial [math]\displaystyle{ -q^2+3 q-4+6 q^{-1} -6 q^{-2} +6 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ a^2 z^6-a^4 z^4+4 a^2 z^4-z^4-2 a^4 z^2+5 a^2 z^2-2 z^2-a^4+2 a^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a^4 z^8+a^2 z^8+a^5 z^7+4 a^3 z^7+3 a z^7-a^4 z^6+2 a^2 z^6+3 z^6-a^5 z^5-9 a^3 z^5-7 a z^5+z^5 a^{-1} +3 a^6 z^4+2 a^4 z^4-9 a^2 z^4-8 z^4+a^7 z^3+4 a^5 z^3+8 a^3 z^3+3 a z^3-2 z^3 a^{-1} -2 a^6 z^2+a^4 z^2+7 a^2 z^2+4 z^2-a^7 z-2 a^5 z-2 a^3 z-a z-a^4-2 a^2 }[/math]
The A2 invariant [math]\displaystyle{ -q^{18}+q^{16}-q^{14}+q^{10}-q^8+2 q^6-q^4+2 q^2+1+ q^{-4} - q^{-6} }[/math]
The G2 invariant [math]\displaystyle{ q^{100}-q^{98}+q^{96}-2 q^{90}+2 q^{88}+2 q^{86}-6 q^{84}+11 q^{82}-16 q^{80}+10 q^{78}-3 q^{76}-13 q^{74}+29 q^{72}-35 q^{70}+27 q^{68}-8 q^{66}-16 q^{64}+35 q^{62}-36 q^{60}+23 q^{58}-20 q^{54}+28 q^{52}-21 q^{50}+2 q^{48}+21 q^{46}-34 q^{44}+32 q^{42}-18 q^{40}-5 q^{38}+27 q^{36}-42 q^{34}+42 q^{32}-31 q^{30}+11 q^{28}+15 q^{26}-35 q^{24}+44 q^{22}-33 q^{20}+15 q^{18}+10 q^{16}-28 q^{14}+31 q^{12}-16 q^{10}-3 q^8+25 q^6-32 q^4+24 q^2-1-22 q^{-2} +36 q^{-4} -35 q^{-6} +22 q^{-8} -3 q^{-10} -16 q^{-12} +25 q^{-14} -22 q^{-16} +16 q^{-18} -5 q^{-20} -3 q^{-22} +5 q^{-24} -7 q^{-26} +4 q^{-28} -2 q^{-30} + q^{-32} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_156.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{13}+2 q^{11}-2 q^9+q^7+2 q- q^{-1} +2 q^{-3} - q^{-5} }[/math]
2 [math]\displaystyle{ -2 q^{34}+2 q^{32}+5 q^{30}-7 q^{28}-2 q^{26}+10 q^{24}-6 q^{22}-6 q^{20}+8 q^{18}-5 q^{14}+q^{12}+5 q^{10}-3 q^8-5 q^6+8 q^4+2 q^2-8+6 q^{-2} +6 q^{-4} -8 q^{-6} - q^{-8} +6 q^{-10} -2 q^{-12} -2 q^{-14} + q^{-16} }[/math]
3 [math]\displaystyle{ q^{71}-q^{67}-5 q^{65}-q^{63}+11 q^{61}+9 q^{59}-10 q^{57}-24 q^{55}+6 q^{53}+35 q^{51}+4 q^{49}-40 q^{47}-22 q^{45}+39 q^{43}+34 q^{41}-28 q^{39}-36 q^{37}+14 q^{35}+35 q^{33}-q^{31}-28 q^{29}-11 q^{27}+20 q^{25}+17 q^{23}-13 q^{21}-27 q^{19}+7 q^{17}+33 q^{15}-39 q^{11}-5 q^9+42 q^7+18 q^5-37 q^3-29 q+30 q^{-1} +35 q^{-3} -12 q^{-5} -37 q^{-7} -3 q^{-9} +30 q^{-11} +15 q^{-13} -16 q^{-15} -18 q^{-17} +3 q^{-19} +14 q^{-21} + q^{-23} -6 q^{-25} -3 q^{-27} +2 q^{-29} +2 q^{-31} - q^{-33} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{18}+q^{16}-q^{14}+q^{10}-q^8+2 q^6-q^4+2 q^2+1+ q^{-4} - q^{-6} }[/math]
