9 23

9_22

9_24

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 23's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 23 at Knotilus!

Symmetrical decorative knot	With crossings on 3x3 grid	Depiction with two axes of symmetry	Mongolian ornament (two crossings are unnecessary)
Mongolian ornament, sum of two 9.23	Logo of the ICMC-USP, Brazil	Other depicture with central symmetry by Alain Esculier

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X5,12,6,13 X7,16,8,17 X13,18,14,1 X17,14,18,15 X15,6,16,7 X11,8,12,9 X9,2,10,3
Gauss code	-1, 9, -2, 1, -3, 7, -4, 8, -9, 2, -8, 3, -5, 6, -7, 4, -6, 5
Dowker-Thistlethwaite code	4 10 12 16 2 8 18 6 14
Conway Notation	[22122]

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

[edit Notes on presentations of 9 23]

Computer Talk[show]

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 23"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 12 16 2 8 18 6 14

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[22122]

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]=

${\textrm {BR}}(4,\{-1,-1,-1,-2,1,-2,-2,-3,2,-3,-3\})$

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, 4}, {3, 9}, {8, 10}, {9, 11}, {10, 5}, {4, 6}, {5, 2}, {1, 3}, {2, 7}, {6, 8}, {7, 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 $\{4,7\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-14][3] Hyperbolic Volume 10.6113 A-Polynomial See Data:9 23/A-polynomial

[edit Notes for 9 23's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $2$ Rasmussen s-Invariant -4

[edit Notes for 9 23's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$4t^{2}-11t+15-11t^{-1}+4t^{-2}$
Conway polynomial	$4z^{4}+5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 45, -4 }
Jones polynomial	$q^{-2}-2q^{-3}+5q^{-4}-6q^{-5}+8q^{-6}-8q^{-7}+6q^{-8}-5q^{-9}+3q^{-10}-q^{-11}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{10}+z^{4}a^{8}-2a^{8}+2z^{4}a^{6}+4z^{2}a^{6}+2a^{6}+z^{4}a^{4}+2z^{2}a^{4}+a^{4}$
Kauffman polynomial (db, data sources)	$z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-7z^{4}a^{12}+3z^{2}a^{12}+3z^{7}a^{11}-5z^{5}a^{11}+za^{11}+z^{8}a^{10}+4z^{6}a^{10}-10z^{4}a^{10}+3z^{2}a^{10}+5z^{7}a^{9}-6z^{5}a^{9}-2z^{3}a^{9}+4za^{9}+z^{8}a^{8}+4z^{6}a^{8}-8z^{4}a^{8}+6z^{2}a^{8}-2a^{8}+2z^{7}a^{7}+2z^{5}a^{7}-6z^{3}a^{7}+4za^{7}+3z^{6}a^{6}-4z^{4}a^{6}+4z^{2}a^{6}-2a^{6}+2z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-2z^{2}a^{4}+a^{4}$
The A2 invariant	$-q^{34}+q^{32}+q^{30}-2q^{28}-2q^{24}-q^{22}+q^{20}+3q^{16}+q^{12}+2q^{10}-q^{8}+q^{6}$
The G2 invariant	$q^{176}-2q^{174}+5q^{172}-8q^{170}+7q^{168}-3q^{166}-6q^{164}+20q^{162}-27q^{160}+30q^{158}-24q^{156}+q^{154}+23q^{152}-47q^{150}+55q^{148}-43q^{146}+18q^{144}+14q^{142}-38q^{140}+48q^{138}-38q^{136}+13q^{134}+12q^{132}-30q^{130}+30q^{128}-10q^{126}-16q^{124}+40q^{122}-43q^{120}+36q^{118}-13q^{116}-28q^{114}+56q^{112}-73q^{110}+66q^{108}-39q^{106}-6q^{104}+43q^{102}-64q^{100}+61q^{98}-44q^{96}+7q^{94}+20q^{92}-36q^{90}+30q^{88}-7q^{86}-16q^{84}+35q^{82}-29q^{80}+11q^{78}+13q^{76}-34q^{74}+47q^{72}-41q^{70}+28q^{68}-2q^{66}-19q^{64}+35q^{62}-35q^{60}+31q^{58}-17q^{56}+4q^{54}+7q^{52}-15q^{50}+16q^{48}-12q^{46}+9q^{44}-2q^{42}-q^{40}+3q^{38}-3q^{36}+3q^{34}-q^{32}+q^{30}$

Further Quantum Invariants[show]

Further quantum knot invariants for 9_23.

A1 Invariants.

