5 2

5_1

6_1

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 5 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 5 2 at Knotilus!
	5_2 is also known as the 3-twist knot. The Bowstring knot of practical knot tying deforms to 5_2.

3D depiction

Simple square depiction

Lissajous curve x=cos(2t+0.2), y=cos(3t+0.7), z=cos(7t); 2 crossings can be removed

Knot presentations

Planar diagram presentation	X1425 X3849 X5,10,6,1 X9,6,10,7 X7283
Gauss code	-1, 5, -2, 1, -3, 4, -5, 2, -4, 3
Dowker-Thistlethwaite code	4 8 10 2 6
Conway Notation	[32]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 6, width is 3, Braid index is 3		[{7, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 6}, {5, 7}, {6, 1}]

[edit Notes on presentations of 5 2]

Knot 5_2.

A graph, knot 5_2.

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["5 2"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3849 X5,10,6,1 X9,6,10,7 X7283

In[5]:=

GaussCode[K]

Out[5]=

-1, 5, -2, 1, -3, 4, -5, 2, -4, 3

In[6]:=

DTCode[K]

Out[6]=

4 8 10 2 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[32]

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,-1,-2,1,-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]]

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, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 6}, {5, 7}, {6, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index [math]\displaystyle{ \{3,4\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-8][1] Hyperbolic Volume 2.82812 A-Polynomial See Data:5 2/A-polynomial

[edit Notes for 5 2'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{ \textrm{ConcordanceGenus}(\textrm{Knot}(5,2)) }[/math] Rasmussen s-Invariant -2

