6 3

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6 2.gif

6_2

7 1.gif

7_1

6 3.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 6 3's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 6 3 at Knotilus!

[edit 6 3 Quick Notes]

The Eskimo bowline knot of practical knot tying deforms to 6_3. The standard bowline is at 6_2.

[edit 6 3 Further Notes and Views]
3D depiction
Irish knot, sum of four 6.3

Knot presentations

Planar diagram presentation X4251 X8493 X12,9,1,10 X10,5,11,6 X6,11,7,12 X2837
Gauss code 1, -6, 2, -1, 4, -5, 6, -2, 3, -4, 5, -3
Dowker-Thistlethwaite code 4 8 10 2 12 6
Conway Notation [2112]


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

Length is 6, width is 3,

Braid index is 3

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

[edit Notes on presentations of 6 3]

Knot 6_3.
A graph, knot 6_3.
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["6 3"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X8493 X12,9,1,10 X10,5,11,6 X6,11,7,12 X2837
In[5]:= GaussCode[K]
Out[5]= 1, -6, 2, -1, 4, -5, 6, -2, 3, -4, 5, -3
In[6]:= DTCode[K]
Out[6]= 4 8 10 2 12 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [2112]
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}(3,\{-1,-1,2,-1,2,2\}) }[/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]= { 3, 6, 3 }
In[11]:= Show[BraidPlot[br]]
BraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.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."
6 3 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 7}, {2, 5}, {4, 6}, {5, 8}, {7, 9}, {8, 4}, {1, 3}, {9, 2}, {6, 1}]
In[14]:= Draw[ap]
6 3 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Fully amphicheiral
Unknotting number 1
3-genus 2
Bridge index 2
Super bridge index [math]\displaystyle{ \{3,4\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [-4][-4]
Hyperbolic Volume 5.69302
A-Polynomial See Data:6 3/A-polynomial

[edit Notes for 6 3'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{ 2 }[/math]
Rasmussen s-Invariant 0