Weight	Invariant
1	$-q^{23}+2q^{21}-2q^{19}+q^{17}-2q^{15}+2q^{11}-q^{9}+3q^{7}-q^{5}+q^{3}$
2	$q^{64}-2q^{62}-2q^{60}+7q^{58}-q^{56}-9q^{54}+9q^{52}+3q^{50}-13q^{48}+6q^{46}+7q^{44}-10q^{42}+2q^{40}+7q^{38}-3q^{36}-6q^{34}+q^{32}+7q^{30}-10q^{28}-4q^{26}+14q^{24}-6q^{22}-6q^{20}+11q^{18}-2q^{16}-3q^{14}+5q^{12}-q^{8}+q^{6}$
3	$-q^{123}+2q^{121}+2q^{119}-3q^{117}-7q^{115}+q^{113}+16q^{111}+4q^{109}-21q^{107}-15q^{105}+22q^{103}+29q^{101}-20q^{99}-41q^{97}+10q^{95}+49q^{93}+3q^{91}-50q^{89}-16q^{87}+50q^{85}+24q^{83}-41q^{81}-30q^{79}+32q^{77}+32q^{75}-23q^{73}-32q^{71}+8q^{69}+33q^{67}+5q^{65}-24q^{63}-25q^{61}+21q^{59}+38q^{57}-10q^{55}-51q^{53}-3q^{51}+53q^{49}+14q^{47}-52q^{45}-23q^{43}+40q^{41}+24q^{39}-28q^{37}-23q^{35}+20q^{33}+16q^{31}-9q^{29}-10q^{27}+8q^{25}+7q^{23}-3q^{21}-3q^{19}+3q^{17}+2q^{15}-q^{11}+q^{9}$
4	$q^{200}-2q^{198}-2q^{196}+3q^{194}+3q^{192}+7q^{190}-8q^{188}-16q^{186}-4q^{184}+8q^{182}+42q^{180}+8q^{178}-38q^{176}-47q^{174}-26q^{172}+80q^{170}+74q^{168}-2q^{166}-91q^{164}-129q^{162}+48q^{160}+141q^{158}+111q^{156}-53q^{154}-222q^{152}-72q^{150}+115q^{148}+222q^{146}+68q^{144}-223q^{142}-187q^{140}+12q^{138}+241q^{136}+171q^{134}-143q^{132}-218q^{130}-80q^{128}+186q^{126}+200q^{124}-53q^{122}-179q^{120}-116q^{118}+108q^{116}+175q^{114}+23q^{112}-124q^{110}-133q^{108}+16q^{106}+128q^{104}+114q^{102}-40q^{100}-145q^{98}-109q^{96}+55q^{94}+212q^{92}+80q^{90}-120q^{88}-226q^{86}-64q^{84}+234q^{82}+193q^{80}-21q^{78}-254q^{76}-175q^{74}+153q^{72}+208q^{70}+81q^{68}-167q^{66}-188q^{64}+43q^{62}+122q^{60}+103q^{58}-58q^{56}-115q^{54}-3q^{52}+36q^{50}+61q^{48}-8q^{46}-44q^{44}+4q^{40}+22q^{38}-2q^{36}-13q^{34}+6q^{32}+7q^{28}-q^{26}-4q^{24}+3q^{22}+2q^{18}-q^{14}+q^{12}$
5	$-q^{295}+2q^{293}+2q^{291}-3q^{289}-3q^{287}-3q^{285}+8q^{281}+16q^{279}+4q^{277}-17q^{275}-29q^{273}-26q^{271}+8q^{269}+54q^{267}+74q^{265}+19q^{263}-70q^{261}-124q^{259}-94q^{257}+38q^{255}+183q^{253}+209q^{251}+46q^{249}-195q^{247}-327q^{245}-217q^{243}+118q^{241}+429q^{239}+433q^{237}+53q^{235}-435q^{233}-637q^{231}-334q^{229}+320q^{227}+792q^{225}+648q^{223}-82q^{221}-822q^{219}-937q^{217}-253q^{215}+725q^{213}+1141q^{211}+595q^{209}-506q^{207}-1226q^{205}-894q^{203}+234q^{201}+1175q^{199}+1097q^{197}+55q^{195}-1039q^{193}-1188q^{191}-273q^{189}+841q^{187}+1168q^{185}+438q^{183}-642q^{181}-1082q^{179}-522q^{177}+464q^{175}+946q^{173}+545q^{171}-302q^{169}-814q^{167}-563q^{165}+174q^{163}+686q^{161}+565q^{159}-13q^{157}-553q^{155}-629q^{153}-153q^{151}+432q^{149}+669q^{147}+394q^{145}-249q^{143}-752q^{141}-648q^{139}+33q^{137}+765q^{135}+919q^{133}+263q^{131}-717q^{129}-1145q^{127}-588q^{125}+552q^{123}+1285q^{121}+894q^{119}-303q^{117}-1273q^{115}-1131q^{113}-4q^{111}+1127q^{109}+1242q^{107}+280q^{105}-859q^{103}-1192q^{101}-493q^{99}+546q^{97}+1019q^{95}+593q^{93}-269q^{91}-766q^{89}-564q^{87}+43q^{85}+502q^{83}+475q^{81}+66q^{79}-295q^{77}-329q^{75}-112q^{73}+136q^{71}+212q^{69}+96q^{67}-57q^{65}-112q^{63}-66q^{61}+16q^{59}+55q^{57}+34q^{55}-2q^{53}-19q^{51}-16q^{49}+3q^{47}+10q^{45}+3q^{43}-2q^{41}+2q^{39}-2q^{37}+2q^{35}+5q^{33}-q^{31}-2q^{29}+2q^{27}+2q^{21}-q^{17}+q^{15}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{34}+q^{32}+q^{30}-2q^{28}-2q^{24}-q^{22}+q^{20}+3q^{16}+q^{12}+2q^{10}-q^{8}+q^{6}$