[edit Notes for 5 2's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t+2 t^{-1} -3 }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 7, -2 }
Jones polynomial	[math]\displaystyle{ - q^{-6} + q^{-5} - q^{-4} +2 q^{-3} - q^{-2} + q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^6+a^4 z^2+a^4+a^2 z^2+a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^7 z^3-2 a^7 z+a^6 z^4-2 a^6 z^2+a^6+2 a^5 z^3-2 a^5 z+a^4 z^4-a^4 z^2+a^4+a^3 z^3+a^2 z^2-a^2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{20}-q^{18}+q^{12}+q^{10}+q^8+q^6+q^2 }[/math]
The G2 invariant	[math]\displaystyle{ q^{100}+q^{96}-q^{94}-q^{92}+q^{90}-q^{88}-q^{84}-q^{82}-q^{78}-q^{76}-q^{74}-q^{72}-q^{68}-q^{66}+q^{64}+q^{60}+q^{56}+q^{54}+2 q^{50}-q^{48}+2 q^{46}+q^{44}+q^{40}+q^{34}+2 q^{24}+q^{20}+q^{14}+q^{10} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 5_2.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{13}+q^7+q^5+q }[/math]
2	[math]\displaystyle{ q^{36}-q^{32}-q^{26}-q^{20}+q^{14}+q^{10}+2 q^8+q^2 }[/math]
3	[math]\displaystyle{ -q^{69}+q^{65}+q^{63}-q^{59}+q^{55}-q^{51}+q^{47}+q^{45}-q^{43}-q^{37}-2 q^{35}-q^{33}-q^{31}+q^{27}+q^{21}+2 q^{19}-q^{15}+q^{13}+2 q^{11}+q^9+q^3 }[/math]
4	[math]\displaystyle{ q^{112}-q^{108}-q^{106}-q^{104}+q^{102}+q^{100}+q^{98}-2 q^{94}+q^{90}+q^{88}-2 q^{84}-q^{82}+2 q^{78}+q^{76}-q^{74}+q^{70}+2 q^{68}+q^{66}-q^{64}-q^{56}-2 q^{54}-q^{52}-q^{50}-q^{48}+q^{44}-q^{42}-q^{40}-q^{38}+2 q^{34}-q^{30}-q^{28}+q^{26}+4 q^{24}+q^{22}-q^{18}+2 q^{14}+q^{12}+q^{10}+q^4 }[/math]
5	[math]\displaystyle{ -q^{165}+q^{161}+q^{159}+q^{157}-q^{153}-2 q^{151}-q^{149}+q^{145}+2 q^{143}+q^{141}-q^{139}-2 q^{137}-q^{135}+q^{133}+2 q^{131}+2 q^{129}-2 q^{125}-3 q^{123}-q^{121}+q^{119}+2 q^{117}+q^{115}-2 q^{113}-2 q^{111}-q^{109}+q^{107}+3 q^{105}+q^{103}-q^{101}-q^{99}+q^{95}+2 q^{93}+q^{91}-q^{89}+q^{85}+q^{83}+q^{81}-q^{77}-q^{73}-q^{71}-q^{69}-q^{63}-2 q^{61}-2 q^{59}-2 q^{57}+2 q^{53}+q^{51}-q^{49}-2 q^{47}-2 q^{45}+q^{43}+3 q^{41}+3 q^{39}-3 q^{35}-2 q^{33}+3 q^{29}+3 q^{27}+2 q^{25}-q^{21}+q^{17}+q^{15}+q^{13}+q^{11}+q^5 }[/math]
6	[math]\displaystyle{ q^{228}-q^{224}-q^{222}-q^{220}+2 q^{214}+2 q^{212}+q^{210}-q^{206}-2 q^{204}-3 q^{202}+q^{198}+2 q^{196}+2 q^{194}+q^{192}-q^{190}-4 q^{188}-2 q^{186}+2 q^{182}+3 q^{180}+4 q^{178}+q^{176}-3 q^{174}-3 q^{172}-3 q^{170}+2 q^{166}+4 q^{164}+2 q^{162}-2 q^{160}-3 q^{158}-4 q^{156}-q^{154}+2 q^{152}+4 q^{150}+2 q^{148}-q^{146}-2 q^{144}-3 q^{142}-q^{140}+q^{138}+3 q^{136}+q^{134}-2 q^{132}-2 q^{130}-2 q^{128}+q^{124}+2 q^{122}+q^{120}-q^{118}+q^{116}+q^{114}+2 q^{112}+2 q^{110}+2 q^{108}+q^{106}-q^{98}-q^{96}-q^{94}+q^{90}-q^{88}-q^{86}-2 q^{84}-2 q^{82}-2 q^{80}+3 q^{76}+q^{74}-2 q^{70}-4 q^{68}-4 q^{66}-q^{64}+4 q^{62}+3 q^{60}+2 q^{58}-2 q^{56}-4 q^{54}-5 q^{52}-q^{50}+5 q^{48}+4 q^{46}+4 q^{44}+q^{42}-2 q^{40}-4 q^{38}-2 q^{36}+2 q^{34}+2 q^{32}+3 q^{30}+2 q^{28}+q^{26}-q^{24}+q^{20}+q^{16}+q^{14}+q^{12}+q^6 }[/math]

A2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{20}-q^{18}+q^{12}+q^{10}+q^8+q^6+q^2 }[/math]
0,2	[math]\displaystyle{ q^{50}+q^{48}+q^{46}-q^{44}-q^{42}-q^{40}-q^{38}-q^{36}-q^{34}-q^{30}-q^{28}-q^{26}+2 q^{20}+2 q^{18}+2 q^{16}+2 q^{14}+2 q^{12}+q^{10}+q^4 }[/math]
1,0	[math]\displaystyle{ -q^{20}-q^{18}+q^{12}+q^{10}+q^8+q^6+q^2 }[/math]
1,1	[math]\displaystyle{ q^{52}+2 q^{48}-2 q^{46}-2 q^{42}+2 q^{34}-2 q^{32}-5 q^{28}-4 q^{24}+2 q^{22}+q^{20}+2 q^{18}+4 q^{16}+2 q^{14}+4 q^{12}+2 q^8+q^4 }[/math]
2,0	[math]\displaystyle{ q^{50}+q^{48}+q^{46}-q^{44}-q^{42}-q^{40}-q^{38}-q^{36}-q^{34}-q^{30}-q^{28}-q^{26}+2 q^{20}+2 q^{18}+2 q^{16}+2 q^{14}+2 q^{12}+q^{10}+q^4 }[/math]
3,0	[math]\displaystyle{ -q^{90}-q^{88}-q^{86}+2 q^{82}+2 q^{80}+2 q^{78}-q^{66}+q^{62}+2 q^{60}+q^{58}-q^{54}-q^{52}-3 q^{50}-4 q^{48}-5 q^{46}-4 q^{44}-3 q^{42}-2 q^{40}+2 q^{36}+3 q^{34}+2 q^{32}+3 q^{30}+2 q^{28}+3 q^{26}+2 q^{24}+q^{22}+q^{20}+2 q^{18}+3 q^{16}+2 q^{14}+q^6 }[/math]