[edit Notes for 6 3's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^2-3 t+5-3 t^{-1} + t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 13, 0 }
Jones polynomial [math]\displaystyle{ -q^3+2 q^2-2 q+3-2 q^{-1} +2 q^{-2} - q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^4-a^2 z^2-z^2 a^{-2} +3 z^2-a^2- a^{-2} +3 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a z^5+z^5 a^{-1} +2 a^2 z^4+2 z^4 a^{-2} +4 z^4+a^3 z^3+a z^3+z^3 a^{-1} +z^3 a^{-3} -3 a^2 z^2-3 z^2 a^{-2} -6 z^2-a^3 z-2 a z-2 z a^{-1} -z a^{-3} +a^2+ a^{-2} +3 }[/math]
The A2 invariant [math]\displaystyle{ -q^{10}+2 q^2+1+2 q^{-2} - q^{-10} }[/math]
The G2 invariant [math]\displaystyle{ q^{52}-q^{50}+2 q^{48}-2 q^{46}-q^{44}+q^{42}-3 q^{40}+4 q^{38}-4 q^{36}+q^{34}-3 q^{30}+3 q^{28}-3 q^{26}+q^{24}+q^{22}-2 q^{20}+q^{18}+q^{16}-q^{14}+4 q^{12}-3 q^{10}+3 q^8+q^6-q^4+6 q^2-5+6 q^{-2} - q^{-4} + q^{-6} +3 q^{-8} -3 q^{-10} +4 q^{-12} - q^{-14} + q^{-16} + q^{-18} -2 q^{-20} + q^{-22} + q^{-24} -3 q^{-26} +3 q^{-28} -3 q^{-30} + q^{-34} -4 q^{-36} +4 q^{-38} -3 q^{-40} + q^{-42} - q^{-44} -2 q^{-46} +2 q^{-48} - q^{-50} + q^{-52} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 6_3.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^7+q^5+q+ q^{-1} + q^{-5} - q^{-7} }[/math]
2 [math]\displaystyle{ q^{20}-q^{18}-2 q^{16}+2 q^{14}-2 q^{10}+2 q^8+q^6-q^4+q^2+1+ q^{-2} - q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} +2 q^{-14} -2 q^{-16} - q^{-18} + q^{-20} }[/math]
3 [math]\displaystyle{ -q^{39}+q^{37}+2 q^{35}-3 q^{31}-2 q^{29}+4 q^{27}+2 q^{25}-4 q^{23}-4 q^{21}+3 q^{19}+4 q^{17}-2 q^{15}-4 q^{13}+3 q^{11}+4 q^9-2 q^5+q+ q^{-1} -2 q^{-5} +4 q^{-9} +3 q^{-11} -4 q^{-13} -2 q^{-15} +4 q^{-17} +3 q^{-19} -4 q^{-21} -4 q^{-23} +2 q^{-25} +4 q^{-27} -2 q^{-29} -3 q^{-31} +2 q^{-35} + q^{-37} - q^{-39} }[/math]
4 [math]\displaystyle{ q^{64}-q^{62}-2 q^{60}+q^{56}+5 q^{54}-4 q^{50}-4 q^{48}-2 q^{46}+10 q^{44}+5 q^{42}-4 q^{40}-9 q^{38}-8 q^{36}+9 q^{34}+9 q^{32}+q^{30}-9 q^{28}-11 q^{26}+7 q^{24}+10 q^{22}+4 q^{20}-6 q^{18}-9 q^{16}+3 q^{14}+6 q^{12}+4 q^{10}-2 q^8-5 q^6+2 q^2+3+2 q^{-2} -5 q^{-6} -2 q^{-8} +4 q^{-10} +6 q^{-12} +3 q^{-14} -9 q^{-16} -6 q^{-18} +4 q^{-20} +10 q^{-22} +7 q^{-24} -11 q^{-26} -9 q^{-28} + q^{-30} +9 q^{-32} +9 q^{-34} -8 q^{-36} -9 q^{-38} -4 q^{-40} +5 q^{-42} +10 q^{-44} -2 q^{-46} -4 q^{-48} -4 q^{-50} +5 q^{-54} + q^{-56} -2 q^{-60} - q^{-62} + q^{-64} }[/math]