1,1 [math]\displaystyle{ q^{52}-2 q^{50}-2 q^{48}+2 q^{46}+2 q^{44}-4 q^{42}+18 q^{40}-36 q^{38}+62 q^{36}-90 q^{34}+104 q^{32}-114 q^{30}+99 q^{28}-68 q^{26}+26 q^{24}+32 q^{22}-84 q^{20}+132 q^{18}-170 q^{16}+186 q^{14}-194 q^{12}+176 q^{10}-138 q^8+94 q^6-32 q^4-16 q^2+70-96 q^{-2} +107 q^{-4} -104 q^{-6} +84 q^{-8} -62 q^{-10} +40 q^{-12} -22 q^{-14} +10 q^{-16} -4 q^{-18} + q^{-20} }[/math]
2,0 [math]\displaystyle{ -q^{48}+q^{42}+3 q^{40}-2 q^{38}-q^{36}+q^{34}+2 q^{32}-2 q^{30}-5 q^{28}+3 q^{26}+q^{24}-3 q^{22}-q^{20}+3 q^{18}-q^{16}-q^{14}+2 q^{12}+q^6+5 q^4-q^2+5 q^{-2} + q^{-4} -4 q^{-6} - q^{-8} +2 q^{-10} + q^{-12} -2 q^{-14} - q^{-16} + q^{-18} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{42}-q^{40}-q^{38}+4 q^{36}-3 q^{34}-4 q^{32}+7 q^{30}-4 q^{28}-4 q^{26}+6 q^{24}-3 q^{22}-2 q^{20}+2 q^{18}+q^{16}-q^{12}+3 q^{10}+3 q^8-5 q^6+3 q^4+6 q^2-5+3 q^{-2} +4 q^{-4} -5 q^{-6} +2 q^{-8} -2 q^{-12} + q^{-14} }[/math]
1,0,0 [math]\displaystyle{ -q^{23}+q^{21}-2 q^{19}+q^{17}-q^{15}+q^{13}+q^9+q^7+2 q^3+2 q^{-1} - q^{-3} + q^{-5} - q^{-7} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ 2 q^{52}-q^{50}-2 q^{48}+4 q^{46}+q^{44}-6 q^{42}+4 q^{38}-4 q^{36}-6 q^{34}+3 q^{32}+3 q^{30}-5 q^{28}+8 q^{24}-3 q^{22}-5 q^{20}+8 q^{18}-6 q^{14}+4 q^{12}+7 q^{10}-3 q^8-q^6+6 q^4+4 q^2-3+ q^{-2} +4 q^{-4} -2 q^{-6} -2 q^{-8} + q^{-10} - q^{-12} - q^{-14} + q^{-16} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{28}+q^{26}-2 q^{24}-q^{18}+q^{16}+2 q^{12}+2 q^8+2 q^4+1+ q^{-2} - q^{-4} + q^{-6} - q^{-8} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{42}+q^{40}-3 q^{38}+6 q^{36}-7 q^{34}+8 q^{32}-7 q^{30}+6 q^{28}-4 q^{26}+3 q^{22}-8 q^{20}+10 q^{18}-13 q^{16}+14 q^{14}-13 q^{12}+13 q^{10}-7 q^8+5 q^6+q^4-2 q^2+5-7 q^{-2} +8 q^{-4} -7 q^{-6} +6 q^{-8} -4 q^{-10} +2 q^{-12} - q^{-14} }[/math]
1,0 [math]\displaystyle{ q^{68}-q^{64}-2 q^{62}+5 q^{58}+2 q^{56}-5 q^{54}-6 q^{52}+2 q^{50}+8 q^{48}+q^{46}-8 q^{44}-4 q^{42}+6 q^{40}+5 q^{38}-4 q^{36}-6 q^{34}+2 q^{32}+6 q^{30}-6 q^{26}-q^{24}+5 q^{22}+3 q^{20}-3 q^{18}-4 q^{16}+4 q^{14}+5 q^{12}-2 q^{10}-7 q^8+2 q^6+8 q^4+4 q^2-6-6 q^{-2} +5 q^{-4} +8 q^{-6} - q^{-8} -6 q^{-10} -2 q^{-12} +4 q^{-14} +2 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-24} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{58}-q^{56}+q^{54}-2 q^{52}+5 q^{50}-6 q^{48}+4 q^{46}-6 q^{44}+7 q^{42}-6 q^{40}+3 q^{38}-4 q^{36}+q^{34}+2 q^{32}-4 q^{30}+4 q^{28}-8 q^{26}+10 q^{24}-9 q^{22}+11 q^{20}-10 q^{18}+11 q^{16}-7 q^{14}+9 q^{12}-5 q^{10}+3 q^8+q^6+3 q^2-3+7 q^{-2} -6 q^{-4} +6 