9 7

9_6

9_8

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 7's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 7 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3,12,4,13 X5,16,6,17 X7,18,8,1 X17,6,18,7 X9,14,10,15 X13,10,14,11 X15,8,16,9 X11,2,12,3
Gauss code	-1, 9, -2, 1, -3, 5, -4, 8, -6, 7, -9, 2, -7, 6, -8, 3, -5, 4
Dowker-Thistlethwaite code	4 12 16 18 14 2 10 8 6
Conway Notation	[342]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 11, width is 4, Braid index is 4		[{11, 2}, {1, 9}, {8, 10}, {9, 11}, {10, 7}, {6, 8}, {7, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 1}]

[edit Notes on presentations of 9 7]

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 7"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3,12,4,13 X5,16,6,17 X7,18,8,1 X17,6,18,7 X9,14,10,15 X13,10,14,11 X15,8,16,9 X11,2,12,3

In[5]:=

GaussCode[K]

Out[5]=

-1, 9, -2, 1, -3, 5, -4, 8, -6, 7, -9, 2, -7, 6, -8, 3, -5, 4

In[6]:=

DTCode[K]

Out[6]=

4 12 16 18 14 2 10 8 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[342]

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,-1,-2,1,-2,-3,2,-3,-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]]

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[{11, 2}, {1, 9}, {8, 10}, {9, 11}, {10, 7}, {6, 8}, {7, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-14][3] Hyperbolic Volume 8.01486 A-Polynomial See Data:9 7/A-polynomial

[edit Notes for 9 7'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{ 2 }[/math] Rasmussen s-Invariant -4