5 [math]\displaystyle{ -q^{95}+q^{93}+2 q^{91}-q^{87}-3 q^{85}-3 q^{83}+6 q^{79}+6 q^{77}+q^{75}-6 q^{73}-10 q^{71}-6 q^{69}+5 q^{67}+17 q^{65}+13 q^{63}-3 q^{61}-17 q^{59}-20 q^{57}-5 q^{55}+17 q^{53}+25 q^{51}+9 q^{49}-15 q^{47}-27 q^{45}-17 q^{43}+12 q^{41}+27 q^{39}+19 q^{37}-6 q^{35}-25 q^{33}-19 q^{31}+4 q^{29}+21 q^{27}+17 q^{25}-16 q^{21}-16 q^{19}+11 q^{15}+11 q^{13}+3 q^{11}-6 q^9-8 q^7-3 q^5+3 q^3+6 q+6 q^{-1} +3 q^{-3} -3 q^{-5} -8 q^{-7} -6 q^{-9} +3 q^{-11} +11 q^{-13} +11 q^{-15} -16 q^{-19} -16 q^{-21} +17 q^{-25} +21 q^{-27} +4 q^{-29} -19 q^{-31} -25 q^{-33} -6 q^{-35} +19 q^{-37} +27 q^{-39} +12 q^{-41} -17 q^{-43} -27 q^{-45} -15 q^{-47} +9 q^{-49} +25 q^{-51} +17 q^{-53} -5 q^{-55} -20 q^{-57} -17 q^{-59} -3 q^{-61} +13 q^{-63} +17 q^{-65} +5 q^{-67} -6 q^{-69} -10 q^{-71} -6 q^{-73} + q^{-75} +6 q^{-77} +6 q^{-79} -3 q^{-83} -3 q^{-85} - q^{-87} +2 q^{-91} + q^{-93} - q^{-95} }[/math]
6 [math]\displaystyle{ q^{132}-q^{130}-2 q^{128}+q^{124}+3 q^{122}+q^{120}+3 q^{118}-2 q^{116}-8 q^{114}-5 q^{112}-q^{110}+6 q^{108}+8 q^{106}+14 q^{104}+q^{102}-14 q^{100}-19 q^{98}-15 q^{96}+13 q^{92}+38 q^{90}+25 q^{88}-4 q^{86}-29 q^{84}-41 q^{82}-28 q^{80}-q^{78}+49 q^{76}+53 q^{74}+26 q^{72}-18 q^{70}-53 q^{68}-58 q^{66}-30 q^{64}+39 q^{62}+64 q^{60}+53 q^{58}+7 q^{56}-43 q^{54}-66 q^{52}-49 q^{50}+18 q^{48}+53 q^{46}+57 q^{44}+22 q^{42}-24 q^{40}-53 q^{38}-47 q^{36}+4 q^{34}+33 q^{32}+43 q^{30}+21 q^{28}-9 q^{26}-32 q^{24}-33 q^{22}-3 q^{20}+14 q^{18}+23 q^{16}+15 q^{14}+q^{12}-13 q^{10}-17 q^8-7 q^6+2 q^4+11 q^2+13+11 q^{-2} +2 q^{-4} -7 q^{-6} -17 q^{-8} -13 q^{-10} + q^{-12} +15 q^{-14} +23 q^{-16} +14 q^{-18} -3 q^{-20} -33 q^{-22} -32 q^{-24} -9 q^{-26} +21 q^{-28} +43 q^{-30} +33 q^{-32} +4 q^{-34} -47 q^{-36} -53 q^{-38} -24 q^{-40} +22 q^{-42} +57 q^{-44} +53 q^{-46} +18 q^{-48} -49 q^{-50} -66 q^{-52} -43 q^{-54} +7 q^{-56} +53 q^{-58} +64 q^{-60} +39 q^{-62} -30 q^{-64} -58 q^{-66} -53 q^{-68} -18 q^{-70} +26 q^{-72} +53 q^{-74} +49 q^{-76} - q^{-78} -28 q^{-80} -41 q^{-82} -29 q^{-84} -4 q^{-86} +25 q^{-88} +38 q^{-90} +13 q^{-92} -15 q^{-96} -19 q^{-98} -14 q^{-100} + q^{-102} +14 q^{-104} +8 q^{-106} +6 q^{-108} - q^{-110} -5 q^{-112} -8 q^{-114} -2 q^{-116} +3 q^{-118} + q^{-120} +3 q^{-122} + q^{-124} -2 q^{-128} - q^{-130} + q^{-132} }[/math]