A3 Invariants.

Weight	Invariant
0,0,1	[math]\displaystyle{ -q^{27}-q^{25}-q^{23}+q^{17}+q^{15}+q^{13}+q^{11}+q^9+q^7+q^3 }[/math]
0,1,0	[math]\displaystyle{ q^{42}+q^{38}-2 q^{34}-2 q^{32}-2 q^{30}-2 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^4 }[/math]
1,0,0	[math]\displaystyle{ -q^{27}-q^{25}-q^{23}+q^{17}+q^{15}+q^{13}+q^{11}+q^9+q^7+q^3 }[/math]
1,0,1	[math]\displaystyle{ q^{68}+2 q^{64}-q^{60}-3 q^{56}+q^{54}-q^{52}+q^{50}+3 q^{48}-q^{46}+q^{44}-2 q^{42}-4 q^{40}-4 q^{38}-4 q^{36}-5 q^{34}-3 q^{32}-q^{30}+5 q^{26}+3 q^{24}+7 q^{22}+6 q^{20}+3 q^{18}+5 q^{16}+q^{14}+2 q^{12}+2 q^{10}+q^6 }[/math]

A4 Invariants.

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

A5 Invariants.

Weight	Invariant
0,0,0,0,1	[math]\displaystyle{ -q^{41}-q^{39}-q^{37}-q^{35}-q^{33}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^9+q^5 }[/math]
1,0,0,0,0	[math]\displaystyle{ -q^{41}-q^{39}-q^{37}-q^{35}-q^{33}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^9+q^5 }[/math]

A6 Invariants.

Weight	Invariant
0,0,0,0,0,1	[math]\displaystyle{ -q^{48}-q^{46}-q^{44}-q^{42}-q^{40}-q^{38}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^6 }[/math]
1,0,0,0,0,0	[math]\displaystyle{ -q^{48}-q^{46}-q^{44}-q^{42}-q^{40}-q^{38}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^6 }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{42}-q^{38}+q^{26}-q^{24}+q^{22}+q^{18}+q^{14}+q^{12}+q^8+q^4 }[/math]
1,0	[math]\displaystyle{ q^{68}+q^{60}-q^{56}-q^{54}-q^{52}-q^{50}-q^{48}-q^{46}-q^{44}+q^{34}+q^{32}+q^{30}+q^{26}+q^{24}+q^{22}+q^{18}+q^{16}+q^{14}+q^6 }[/math]

B3 Invariants.

Weight	Invariant
1,0,0	[math]\displaystyle{ q^{100}+q^{92}-q^{82}-q^{80}-q^{78}-q^{76}-q^{74}-q^{72}-q^{70}-q^{68}-q^{66}-q^{64}+q^{56}+q^{54}+q^{50}+q^{48}+q^{46}+q^{42}+q^{40}+q^{38}+q^{34}+q^{30}+q^{26}+q^{24}+q^{22}+q^{18}+q^{10} }[/math]

B4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{132}+q^{124}-q^{110}-q^{106}-q^{104}-q^{102}-q^{100}-q^{98}-q^{96}-q^{94}-q^{92}-q^{90}-q^{88}-q^{86}+q^{74}+q^{72}+q^{70}+q^{66}+q^{64}+q^{62}+q^{58}+q^{56}+q^{54}+q^{50}+q^{46}+q^{42}+q^{38}+q^{34}+q^{32}+q^{30}+q^{26}+q^{22}+q^{14} }[/math]

C3 Invariants.