[edit Notes for 9 7's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 3 t^2-7 t+9-7 t^{-1} +3 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 3 z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 29, -4 }
Jones polynomial	[math]\displaystyle{ q^{-2} - q^{-3} +3 q^{-4} -4 q^{-5} +5 q^{-6} -5 q^{-7} +4 q^{-8} -3 q^{-9} +2 q^{-10} - q^{-11} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^{10}-a^{10}+z^4 a^8+2 z^2 a^8+a^8+z^4 a^6+z^2 a^6-a^6+z^4 a^4+3 z^2 a^4+2 a^4 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^5 a^{13}-3 z^3 a^{13}+z a^{13}+2 z^6 a^{12}-6 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-6 z^5 a^{11}+5 z^3 a^{11}-2 z a^{11}+z^8 a^{10}-2 z^6 a^{10}+2 z^4 a^{10}-2 z^2 a^{10}+a^{10}+3 z^7 a^9-9 z^5 a^9+11 z^3 a^9-3 z a^9+z^8 a^8-3 z^6 a^8+7 z^4 a^8-4 z^2 a^8+a^8+z^7 a^7-z^5 a^7+2 z^3 a^7-z a^7+z^6 a^6-2 z^2 a^6+a^6+z^5 a^5-z^3 a^5-z a^5+z^4 a^4-3 z^2 a^4+2 a^4 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{34}-q^{28}+q^{26}-q^{18}+q^{16}+q^{12}+2 q^{10}+q^6 }[/math]
The G2 invariant	[math]\displaystyle{ q^{176}-q^{174}+2 q^{172}-3 q^{170}+2 q^{168}-q^{166}-2 q^{164}+7 q^{162}-8 q^{160}+9 q^{158}-7 q^{156}+6 q^{152}-12 q^{150}+13 q^{148}-11 q^{146}+4 q^{144}+4 q^{142}-10 q^{140}+10 q^{138}-6 q^{136}+5 q^{132}-9 q^{130}+5 q^{128}-7 q^{124}+11 q^{122}-12 q^{120}+8 q^{118}-q^{116}-8 q^{114}+13 q^{112}-16 q^{110}+15 q^{108}-7 q^{106}-q^{104}+9 q^{102}-13 q^{100}+14 q^{98}-7 q^{96}+q^{94}+5 q^{92}-7 q^{90}+5 q^{88}+q^{86}-6 q^{84}+8 q^{82}-7 q^{80}+4 q^{76}-9 q^{74}+10 q^{72}-8 q^{70}+4 q^{68}-q^{66}-4 q^{64}+6 q^{62}-6 q^{60}+7 q^{58}-3 q^{56}+3 q^{54}+q^{52}-q^{50}+4 q^{48}-3 q^{46}+4 q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+2 q^{34}+q^{30} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_7.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{23}+q^{21}-q^{19}+q^{17}-q^{15}+q^{11}-q^9+2 q^7+q^3 }[/math]
2	[math]\displaystyle{ q^{64}-q^{62}-q^{60}+3 q^{58}-q^{56}-4 q^{54}+3 q^{52}+2 q^{50}-4 q^{48}+2 q^{46}+3 q^{44}-4 q^{42}+2 q^{38}-q^{36}-2 q^{34}+3 q^{30}-4 q^{28}-2 q^{26}+5 q^{24}-2 q^{22}-2 q^{20}+4 q^{18}+2 q^{12}+q^6 }[/math]
3	[math]\displaystyle{ -q^{123}+q^{121}+q^{119}-q^{117}-2 q^{115}+q^{113}+5 q^{111}-7 q^{107}-3 q^{105}+6 q^{103}+7 q^{101}-4 q^{99}-10 q^{97}+q^{95}+10 q^{93}+4 q^{91}-10 q^{89}-7 q^{87}+9 q^{85}+9 q^{83}-6 q^{81}-9 q^{79}+5 q^{77}+8 q^{75}-3 q^{73}-8 q^{71}+6 q^{67}+2 q^{65}-3 q^{63}-6 q^{61}+3 q^{59}+9 q^{57}-12 q^{53}-4 q^{51}+10 q^{49}+6 q^{47}-11 q^{45}-7 q^{43}+5 q^{41}+7 q^{39}-2 q^{37}-5 q^{35}+q^{33}+2 q^{31}+q^{29}+q^{25}+q^{21}+q^{19}+2 q^{17}+q^9 }[/math]