A2 Invariants.

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

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{22}-q^{20}+q^{16}-3 q^{14}-q^{12}-2 q^8+3 q^4+3 q^2+4+3 q^{-2} +3 q^{-4} -2 q^{-8} - q^{-12} -3 q^{-14} + q^{-16} - q^{-20} + q^{-22} }[/math]
1,0,0 [math]\displaystyle{ -q^{13}-q^9+2 q^3+2 q+2 q^{-1} +2 q^{-3} - q^{-9} - q^{-13} }[/math]
1,0,1 [math]\displaystyle{ q^{36}-2 q^{34}+3 q^{32}-q^{30}-3 q^{28}+7 q^{26}-9 q^{24}+6 q^{22}-q^{20}-9 q^{18}+10 q^{16}-15 q^{14}+5 q^{12}-q^{10}-7 q^8+11 q^6+8 q^2+9+8 q^{-2} +11 q^{-6} -7 q^{-8} - q^{-10} +5 q^{-12} -15 q^{-14} +10 q^{-16} -9 q^{-18} - q^{-20} +6 q^{-22} -9 q^{-24} +7 q^{-26} -3 q^{-28} - q^{-30} +3 q^{-32} -2 q^{-34} + q^{-36} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{28}+q^{22}-3 q^{18}-3 q^{16}-2 q^{14}-3 q^{12}-3 q^{10}+4 q^6+3 q^4+6 q^2+8+6 q^{-2} +3 q^{-4} +4 q^{-6} -3 q^{-10} -3 q^{-12} -2 q^{-14} -3 q^{-16} -3 q^{-18} + q^{-22} + q^{-28} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{16}-q^{12}-q^{10}+2 q^4+2 q^2+3+2 q^{-2} +2 q^{-4} - q^{-10} - q^{-12} - q^{-16} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{22}+q^{20}-2 q^{18}+q^{16}-q^{14}+q^{12}+2 q^6-q^4+3 q^2-2+3 q^{-2} - q^{-4} +2 q^{-6} + q^{-12} - q^{-14} + q^{-16} -2 q^{-18} + q^{-20} - q^{-22} }[/math]
1,0 [math]\displaystyle{ q^{36}-q^{32}-q^{30}+q^{28}+q^{26}-q^{24}-2 q^{22}-q^{20}+q^{18}-q^{14}-q^{12}+q^{10}+q^8+q^6+2 q^2+3+2 q^{-2} + q^{-6} + q^{-8} + q^{-10} - q^{-12} - q^{-14} + q^{-18} - q^{-20} -2 q^{-22} - q^{-24} + q^{-26} + q^{-28} - q^{-30} - q^{-32} + q^{-36} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{30}-q^{28}+q^{26}-q^{24}+q^{22}-2 q^{20}-q^{18}-2 q^{16}-q^{14}-q^{12}-2 q^{10}+q^8+q^6+5 q^4+2 q^2+6+2 q^{-2} +5 q^{-4} + q^{-6} + q^{-8} -2 q^{-10} - q^{-12} - q^{-14} -2 q^{-16} - q^{-18} -2 q^{-20} + q^{-22} - q^{-24} + q^{-26} - q^{-28} + q^{-30} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{52}-q^{50}+2 q^{48}-2 q^{46}-q^{44}+q^{42}-3 q^{40}+4 q^{38}-4 q^{36}+q^{34}-3 q^{30}+3 q^{28}-3 q^{26}+q^{24}+q^{22}-2 q^{20}+q^{18}+q^{16}-q^{14}+4 q^{12}-3 q^{10}+3 q^8+q^6-q^4+6 q^2-5+6 q^{-2} - q^{-4} + q^{-6} +3 q^{-8} -3 q^{-10} +4 q^{-12} - q^{-14} + q^{-16} + q^{-18} -2 q^{-20} + q^{-22} + q^{-24} -3 q^{-26} +3 q^{-28} -3 q^{-30} + q^{-34} -4 q^{-36} +4 q^{-38} -3 q^{-40} + q^{-42} - q^{-44} -2 q^{-46} +2 q^{-48} - q^{-50} + q^{-52} }[/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["6 3"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^2-3 t+5-3 t^{-1} + t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 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, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^3+2 q^2-2 q+3-2 q^{-1} +2 q^{-2} - q^{-3} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^4-a^2 z^2-z^2 a^{-2} +3 z^2-a^2- a^{-2} +3 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a z^5+z^5 a^{-1} +2 a^2 z^4+2 z^4 a^{-2} +4 z^4+a^3 z^3+a z^3+z^3 a^{-1} +z^3 a^{-3} -3 a^2 z^2-3 z^2 a^{-2} -6 z^2-a^3 z-2 a z-2 z a^{-1} -z a^{-3} +a^2+ a^{-2} +3 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n12,}