Weight	Invariant
1,0,0	[math]\displaystyle{ -q^{58}-q^{54}-q^{48}+q^{30}+q^{26}+q^{22}+q^{20}+q^{18}+q^{16}+q^{12}+q^{10}+q^6 }[/math]

C4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ -q^{74}-q^{70}-q^{62}-q^{60}-q^{48}+q^{42}+q^{38}+q^{34}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{16}+q^{14}+q^{12}+q^8 }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{58}+q^{54}-q^{48}-2 q^{46}-2 q^{44}-2 q^{42}-2 q^{40}-2 q^{38}+2 q^{32}+q^{30}+2 q^{28}+q^{26}+2 q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+2 q^{14}+q^{12}+q^{10}+q^6 }[/math]

G2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{204}+q^{192}+q^{190}+q^{188}-q^{186}-q^{184}-q^{176}+q^{172}-2 q^{168}-q^{166}-q^{156}+q^{154}+q^{152}+q^{142}+q^{140}+q^{138}+2 q^{136}+q^{134}-q^{132}-2 q^{130}-q^{122}-q^{120}-q^{118}-2 q^{116}-2 q^{114}-3 q^{112}-2 q^{110}-q^{108}-2 q^{106}-2 q^{104}-q^{102}-q^{100}-q^{98}-q^{96}-2 q^{94}-2 q^{92}+q^{88}+q^{86}+2 q^{84}+q^{82}+q^{78}+2 q^{74}+2 q^{72}+2 q^{70}+2 q^{68}+3 q^{66}+2 q^{64}+q^{62}+q^{60}+2 q^{58}+2 q^{56}+q^{54}+q^{50}+3 q^{48}+q^{46}+q^{36}+q^{34}+q^{32}+q^{30}+q^{18} }[/math]
1,0	[math]\displaystyle{ q^{100}+q^{96}-q^{94}-q^{92}+q^{90}-q^{88}-q^{84}-q^{82}-q^{78}-q^{76}-q^{74}-q^{72}-q^{68}-q^{66}+q^{64}+q^{60}+q^{56}+q^{54}+2 q^{50}-q^{48}+2 q^{46}+q^{44}+q^{40}+q^{34}+2 q^{24}+q^{20}+q^{14}+q^{10} }[/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["5 2"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t+2 t^{-1} -3 }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 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]=

{ 7, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ - q^{-6} + q^{-5} - q^{-4} +2 q^{-3} - 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{ -a^6+a^4 z^2+a^4+a^2 z^2+a^2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^7 z^3-2 a^7 z+a^6 z^4-2 a^6 z^2+a^6+2 a^5 z^3-2 a^5 z+a^4 z^4-a^4 z^2+a^4+a^3 z^3+a^2 z^2-a^2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n57,}

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["5 2"];

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{ 2 t+2 t^{-1} -3 }[/math], [math]\displaystyle{ - q^{-6} + q^{-5} - q^{-4} +2 q^{-3} - 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]=

{K11n57,}

Vassiliev invariants

V2 and V3:

(2, -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 [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{268}{3} }[/math] [math]\displaystyle{ \frac{44}{3} }[/math] [math]\displaystyle{ -192 }[/math] [math]\displaystyle{ -368 }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ -56 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ \frac{2144}{3} }[/math] [math]\displaystyle{ \frac{352}{3} }[/math] [math]\displaystyle{ \frac{22951}{15} }[/math] [math]\displaystyle{ -\frac{28}{5} }[/math] [math]\displaystyle{ \frac{29764}{45} }[/math] [math]\displaystyle{ \frac{137}{9} }[/math] [math]\displaystyle{ \frac{1351}{15} }[/math]