4	[math]\displaystyle{ q^{200}-q^{198}-q^{196}+q^{194}+2 q^{190}-3 q^{188}-3 q^{186}+2 q^{184}+2 q^{182}+9 q^{180}-3 q^{178}-11 q^{176}-5 q^{174}-q^{172}+18 q^{170}+9 q^{168}-5 q^{166}-13 q^{164}-20 q^{162}+9 q^{160}+18 q^{158}+16 q^{156}-q^{154}-31 q^{152}-16 q^{150}+5 q^{148}+32 q^{146}+26 q^{144}-21 q^{142}-32 q^{140}-18 q^{138}+26 q^{136}+40 q^{134}-3 q^{132}-30 q^{130}-30 q^{128}+13 q^{126}+36 q^{124}+6 q^{122}-19 q^{120}-25 q^{118}+4 q^{116}+24 q^{114}+10 q^{112}-10 q^{110}-17 q^{108}-3 q^{106}+11 q^{104}+16 q^{102}+q^{100}-11 q^{98}-18 q^{96}-4 q^{94}+27 q^{92}+18 q^{90}-4 q^{88}-33 q^{86}-27 q^{84}+25 q^{82}+35 q^{80}+17 q^{78}-31 q^{76}-44 q^{74}+6 q^{72}+28 q^{70}+31 q^{68}-8 q^{66}-35 q^{64}-12 q^{62}+5 q^{60}+24 q^{58}+9 q^{56}-13 q^{54}-9 q^{52}-8 q^{50}+8 q^{48}+9 q^{46}-q^{42}-7 q^{40}+q^{38}+4 q^{36}+2 q^{34}+3 q^{32}-3 q^{30}+q^{26}+q^{24}+3 q^{22}+q^{12} }[/math]
5	[math]\displaystyle{ -q^{295}+q^{293}+q^{291}-q^{289}+q^{281}+2 q^{279}-2 q^{277}-5 q^{275}-2 q^{273}+q^{271}+6 q^{269}+9 q^{267}+5 q^{265}-9 q^{263}-17 q^{261}-11 q^{259}+q^{257}+19 q^{255}+24 q^{253}+13 q^{251}-12 q^{249}-30 q^{247}-27 q^{245}-9 q^{243}+20 q^{241}+40 q^{239}+35 q^{237}+2 q^{235}-35 q^{233}-54 q^{231}-39 q^{229}+11 q^{227}+61 q^{225}+75 q^{223}+27 q^{221}-51 q^{219}-96 q^{217}-73 q^{215}+18 q^{213}+105 q^{211}+112 q^{209}+21 q^{207}-92 q^{205}-132 q^{203}-62 q^{201}+67 q^{199}+140 q^{197}+92 q^{195}-40 q^{193}-132 q^{191}-105 q^{189}+11 q^{187}+112 q^{185}+107 q^{183}+6 q^{181}-91 q^{179}-95 q^{177}-15 q^{175}+67 q^{173}+81 q^{171}+19 q^{169}-50 q^{167}-63 q^{165}-18 q^{163}+35 q^{161}+48 q^{159}+20 q^{157}-20 q^{155}-44 q^{153}-26 q^{151}+8 q^{149}+35 q^{147}+43 q^{145}+12 q^{143}-36 q^{141}-62 q^{139}-36 q^{137}+31 q^{135}+86 q^{133}+72 q^{131}-15 q^{129}-102 q^{127}-111 q^{125}-14 q^{123}+111 q^{121}+142 q^{119}+50 q^{117}-93 q^{115}-160 q^{113}-92 q^{111}+65 q^{109}+157 q^{107}+110 q^{105}-18 q^{103}-127 q^{101}-124 q^{99}-19 q^{97}+87 q^{95}+109 q^{93}+40 q^{91}-44 q^{89}-84 q^{87}-53 q^{85}+11 q^{83}+55 q^{81}+45 q^{79}+5 q^{77}-27 q^{75}-35 q^{73}-14 q^{71}+13 q^{69}+22 q^{67}+13 q^{65}-3 q^{63}-15 q^{61}-11 q^{59}+q^{57}+7 q^{55}+8 q^{53}+4 q^{51}-6 q^{49}-5 q^{47}+2 q^{43}+4 q^{41}+5 q^{39}-q^{37}-2 q^{35}+q^{29}+3 q^{27}+q^{25}+q^{15} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{34}-q^{28}+q^{26}-q^{18}+q^{16}+q^{12}+2 q^{10}+q^6 }[/math]
1,1	[math]\displaystyle{ q^{92}-2 q^{90}+4 q^{88}-8 q^{86}+15 q^{84}-20 q^{82}+26 q^{80}-34 q^{78}+35 q^{76}-34 q^{74}+28 q^{72}-16 q^{70}+3 q^{68}+14 q^{66}-30 q^{64}+42 q^{62}-53 q^{60}+58 q^{58}-58 q^{56}+56 q^{54}-46 q^{52}+36 q^{50}-22 q^{48}+6 q^{46}+q^{44}-18 q^{42}+18 q^{40}-24 q^{38}+21 q^{36}-20 q^{34}+20 q^{32}-14 q^{30}+16 q^{28}-10 q^{26}+12 q^{24}-6 q^{22}+8 q^{20}-2 q^{18}+4 q^{16}+q^{12} }[/math]
2,0	[math]\displaystyle{ q^{86}+q^{78}-3 q^{74}-q^{72}+q^{70}+q^{68}-q^{66}+3 q^{62}+q^{60}-2 q^{58}-q^{56}+q^{54}-q^{52}-q^{50}-q^{46}-3 q^{44}-2 q^{42}-3 q^{38}-q^{36}+4 q^{34}+2 q^{32}-q^{30}+2 q^{28}+4 q^{26}+2 q^{24}-q^{22}+2 q^{20}+2 q^{18}+q^{12} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{74}-q^{72}+q^{68}-2 q^{66}+2 q^{64}+2 q^{62}-3 q^{60}+2 q^{58}+q^{56}-5 q^{54}-q^{52}+q^{50}-3 q^{48}-q^{46}+q^{44}+q^{42}+4 q^{36}-2 q^{34}-3 q^{32}+3 q^{30}-2 q^{28}-3 q^{26}+4 q^{24}+2 q^{22}+q^{20}+3 q^{18}+2 q^{16}+q^{12} }[/math]