1,1	$q^{92}-4q^{90}+12q^{88}-28q^{86}+50q^{84}-80q^{82}+118q^{80}-154q^{78}+184q^{76}-194q^{74}+186q^{72}-156q^{70}+93q^{68}-18q^{66}-74q^{64}+166q^{62}-244q^{60}+316q^{58}-352q^{56}+364q^{54}-340q^{52}+284q^{50}-212q^{48}+120q^{46}-36q^{44}-54q^{42}+108q^{40}-152q^{38}+165q^{36}-166q^{34}+154q^{32}-124q^{30}+105q^{28}-72q^{26}+56q^{24}-32q^{22}+23q^{20}-10q^{18}+6q^{16}-2q^{14}+q^{12}$
2,0	$q^{86}-q^{84}-2q^{82}+4q^{78}+3q^{76}-5q^{74}-2q^{72}+4q^{70}+2q^{68}-5q^{66}-3q^{64}+7q^{62}+2q^{60}-4q^{58}+5q^{54}-2q^{52}-4q^{50}-q^{48}-4q^{46}-6q^{44}-q^{42}+3q^{40}-6q^{38}-q^{36}+10q^{34}+4q^{32}-5q^{30}+2q^{28}+8q^{26}+2q^{24}-4q^{22}+2q^{20}+4q^{18}-q^{16}-q^{14}+q^{12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{74}-2q^{72}+q^{70}+3q^{68}-6q^{66}+4q^{64}+4q^{62}-10q^{60}+5q^{58}+5q^{56}-9q^{54}+3q^{52}+8q^{50}-5q^{48}-2q^{46}-2q^{42}-7q^{40}-6q^{38}+8q^{36}-4q^{34}-4q^{32}+13q^{30}-5q^{26}+10q^{24}+q^{22}-3q^{20}+4q^{18}+q^{16}-q^{14}+q^{12}$
1,0,0	$-q^{45}+q^{43}+q^{39}-2q^{37}-3q^{33}-q^{31}-q^{29}+q^{27}+q^{25}+q^{23}+3q^{21}+2q^{17}+2q^{13}-q^{11}+q^{9}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{96}-q^{94}-2q^{92}+3q^{90}+3q^{88}-4q^{86}-2q^{84}+6q^{82}+q^{80}-8q^{78}-q^{76}+9q^{74}-6q^{70}+7q^{68}+6q^{66}-6q^{64}-3q^{62}+2q^{60}-8q^{58}-11q^{56}-2q^{54}-2q^{52}-10q^{50}-2q^{48}+11q^{46}+2q^{44}-3q^{42}+8q^{40}+10q^{38}-q^{34}+5q^{32}+5q^{30}-q^{28}+3q^{24}+q^{22}-q^{20}+q^{18}$
1,0,0,0	$-q^{56}+q^{54}+q^{48}-2q^{46}-3q^{42}-2q^{40}-q^{38}-q^{36}+q^{34}+q^{32}+2q^{30}+q^{28}+3q^{26}+2q^{22}+q^{20}+2q^{16}-q^{14}+q^{12}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{74}+2q^{72}-5q^{70}+7q^{68}-8q^{66}+10q^{64}-10q^{62}+10q^{60}-7q^{58}+3q^{56}+q^{54}-7q^{52}+10q^{50}-17q^{48}+18q^{46}-20q^{44}+18q^{42}-15q^{40}+12q^{38}-6q^{36}+2q^{34}+4q^{32}-7q^{30}+10q^{28}-9q^{26}+10q^{24}-7q^{22}+7q^{20}-4q^{18}+3q^{16}-q^{14}+q^{12}$
1,0	$q^{120}-2q^{116}-2q^{114}+3q^{112}+5q^{110}-2q^{108}-7q^{106}-2q^{104}+9q^{102}+7q^{100}-7q^{98}-11q^{96}+q^{94}+11q^{92}+4q^{90}-8q^{88}-7q^{86}+5q^{84}+8q^{82}-7q^{78}+6q^{74}+q^{72}-9q^{70}-5q^{68}+5q^{66}+3q^{64}-7q^{62}-8q^{60}+4q^{58}+8q^{56}-q^{54}-10q^{52}-q^{50}+11q^{48}+8q^{46}-5q^{44}-8q^{42}+q^{40}+10q^{38}+5q^{36}-3q^{34}-5q^{32}+q^{30}+4q^{28}+2q^{26}-q^{24}-q^{22}+q^{18}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{102}-2q^{100}+3q^{98}-4q^{96}+6q^{94}-7q^{92}+7q^{90}-8q^{88}+9q^{86}-8q^{84}+5q^{82}-4q^{80}+3q^{78}-4q^{74}+7q^{72}-6q^{70}+12q^{68}-14q^{66}+13q^{64}-16q^{62}+13q^{60}-18q^{58}+6q^{56}-13q^{54}+6q^{52}-4q^{50}+q^{48}+2q^{46}-q^{44}+11q^{42}-4q^{40}+8q^{38}-6q^{36}+10q^{34}-4q^{32}+6q^{30}-4q^{28}+5q^{26}-q^{24}+2q^{22}-q^{20}+q^{18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{176}-2q^{174}+5q^{172}-8q^{170}+7q^{168}-3q^{166}-6q^{164}+20q^{162}-27q^{160}+30q^{158}-24q^{156}+q^{154}+23q^{152}-47q^{150}+55q^{148}-43q^{146}+18q^{144}+14q^{142}-38q^{140}+48q^{138}-38q^{136}+13q^{134}+12q^{132}-30q^{130}+30q^{128}-10q^{126}-16q^{124}+40q^{122}-43q^{120}+36q^{118}-13q^{116}-28q^{114}+56q^{112}-73q^{110}+66q^{108}-39q^{106}-6q^{104}+43q^{102}-64q^{100}+61q^{98}-44q^{96}+7q^{94}+20q^{92}-36q^{90}+30q^{88}-7q^{86}-16q^{84}+35q^{82}-29q^{80}+11q^{78}+13q^{76}-34q^{74}+47q^{72}-41q^{70}+28q^{68}-2q^{66}-19q^{64}+35q^{62}-35q^{60}+31q^{58}-17q^{56}+4q^{54}+7q^{52}-15q^{50}+16q^{48}-12q^{46}+9q^{44}-2q^{42}-q^{40}+3q^{38}-3q^{36}+3q^{34}-q^{32}+q^{30}$