q^{-6} -6 q^{-8} +5 q^{-10} -4 q^{-12} +2 q^{-14} -2 q^{-16} + q^{-18} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{100}-q^{98}+q^{96}-2 q^{90}+2 q^{88}+2 q^{86}-6 q^{84}+11 q^{82}-16 q^{80}+10 q^{78}-3 q^{76}-13 q^{74}+29 q^{72}-35 q^{70}+27 q^{68}-8 q^{66}-16 q^{64}+35 q^{62}-36 q^{60}+23 q^{58}-20 q^{54}+28 q^{52}-21 q^{50}+2 q^{48}+21 q^{46}-34 q^{44}+32 q^{42}-18 q^{40}-5 q^{38}+27 q^{36}-42 q^{34}+42 q^{32}-31 q^{30}+11 q^{28}+15 q^{26}-35 q^{24}+44 q^{22}-33 q^{20}+15 q^{18}+10 q^{16}-28 q^{14}+31 q^{12}-16 q^{10}-3 q^8+25 q^6-32 q^4+24 q^2-1-22 q^{-2} +36 q^{-4} -35 q^{-6} +22 q^{-8} -3 q^{-10} -16 q^{-12} +25 q^{-14} -22 q^{-16} +16 q^{-18} -5 q^{-20} -3 q^{-22} +5 q^{-24} -7 q^{-26} +4 q^{-28} -2 q^{-30} + q^{-32} }[/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["10 156"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^6+2 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]= { 35, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^2+3 q-4+6 q^{-1} -6 q^{-2} +6 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ a^2 z^6-a^4 z^4+4 a^2 z^4-z^4-2 a^4 z^2+5 a^2 z^2-2 z^2-a^4+2 a^2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a^4 z^8+a^2 z^8+a^5 z^7+4 a^3 z^7+3 a z^7-a^4 z^6+2 a^2 z^6+3 z^6-a^5 z^5-9 a^3 z^5-7 a z^5+z^5 a^{-1} +3 a^6 z^4+2 a^4 z^4-9 a^2 z^4-8 z^4+a^7 z^3+4 a^5 z^3+8 a^3 z^3+3 a z^3-2 z^3 a^{-1} -2 a^6 z^2+a^4 z^2+7 a^2 z^2+4 z^2-a^7 z-2 a^5 z-2 a^3 z-a z-a^4-2 a^2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_16, K11n15, K11n56, K11n58,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {8_16,}

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["10 156"];
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^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3} }[/math], [math]\displaystyle{ -q^2+3 q-4+6 q^{-1} -6 q^{-2} +6 q^{-3} -5 q^{-4} +3 q^{-5} - q^{-6} }[/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]= {8_16, K11n15, K11n56, K11n58,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {8_16,}

Vassiliev invariants

V2 and V3: (1, -1)
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{ -8 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{14}{3} }[/math] [math]\displaystyle{ -\frac{38}{3} }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ \frac{16}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 56 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{56}{3} }[/math] [math]\displaystyle{ -\frac{152}{3} }[/math] [math]\displaystyle{ -\frac{929}{30} }[/math] [math]\displaystyle{ \frac{1086}{5} }[/math] [math]\displaystyle{ -\frac{13258}{45} }[/math] [math]\displaystyle{ -\frac{895}{18} }[/math] [math]\displaystyle{ -\frac{1889}{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]-2 is the signature of 10 156. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123χ
5        1-1
3       2 2
1      21 -1
-1     42  2
-3    33   0
-5   33    0
-7  23     1
-9 13      -2
-11 2       2
-131        -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=-1 }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=3 }[/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^7-3 q^6+9 q^4-10 q^3-7 q^2+23 q-10-21 q^{-1} +33 q^{-2} -4 q^{-3} -34 q^{-4} +35 q^{-5} +4 q^{-6} -38 q^{-7} +29 q^{-8} +9 q^{-9} -30 q^{-10} +15 q^{-11} +9 q^{-12} -14 q^{-13} +3 q^{-14} +4 q^{-15} -2 q^{-16} }[/math]
3 [math]\displaystyle{ -q^{15}+3 q^{14}-5 q^{12}-4 q^{11}+10 q^{10}+13 q^9-16 q^8-25 q^7+12 q^6+44 q^5-q^4-58 q^3-22 q^2+69 q+46-63 q^{-1} -81 q^{-2} +61 q^{-3} +101 q^{-4} -39 q^{-5} -128 q^{-6} +27 q^{-7} +140 q^{-8} -6 q^{-9} -154 q^{-10} -7 q^{-11} +154 q^{-12} +24 q^{-13} -151 q^{-14} -38 q^{-15} +137 q^{-16} +51 q^{-17} -115 q^{-18} -59 q^{-19} +87 q^{-20} +59 q^{-21} -53 q^{-22} -54 q^{-23} +26 q^{-24} +41 q^{-25} -9 q^{-26} -23 q^{-27} -3 q^{-28} +11 q^{-29} +5 q^{-30} -4 q^{-31} - q^{-32} - q^{-33} + q^{-34} }[/math]
4 [math]\displaystyle{ q^{26}-3 q^{25}+5 q^{23}+4 q^{21}-17 q^{20}-6 q^{19}+17 q^{18}+11 q^{17}+31 q^{16}-46 q^{15}-47 q^{14}+5 q^{13}+27 q^{12}+121 q^{11}-25 q^{10}-96 q^9-89 q^8-50 q^7+222 q^6+108 q^5-26 q^4-189 q^3-268 q^2+183 q+253+208 q^{-1} -148 q^{-2} -510 q^{-3} -21 q^{-4} +273 q^{-5} +477 q^{-6} +34 q^{-7} -646 q^{-8} -267 q^{-9} +180 q^{-10} +667 q^{-11} +238 q^{-12} -677 q^{-13} -453 q^{-14} +66 q^{-15} +763 q^{-16} +390 q^{-17} -646 q^{-18} -572 q^{-19} -38 q^{-20} +782 q^{-21} +498 q^{-22} -549 q^{-23} -624 q^{-24} -159 q^{-25} +689 q^{-26} +566 q^{-27} -348 q^{-28} -572 q^{-29} -289 q^{-30} +458 q^{-31} +533 q^{-32} -89 q^{-33} -378 q^{-34} -332 q^{-35} +163 q^{-36} +357 q^{-37} +83 q^{-38} -132 q^{-39} -228 q^{-40} -22 q^{-41} +135 q^{-42} +86 q^{-43} +8 q^{-44} -78 q^{-45} -44 q^{-46} +16 q^{-47} +25 q^{-48} +19 q^{-49} -7 q^{-50} -11 q^{-51} -2 q^{-52} +3 q^{-54} + q^{-55} - q^{-56} }[/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).

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