1,0,0	[math]\displaystyle{ -q^{45}-q^{41}-q^{37}+q^{35}+q^{31}-q^{25}-q^{23}+q^{21}+2 q^{17}+q^{15}+2 q^{13}+q^9 }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{96}-q^{92}+q^{90}+q^{88}-2 q^{86}+4 q^{82}+2 q^{80}-2 q^{78}+2 q^{74}-3 q^{72}-6 q^{70}-q^{68}-2 q^{66}-5 q^{64}-q^{62}+q^{60}-q^{58}+4 q^{54}+3 q^{52}+q^{48}+3 q^{46}-2 q^{44}-5 q^{42}-q^{40}-q^{36}+3 q^{32}+5 q^{30}+3 q^{28}+3 q^{26}+3 q^{24}+2 q^{22}+q^{18} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{56}-q^{52}-q^{50}-q^{46}+q^{44}+q^{40}+q^{38}-q^{32}-q^{30}-q^{28}+q^{26}+2 q^{22}+2 q^{20}+q^{18}+2 q^{16}+q^{12} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{74}+q^{72}-2 q^{70}+3 q^{68}-4 q^{66}+4 q^{64}-4 q^{62}+3 q^{60}-2 q^{58}+q^{56}+q^{54}-3 q^{52}+5 q^{50}-7 q^{48}+7 q^{46}-7 q^{44}+7 q^{42}-6 q^{40}+4 q^{38}-2 q^{36}+q^{32}-3 q^{30}+4 q^{28}-3 q^{26}+4 q^{24}-2 q^{22}+3 q^{20}-q^{18}+2 q^{16}+q^{12} }[/math]
1,0	[math]\displaystyle{ q^{120}-q^{116}-q^{114}+q^{112}+2 q^{110}-q^{108}-3 q^{106}+4 q^{102}+3 q^{100}-3 q^{98}-4 q^{96}+q^{94}+4 q^{92}+q^{90}-4 q^{88}-3 q^{86}+q^{84}+2 q^{82}-q^{80}-3 q^{78}+2 q^{74}-3 q^{70}-q^{68}+3 q^{66}+2 q^{64}-2 q^{62}-2 q^{60}+2 q^{58}+3 q^{56}-q^{54}-4 q^{52}-q^{50}+4 q^{48}+2 q^{46}-3 q^{44}-3 q^{42}+4 q^{38}+2 q^{36}-q^{32}+q^{30}+2 q^{28}+2 q^{26}+q^{18} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{102}-q^{100}+q^{98}-2 q^{96}+3 q^{94}-3 q^{92}+3 q^{90}-3 q^{88}+4 q^{86}-2 q^{84}+2 q^{82}-q^{80}-q^{76}-4 q^{74}+q^{72}-5 q^{70}+3 q^{68}-7 q^{66}+5 q^{64}-5 q^{62}+7 q^{60}-4 q^{58}+5 q^{56}-2 q^{54}+4 q^{52}-q^{46}-3 q^{44}+2 q^{42}-4 q^{40}+q^{38}-3 q^{36}+4 q^{34}+4 q^{30}+q^{28}+4 q^{26}+q^{24}+2 q^{22}+q^{18} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{176}-q^{174}+2 q^{172}-3 q^{170}+2 q^{168}-q^{166}-2 q^{164}+7 q^{162}-8 q^{160}+9 q^{158}-7 q^{156}+6 q^{152}-12 q^{150}+13 q^{148}-11 q^{146}+4 q^{144}+4 q^{142}-10 q^{140}+10 q^{138}-6 q^{136}+5 q^{132}-9 q^{130}+5 q^{128}-7 q^{124}+11 q^{122}-12 q^{120}+8 q^{118}-q^{116}-8 q^{114}+13 q^{112}-16 q^{110}+15 q^{108}-7 q^{106}-q^{104}+9 q^{102}-13 q^{100}+14 q^{98}-7 q^{96}+q^{94}+5 q^{92}-7 q^{90}+5 q^{88}+q^{86}-6 q^{84}+8 q^{82}-7 q^{80}+4 q^{76}-9 q^{74}+10 q^{72}-8 q^{70}+4 q^{68}-q^{66}-4 q^{64}+6 q^{62}-6 q^{60}+7 q^{58}-3 q^{56}+3 q^{54}+q^{52}-q^{50}+4 q^{48}-3 q^{46}+4 q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+2 q^{34}+q^{30} }[/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 7"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 3 t^2-7 t+9-7 t^{-1} +3 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 3 z^4+5 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]=