.

Computer Talk[show]

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 23"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$4t^{2}-11t+15-11t^{-1}+4t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$4z^{4}+5z^{2}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 45, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{-2}-2q^{-3}+5q^{-4}-6q^{-5}+8q^{-6}-8q^{-7}+6q^{-8}-5q^{-9}+3q^{-10}-q^{-11}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{2}a^{10}+z^{4}a^{8}-2a^{8}+2z^{4}a^{6}+4z^{2}a^{6}+2a^{6}+z^{4}a^{4}+2z^{2}a^{4}+a^{4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-7z^{4}a^{12}+3z^{2}a^{12}+3z^{7}a^{11}-5z^{5}a^{11}+za^{11}+z^{8}a^{10}+4z^{6}a^{10}-10z^{4}a^{10}+3z^{2}a^{10}+5z^{7}a^{9}-6z^{5}a^{9}-2z^{3}a^{9}+4za^{9}+z^{8}a^{8}+4z^{6}a^{8}-8z^{4}a^{8}+6z^{2}a^{8}-2a^{8}+2z^{7}a^{7}+2z^{5}a^{7}-6z^{3}a^{7}+4za^{7}+3z^{6}a^{6}-4z^{4}a^{6}+4z^{2}a^{6}-2a^{6}+2z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-2z^{2}a^{4}+a^{4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Computer Talk[show]

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 23"];