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 [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 5 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 χ -1           1 1 -3         1 1 0 -5       1     1 -7       1     1 -9   1 1       0 -11             0 -13 1           -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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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^{-2} - q^{-3} +3 q^{-5} -2 q^{-6} - q^{-7} +4 q^{-8} -3 q^{-9} - q^{-10} +3 q^{-11} -2 q^{-12} - q^{-13} +2 q^{-14} - q^{-15} - q^{-16} + q^{-17} }[/math]
3	[math]\displaystyle{ q^{-3} - q^{-4} + q^{-6} +2 q^{-7} -2 q^{-8} -2 q^{-9} +2 q^{-10} +4 q^{-11} -3 q^{-12} -3 q^{-13} +2 q^{-14} +5 q^{-15} -4 q^{-16} -4 q^{-17} +2 q^{-18} +4 q^{-19} -3 q^{-20} -3 q^{-21} +2 q^{-22} +3 q^{-23} - q^{-24} -3 q^{-25} + q^{-26} +2 q^{-27} -2 q^{-29} + q^{-31} + q^{-32} - q^{-33} }[/math]
4	[math]\displaystyle{ q^{-4} - q^{-5} + q^{-7} +2 q^{-9} -3 q^{-10} - q^{-11} +2 q^{-12} + q^{-13} +5 q^{-14} -6 q^{-15} -3 q^{-16} +2 q^{-17} +2 q^{-18} +7 q^{-19} -8 q^{-20} -4 q^{-21} +2 q^{-22} +2 q^{-23} +9 q^{-24} -9 q^{-25} -5 q^{-26} +2 q^{-27} +2 q^{-28} +8 q^{-29} -8 q^{-30} -4 q^{-31} +2 q^{-32} +2 q^{-33} +7 q^{-34} -6 q^{-35} -3 q^{-36} + q^{-37} + q^{-38} +6 q^{-39} -4 q^{-40} -2 q^{-41} - q^{-42} +5 q^{-44} -2 q^{-45} - q^{-46} - q^{-47} - q^{-48} +3 q^{-49} - q^{-52} - q^{-53} + q^{-54} }[/math]
5	[math]\displaystyle{ q^{-5} - q^{-6} + q^{-8} + q^{-11} -2 q^{-12} - q^{-13} +2 q^{-14} +2 q^{-15} + q^{-16} + q^{-17} -5 q^{-18} -3 q^{-19} + q^{-20} +5 q^{-21} +4 q^{-22} + q^{-23} -7 q^{-24} -6 q^{-25} + q^{-26} +6 q^{-27} +6 q^{-28} +2 q^{-29} -9 q^{-30} -8 q^{-31} + q^{-32} +6 q^{-33} +7 q^{-34} +3 q^{-35} -9 q^{-36} -9 q^{-37} + q^{-38} +6 q^{-39} +8 q^{-40} +2 q^{-41} -8 q^{-42} -8 q^{-43} + q^{-44} +6 q^{-45} +7 q^{-46} + q^{-47} -6 q^{-48} -7 q^{-49} +5 q^{-51} +6 q^{-52} + q^{-53} -4 q^{-54} -5 q^{-55} -2 q^{-56} +3 q^{-57} +5 q^{-58} + q^{-59} - q^{-60} -4 q^{-61} -3 q^{-62} + q^{-63} +3 q^{-64} +2 q^{-65} + q^{-66} -2 q^{-67} -3 q^{-68} + q^{-70} + q^{-71} +2 q^{-72} -2 q^{-74} - q^{-75} + q^{-78} + q^{-79} - q^{-80} }[/math]