{ 29, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{-2} - q^{-3} +3 q^{-4} -4 q^{-5} +5 q^{-6} -5 q^{-7} +4 q^{-8} -3 q^{-9} +2 q^{-10} - q^{-11} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^2 a^{10}-a^{10}+z^4 a^8+2 z^2 a^8+a^8+z^4 a^6+z^2 a^6-a^6+z^4 a^4+3 z^2 a^4+2 a^4 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^5 a^{13}-3 z^3 a^{13}+z a^{13}+2 z^6 a^{12}-6 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-6 z^5 a^{11}+5 z^3 a^{11}-2 z a^{11}+z^8 a^{10}-2 z^6 a^{10}+2 z^4 a^{10}-2 z^2 a^{10}+a^{10}+3 z^7 a^9-9 z^5 a^9+11 z^3 a^9-3 z a^9+z^8 a^8-3 z^6 a^8+7 z^4 a^8-4 z^2 a^8+a^8+z^7 a^7-z^5 a^7+2 z^3 a^7-z a^7+z^6 a^6-2 z^2 a^6+a^6+z^5 a^5-z^3 a^5-z a^5+z^4 a^4-3 z^2 a^4+2 a^4 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["9 7"];

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{ 3 t^2-7 t+9-7 t^{-1} +3 t^{-2} }[/math], [math]\displaystyle{ q^{-2} - q^{-3} +3 q^{-4} -4 q^{-5} +5 q^{-6} -5 q^{-7} +4 q^{-8} -3 q^{-9} +2 q^{-10} - q^{-11} }[/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]=