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]=

{ $4t^{2}-11t+15-11t^{-1}+4t^{-2}$, $q^{-2}-2q^{-3}+5q^{-4}-6q^{-5}+8q^{-6}-8q^{-7}+6q^{-8}-5q^{-9}+3q^{-10}-q^{-11}$ }

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, -11)

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 $20$ $-88$ $200$ ${\frac {1462}{3}}$ ${\frac {194}{3}}$ $-1760$ $-{\frac {9040}{3}}$ $-{\frac {1504}{3}}$ $-344$ ${\frac {4000}{3}}$ $3872$ ${\frac {29240}{3}}$ ${\frac {3880}{3}}$ ${\frac {115087}{6}}$ ${\frac {2986}{3}}$ ${\frac {58214}{9}}$ ${\frac {2389}{18}}$ ${\frac {4687}{6}}$

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-4 is the signature of 9 23. 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                 2 1 -1 -7               3     3 -9             3 2     -1 -11           5 3       2 -13         3 3         0 -15       3 5           -2 -17     2 3             1 -19   1 3               -2 -21   2                 2 -23 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)[show]

$n$	$J_{n}$
2	$q^{-4}-2q^{-5}+q^{-6}+6q^{-7}-10q^{-8}+2q^{-9}+19q^{-10}-27q^{-11}+2q^{-12}+39q^{-13}-45q^{-14}-4q^{-15}+56q^{-16}-51q^{-17}-11q^{-18}+59q^{-19}-41q^{-20}-16q^{-21}+47q^{-22}-24q^{-23}-17q^{-24}+28q^{-25}-8q^{-26}-11q^{-27}+10q^{-28}-3q^{-30}+q^{-31}$
3	$q^{-6}-2q^{-7}+q^{-8}+2q^{-9}+2q^{-10}-8q^{-11}+q^{-12}+12q^{-13}+3q^{-14}-26q^{-15}+2q^{-16}+37q^{-17}+7q^{-18}-69q^{-19}-3q^{-20}+89q^{-21}+23q^{-22}-132q^{-23}-32q^{-24}+155q^{-25}+62q^{-26}-188q^{-27}-80q^{-28}+196q^{-29}+110q^{-30}-205q^{-31}-126q^{-32}+197q^{-33}+139q^{-34}-177q^{-35}-151q^{-36}+157q^{-37}+148q^{-38}-122q^{-39}-151q^{-40}+95q^{-41}+137q^{-42}-57q^{-43}-125q^{-44}+29q^{-45}+103q^{-46}-4q^{-47}-79q^{-48}-10q^{-49}+52q^{-50}+17q^{-51}-30q^{-52}-17q^{-53}+15q^{-54}+11q^{-55}-5q^{-56}-5q^{-57}+3q^{-59}-q^{-60}$
4	$q^{-8}-2q^{-9}+q^{-10}+2q^{-11}-2q^{-12}+4q^{-13}-9q^{-14}+4q^{-15}+10q^{-16}-9q^{-17}+10q^{-18}-28q^{-19}+15q^{-20}+34q^{-21}-27q^{-22}+6q^{-23}-72q^{-24}+51q^{-25}+103q^{-26}-52q^{-27}-33q^{-28}-184q^{-29}+108q^{-30}+264q^{-31}-33q^{-32}-112q^{-33}-415q^{-34}+129q^{-35}+512q^{-36}+94q^{-37}-167q^{-38}-743q^{-39}+50q^{-40}+745q^{-41}+308q^{-42}-126q^{-43}-1041q^{-44}-112q^{-45}+851q^{-46}+508q^{-47}+6q^{-48}-1198q^{-49}-276q^{-50}+815q^{-51}+613q^{-52}+160q^{-53}-1184q^{-54}-388q^{-55}+666q^{-56}+622q^{-57}+307q^{-58}-1032q^{-59}-455q^{-60}+442q^{-61}+559q^{-62}+433q^{-63}-779q^{-64}-469q^{-65}+176q^{-66}+421q^{-67}+508q^{-68}-465q^{-69}-399q^{-70}-53q^{-71}+222q^{-72}+472q^{-73}-174q^{-74}-245q^{-75}-160q^{-76}+35q^{-77}+322q^{-78}-5q^{-79}-81q^{-80}-130q^{-81}-58q^{-82}+145q^{-83}+33q^{-84}+8q^{-85}-54q^{-86}-52q^{-87}+39q^{-88}+12q^{-89}+17q^{-90}-8q^{-91}-18q^{-92}+5q^{-93}+5q^{-95}-3q^{-97}+q^{-98}$

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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