6	[math]\displaystyle{ q^{-6} - q^{-7} + q^{-9} - q^{-12} +2 q^{-13} -2 q^{-14} - q^{-15} +3 q^{-16} + q^{-17} + q^{-18} -2 q^{-19} +2 q^{-20} -6 q^{-21} -3 q^{-22} +5 q^{-23} +4 q^{-24} +4 q^{-25} -2 q^{-26} +3 q^{-27} -12 q^{-28} -7 q^{-29} +6 q^{-30} +6 q^{-31} +8 q^{-32} - q^{-33} +4 q^{-34} -17 q^{-35} -10 q^{-36} +6 q^{-37} +8 q^{-38} +10 q^{-39} +6 q^{-41} -20 q^{-42} -12 q^{-43} +6 q^{-44} +8 q^{-45} +11 q^{-46} +8 q^{-48} -21 q^{-49} -13 q^{-50} +6 q^{-51} +8 q^{-52} +12 q^{-53} +8 q^{-55} -20 q^{-56} -12 q^{-57} +6 q^{-58} +8 q^{-59} +11 q^{-60} +6 q^{-62} -18 q^{-63} -11 q^{-64} +5 q^{-65} +7 q^{-66} +9 q^{-67} +6 q^{-69} -15 q^{-70} -9 q^{-71} +3 q^{-72} +5 q^{-73} +7 q^{-74} + q^{-75} +7 q^{-76} -12 q^{-77} -7 q^{-78} + q^{-79} +2 q^{-80} +4 q^{-81} +2 q^{-82} +8 q^{-83} -8 q^{-84} -5 q^{-85} - q^{-86} + q^{-88} +2 q^{-89} +8 q^{-90} -4 q^{-91} -2 q^{-92} -2 q^{-93} - q^{-94} - q^{-95} +6 q^{-97} - q^{-98} - q^{-100} - q^{-101} -2 q^{-102} - q^{-103} +3 q^{-104} + q^{-106} - q^{-109} - q^{-110} + q^{-111} }[/math]
7	[math]\displaystyle{ q^{-7} - q^{-8} + q^{-10} - q^{-13} +2 q^{-15} -2 q^{-16} +2 q^{-18} + q^{-19} + q^{-20} -2 q^{-21} -2 q^{-22} +2 q^{-23} -5 q^{-24} +4 q^{-26} +4 q^{-27} +5 q^{-28} -2 q^{-29} -4 q^{-30} -2 q^{-31} -10 q^{-32} -2 q^{-33} +6 q^{-34} +6 q^{-35} +12 q^{-36} +2 q^{-37} -6 q^{-38} -5 q^{-39} -17 q^{-40} -5 q^{-41} +6 q^{-42} +9 q^{-43} +18 q^{-44} +5 q^{-45} -6 q^{-46} -7 q^{-47} -22 q^{-48} -9 q^{-49} +7 q^{-50} +10 q^{-51} +21 q^{-52} +7 q^{-53} -5 q^{-54} -6 q^{-55} -25 q^{-56} -11 q^{-57} +7 q^{-58} +10 q^{-59} +23 q^{-60} +7 q^{-61} -4 q^{-62} -5 q^{-63} -26 q^{-64} -12 q^{-65} +7 q^{-66} +9 q^{-67} +24 q^{-68} +7 q^{-69} -3 q^{-70} -6 q^{-71} -25 q^{-72} -11 q^{-73} +7 q^{-74} +9 q^{-75} +23 q^{-76} +7 q^{-77} -4 q^{-78} -7 q^{-79} -23 q^{-80} -10 q^{-81} +6 q^{-82} +8 q^{-83} +21 q^{-84} +7 q^{-85} -5 q^{-86} -6 q^{-87} -20 q^{-88} -8 q^{-89} +4 q^{-90} +6 q^{-91} +18 q^{-92} +7 q^{-93} -3 q^{-94} -4 q^{-95} -16 q^{-96} -7 q^{-97} +2 q^{-98} +2 q^{-99} +14 q^{-100} +8 q^{-101} - q^{-102} - q^{-103} -12 q^{-104} -6 q^{-105} - q^{-106} -2 q^{-107} +10 q^{-108} +7 q^{-109} + q^{-110} +2 q^{-111} -6 q^{-112} -5 q^{-113} -3 q^{-114} -5 q^{-115} +6 q^{-116} +5 q^{-117} + q^{-118} +5 q^{-119} -2 q^{-120} -2 q^{-121} -3 q^{-122} -6 q^{-123} +2 q^{-124} +2 q^{-125} +4 q^{-127} + q^{-128} + q^{-129} - q^{-130} -5 q^{-131} - q^{-134} +2 q^{-135} + q^{-136} +2 q^{-137} + q^{-138} -2 q^{-139} - q^{-140} - q^{-142} + q^{-145} + q^{-146} - q^{-147} }[/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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