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

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

Vassiliev invariants

V2 and V3: (1, 0)
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{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{14}{3} }[/math] [math]\displaystyle{ -\frac{14}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{56}{3} }[/math] [math]\displaystyle{ -\frac{56}{3} }[/math] [math]\displaystyle{ \frac{511}{30} }[/math] [math]\displaystyle{ \frac{418}{15} }[/math] [math]\displaystyle{ -\frac{1858}{45} }[/math] [math]\displaystyle{ \frac{65}{18} }[/math] [math]\displaystyle{ -\frac{449}{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]0 is the signature of 6 3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-10123χ
7      1-1
5     1 1
3    11 0
1   21  1
-1  12   1
-3 11    0
-5 1     1
-71      -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math]
[math]\displaystyle{ r=-3 }[/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=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}_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^9-2 q^8-q^7+5 q^6-4 q^5-3 q^4+9 q^3-5 q^2-5 q+11-5 q^{-1} -5 q^{-2} +9 q^{-3} -3 q^{-4} -4 q^{-5} +5 q^{-6} - q^{-7} -2 q^{-8} + q^{-9} }[/math]
3 [math]\displaystyle{ -q^{18}+2 q^{17}+q^{16}-2 q^{15}-4 q^{14}+3 q^{13}+7 q^{12}-4 q^{11}-10 q^{10}+3 q^9+14 q^8-3 q^7-16 q^6+q^5+21 q^4-2 q^3-20 q^2-q+23- q^{-1} -20 q^{-2} -2 q^{-3} +21 q^{-4} + q^{-5} -16 q^{-6} -3 q^{-7} +14 q^{-8} +3 q^{-9} -10 q^{-10} -4 q^{-11} +7 q^{-12} +3 q^{-13} -4 q^{-14} -2 q^{-15} + q^{-16} +2 q^{-17} - q^{-18} }[/math]
4 [math]\displaystyle{ q^{30}-2 q^{29}-q^{28}+2 q^{27}+q^{26}+5 q^{25}-7 q^{24}-5 q^{23}+2 q^{22}+3 q^{21}+17 q^{20}-12 q^{19}-14 q^{18}-3 q^{17}+4 q^{16}+34 q^{15}-12 q^{14}-22 q^{13}-13 q^{12}+2 q^{11}+52 q^{10}-9 q^9-28 q^8-23 q^7-q^6+64 q^5-6 q^4-30 q^3-29 q^2-4 q+69-4 q^{-1} -29 q^{-2} -30 q^{-3} -6 q^{-4} +64 q^{-5} - q^{-6} -23 q^{-7} -28 q^{-8} -9 q^{-9} +52 q^{-10} +2 q^{-11} -13 q^{-12} -22 q^{-13} -12 q^{-14} +34 q^{-15} +4 q^{-16} -3 q^{-17} -14 q^{-18} -12 q^{-19} +17 q^{-20} +3 q^{-21} +2 q^{-22} -5 q^{-23} -7 q^{-24} +5 q^{-25} + q^{-26} +2 q^{-27} - q^{-28} -2 q^{-29} + q^{-30} }[/math]
5 [math]\displaystyle{ -q^{45}+2 q^{44}+q^{43}-2 q^{42}-q^{41}-2 q^{40}-q^{39}+5 q^{38}+7 q^{37}-2 q^{36}-6 q^{35}-9 q^{34}-5 q^{33}+9 q^{32}+18 q^{31}+10 q^{30}-10 q^{29}-25 q^{28}-19 q^{27}+6 q^{26}+33 q^{25}+32 q^{24}-2 q^{23}-41 q^{22}-43 q^{21}-6 q^{20}+43 q^{19}+61 q^{18}+13 q^{17}-49 q^{16}-68 q^{15}-25 q^{14}+49 q^{13}+84 q^{12}+30 q^{11}-53 q^{10}-85 q^9-41 q^8+49 q^7+100 q^6+41 q^5-53 q^4-93 q^3-50 q^2+47 q+105+47 q^{-1} -50 q^{-2} -93 q^{-3} -53 q^{-4} +41 q^{-5} +100 q^{-6} +49 q^{-7} -41 q^{-8} -85 q^{-9} -53 q^{-10} +30 q^{-11} +84 q^{-12} +49 q^{-13} -25 q^{-14} -68 q^{-15} -49 q^{-16} +13 q^{-17} +61 q^{-18} +43 q^{-19} -6 q^{-20} -43 q^{-21} -41 q^{-22} -2 q^{-23} +32 q^{-24} +33 q^{-25} +6 q^{-26} -19 q^{-27} -25 q^{-28} -10 q^{-29} +10 q^{-30} +18 q^{-31} +9 q^{-32} -5 q^{-33} -9 q^{-34} -6 q^{-35} -2 q^{-36} +7 q^{-37} +5 q^{-38} - q^{-39} -2 q^{-40} - q^{-41} -2 q^{-42} + q^{-43} +2 q^{-44} - q^{-45} }[/math]