{}

Vassiliev invariants

V2 and V3:

(5, -12)

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{ 20 }[/math] [math]\displaystyle{ -96 }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ \frac{1654}{3} }[/math] [math]\displaystyle{ \frac{218}{3} }[/math] [math]\displaystyle{ -1920 }[/math] [math]\displaystyle{ -3552 }[/math] [math]\displaystyle{ -576 }[/math] [math]\displaystyle{ -448 }[/math] [math]\displaystyle{ \frac{4000}{3} }[/math] [math]\displaystyle{ 4608 }[/math] [math]\displaystyle{ \frac{33080}{3} }[/math] [math]\displaystyle{ \frac{4360}{3} }[/math] [math]\displaystyle{ \frac{140719}{6} }[/math] [math]\displaystyle{ \frac{1954}{3} }[/math] [math]\displaystyle{ \frac{77486}{9} }[/math] [math]\displaystyle{ \frac{4789}{18} }[/math] [math]\displaystyle{ \frac{6415}{6} }[/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]-4 is the signature of 9 7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                   1 1 -5                 1 1 0 -7               2     2 -9             2 1     -1 -11           3 2       1 -13         2 2         0 -15       2 3           -1 -17     1 2             1 -19   1 2               -1 -21   1                 1 -23 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/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=-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^{-4} - q^{-5} +3 q^{-7} -3 q^{-8} +7 q^{-10} -9 q^{-11} +14 q^{-13} -16 q^{-14} -2 q^{-15} +21 q^{-16} -19 q^{-17} -4 q^{-18} +22 q^{-19} -16 q^{-20} -6 q^{-21} +18 q^{-22} -9 q^{-23} -7 q^{-24} +12 q^{-25} -3 q^{-26} -6 q^{-27} +5 q^{-28} -2 q^{-30} + q^{-31} }[/math]
3	[math]\displaystyle{ q^{-6} - q^{-7} +3 q^{-10} -2 q^{-11} - q^{-13} +4 q^{-14} -3 q^{-15} + q^{-16} +3 q^{-18} -9 q^{-19} +4 q^{-20} +9 q^{-21} + q^{-22} -21 q^{-23} +26 q^{-25} +5 q^{-26} -35 q^{-27} -8 q^{-28} +38 q^{-29} +14 q^{-30} -41 q^{-31} -17 q^{-32} +41 q^{-33} +19 q^{-34} -37 q^{-35} -23 q^{-36} +33 q^{-37} +24 q^{-38} -26 q^{-39} -26 q^{-40} +19 q^{-41} +27 q^{-42} -11 q^{-43} -26 q^{-44} +3 q^{-45} +24 q^{-46} +3 q^{-47} -20 q^{-48} -6 q^{-49} +13 q^{-50} +9 q^{-51} -9 q^{-52} -7 q^{-53} +4 q^{-54} +5 q^{-55} -2 q^{-56} -2 q^{-57} +2 q^{-59} - q^{-60} }[/math]
4	[math]\displaystyle{ q^{-8} - q^{-9} +4 q^{-13} -3 q^{-14} - q^{-16} -3 q^{-17} +10 q^{-18} -4 q^{-19} +2 q^{-20} -4 q^{-21} -11 q^{-22} +16 q^{-23} -3 q^{-24} +11 q^{-25} -5 q^{-26} -27 q^{-27} +15 q^{-28} -7 q^{-29} +33 q^{-30} +10 q^{-31} -46 q^{-32} -2 q^{-33} -30 q^{-34} +60 q^{-35} +49 q^{-36} -49 q^{-37} -24 q^{-38} -80 q^{-39} +73 q^{-40} +97 q^{-41} -31 q^{-42} -34 q^{-43} -132 q^{-44} +67 q^{-45} +126 q^{-46} -9 q^{-47} -25 q^{-48} -163 q^{-49} +53 q^{-50} +133 q^{-51} +3 q^{-52} -10 q^{-53} -168 q^{-54} +39 q^{-55} +119 q^{-56} +10 q^{-57} +10 q^{-58} -154 q^{-59} +19 q^{-60} +90 q^{-61} +16 q^{-62} +35 q^{-63} -124 q^{-64} -4 q^{-65} +47 q^{-66} +16 q^{-67} +62 q^{-68} -81 q^{-69} -18 q^{-70} +3 q^{-71} +2 q^{-72} +73 q^{-73} -34 q^{-74} -12 q^{-75} -24 q^{-76} -19 q^{-77} +58 q^{-78} -4 q^{-79} +5 q^{-80} -22 q^{-81} -28 q^{-82} +29 q^{-83} +3 q^{-84} +13 q^{-85} -8 q^{-86} -19 q^{-87} +10 q^{-88} - q^{-89} +7 q^{-90} -7 q^{-92} +3 q^{-93} - q^{-94} +2 q^{-95} -2 q^{-97} + q^{-98} }[/math]
5	[math]\displaystyle{ q^{-10} - q^{-11} + q^{-15} +3 q^{-16} -3 q^{-17} - q^{-18} -2 q^{-20} +2 q^{-21} +9 q^{-22} -4 q^{-23} -3 q^{-24} -2 q^{-25} -7 q^{-26} + q^{-27} +19 q^{-28} -4 q^{-30} -8 q^{-31} -19 q^{-32} -3 q^{-33} +31 q^{-34} +16 q^{-35} +5 q^{-36} -17 q^{-37} -46 q^{-38} -24 q^{-39} +39 q^{-40} +48 q^{-41} +45 q^{-42} -7 q^{-43} -90 q^{-44} -88 q^{-45} +8 q^{-46} +88 q^{-47} +129 q^{-48} +62 q^{-49} -112 q^{-50} -194 q^{-51} -97 q^{-52} +85 q^{-53} +238 q^{-54} +190 q^{-55} -65 q^{-56} -286 q^{-57} -254 q^{-58} +17 q^{-59} +305 q^{-60} +333 q^{-61} +27 q^{-62} -317 q^{-63} -379 q^{-64} -80 q^{-65} +314 q^{-66} +420 q^{-67} +114 q^{-68} -303 q^{-69} -434 q^{-70} -147 q^{-71} +288 q^{-72} +446 q^{-73} +162 q^{-74} -272 q^{-75} -442 q^{-76} -174 q^{-77} +254 q^{-78} +428 q^{-79} +186 q^{-80} -232 q^{-81} -414 q^{-82} -187 q^{-83} +201 q^{-84} +383 q^{-85} +199 q^{-86} -163 q^{-87} -352 q^{-88} -201 q^{-89} +119 q^{-90} +303 q^{-91} +203 q^{-92} -66 q^{-93} -251 q^{-94} -196 q^{-95} +18 q^{-96} +187 q^{-97} +176 q^{-98} +26 q^{-99} -119 q^{-100} -148 q^{-101} -55 q^{-102} +58 q^{-103} +106 q^{-104} +66 q^{-105} -6 q^{-106} -57 q^{-107} -62 q^{-108} -29 q^{-109} +15 q^{-110} +43 q^{-111} +39 q^{-112} +21 q^{-113} -14 q^{-114} -43 q^{-115} -35 q^{-116} -7 q^{-117} +24 q^{-118} +40 q^{-119} +23 q^{-120} -10 q^{-121} -30 q^{-122} -27 q^{-123} -5 q^{-124} +22 q^{-125} +20 q^{-126} +8 q^{-127} -5 q^{-128} -16 q^{-129} -10 q^{-130} +4 q^{-131} +8 q^{-132} +2 q^{-133} +3 q^{-134} -2 q^{-135} -6 q^{-136} + q^{-137} +3 q^{-138} - q^{-139} + q^{-141} -2 q^{-142} +2 q^{-144} - q^{-145} }[/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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