6 [math]\displaystyle{ q^{63}-2 q^{62}-q^{61}+2 q^{60}+q^{59}+2 q^{58}-2 q^{57}+3 q^{56}-7 q^{55}-7 q^{54}+5 q^{53}+5 q^{52}+9 q^{51}+9 q^{49}-20 q^{48}-22 q^{47}+9 q^{45}+24 q^{44}+13 q^{43}+34 q^{42}-33 q^{41}-51 q^{40}-25 q^{39}-3 q^{38}+37 q^{37}+40 q^{36}+84 q^{35}-29 q^{34}-78 q^{33}-69 q^{32}-38 q^{31}+32 q^{30}+68 q^{29}+153 q^{28}-4 q^{27}-89 q^{26}-115 q^{25}-88 q^{24}+9 q^{23}+85 q^{22}+220 q^{21}+31 q^{20}-85 q^{19}-150 q^{18}-134 q^{17}-20 q^{16}+91 q^{15}+271 q^{14}+60 q^{13}-75 q^{12}-172 q^{11}-164 q^{10}-43 q^9+90 q^8+301 q^7+77 q^6-66 q^5-180 q^4-178 q^3-57 q^2+86 q+311+86 q^{-1} -57 q^{-2} -178 q^{-3} -180 q^{-4} -66 q^{-5} +77 q^{-6} +301 q^{-7} +90 q^{-8} -43 q^{-9} -164 q^{-10} -172 q^{-11} -75 q^{-12} +60 q^{-13} +271 q^{-14} +91 q^{-15} -20 q^{-16} -134 q^{-17} -150 q^{-18} -85 q^{-19} +31 q^{-20} +220 q^{-21} +85 q^{-22} +9 q^{-23} -88 q^{-24} -115 q^{-25} -89 q^{-26} -4 q^{-27} +153 q^{-28} +68 q^{-29} +32 q^{-30} -38 q^{-31} -69 q^{-32} -78 q^{-33} -29 q^{-34} +84 q^{-35} +40 q^{-36} +37 q^{-37} -3 q^{-38} -25 q^{-39} -51 q^{-40} -33 q^{-41} +34 q^{-42} +13 q^{-43} +24 q^{-44} +9 q^{-45} -22 q^{-47} -20 q^{-48} +9 q^{-49} +9 q^{-51} +5 q^{-52} +5 q^{-53} -7 q^{-54} -7 q^{-55} +3 q^{-56} -2 q^{-57} +2 q^{-58} + q^{-59} +2 q^{-60} - q^{-61} -2 q^{-62} + q^{-63} }[/math]
7 [math]\displaystyle{ -q^{84}+2 q^{83}+q^{82}-2 q^{81}-q^{80}-2 q^{79}+2 q^{78}-q^{76}+7 q^{75}+4 q^{74}-4 q^{73}-5 q^{72}-11 q^{71}-q^{70}+3 q^{69}-q^{68}+21 q^{67}+16 q^{66}+q^{65}-9 q^{64}-34 q^{63}-21 q^{62}-8 q^{61}-4 q^{60}+44 q^{59}+49 q^{58}+30 q^{57}+12 q^{56}-59 q^{55}-68 q^{54}-55 q^{53}-41 q^{52}+58 q^{51}+94 q^{50}+98 q^{49}+78 q^{48}-50 q^{47}-118 q^{46}-138 q^{45}-128 q^{44}+28 q^{43}+126 q^{42}+181 q^{41}+195 q^{40}+7 q^{39}-135 q^{38}-224 q^{37}-250 q^{36}-50 q^{35}+119 q^{34}+256 q^{33}+322 q^{32}+101 q^{31}-115 q^{30}-281 q^{29}-371 q^{28}-149 q^{27}+86 q^{26}+299 q^{25}+433 q^{24}+191 q^{23}-75 q^{22}-312 q^{21}-460 q^{20}-232 q^{19}+45 q^{18}+319 q^{17}+506 q^{16}+260 q^{15}-44 q^{14}-321 q^{13}-512 q^{12}-284 q^{11}+14 q^{10}+322 q^9+547 q^8+298 q^7-23 q^6-320 q^5-534 q^4-309 q^3-7 q^2+316 q+559+316 q^{-1} -7 q^{-2} -309 q^{-3} -534 q^{-4} -320 q^{-5} -23 q^{-6} +298 q^{-7} +547 q^{-8} +322 q^{-9} +14 q^{-10} -284 q^{-11} -512 q^{-12} -321 q^{-13} -44 q^{-14} +260 q^{-15} +506 q^{-16} +319 q^{-17} +45 q^{-18} -232 q^{-19} -460 q^{-20} -312 q^{-21} -75 q^{-22} +191 q^{-23} +433 q^{-24} +299 q^{-25} +86 q^{-26} -149 q^{-27} -371 q^{-28} -281 q^{-29} -115 q^{-30} +101 q^{-31} +322 q^{-32} +256 q^{-33} +119 q^{-34} -50 q^{-35} -250 q^{-36} -224 q^{-37} -135 q^{-38} +7 q^{-39} +195 q^{-40} +181 q^{-41} +126 q^{-42} +28 q^{-43} -128 q^{-44} -138 q^{-45} -118 q^{-46} -50 q^{-47} +78 q^{-48} +98 q^{-49} +94 q^{-50} +58 q^{-51} -41 q^{-52} -55 q^{-53} -68 q^{-54} -59 q^{-55} +12 q^{-56} +30 q^{-57} +49 q^{-58} +44 q^{-59} -4 q^{-60} -8 q^{-61} -21 q^{-62} -34 q^{-63} -9 q^{-64} + q^{-65} +16 q^{-66} +21 q^{-67} - q^{-68} +3 q^{-69} - q^{-70} -11 q^{-71} -5 q^{-72} -4 q^{-73} +4 q^{-74} +7 q^{-75} - q^{-76} +2 q^{-78} -2 q^{-79} - q^{-80} -2 q^{-81} + q^{-82} +2 q^{-83} - q^{-84} }[/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.

6 2.gif

6_2

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7_1

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