9 38

9_37

9_39

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 38's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 38 at Knotilus!

Knot presentations

Planar diagram presentation	X1627 X5,14,6,15 X7,18,8,1 X15,8,16,9 X3,10,4,11 X9,4,10,5 X17,12,18,13 X11,16,12,17 X13,2,14,3
Gauss code	-1, 9, -5, 6, -2, 1, -3, 4, -6, 5, -8, 7, -9, 2, -4, 8, -7, 3
Dowker-Thistlethwaite code	6 10 14 18 4 16 2 8 12
Conway Notation	[.2.2.2]

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

[edit Notes on presentations of 9 38]

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 38"];

In[4]:=

PD[K]

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

Out[4]=

X1627 X5,14,6,15 X7,18,8,1 X15,8,16,9 X3,10,4,11 X9,4,10,5 X17,12,18,13 X11,16,12,17 X13,2,14,3

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 10 14 18 4 16 2 8 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[.2.2.2]

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]=

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

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 2 Bridge index 3 Super bridge index $\{4,7\}$ Nakanishi index 2 Maximal Thurston-Bennequin number [-14][3] Hyperbolic Volume 12.9329 A-Polynomial See Data:9 38/A-polynomial

[edit Notes for 9 38'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 38's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$5t^{2}-14t+19-14t^{-1}+5t^{-2}$
Conway polynomial	$5z^{4}+6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 57, -4 }
Jones polynomial	$q^{-2}-3q^{-3}+7q^{-4}-8q^{-5}+10q^{-6}-10q^{-7}+8q^{-8}-6q^{-9}+3q^{-10}-q^{-11}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{10}+z^{4}a^{8}-z^{2}a^{8}-3a^{8}+3z^{4}a^{6}+7z^{2}a^{6}+4a^{6}+z^{4}a^{4}+z^{2}a^{4}$
Kauffman polynomial (db, data sources)	$z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-6z^{4}a^{12}+3z^{2}a^{12}+4z^{7}a^{11}-7z^{5}a^{11}+3z^{3}a^{11}-za^{11}+2z^{8}a^{10}+3z^{6}a^{10}-10z^{4}a^{10}+3z^{2}a^{10}+9z^{7}a^{9}-15z^{5}a^{9}+5z^{3}a^{9}+za^{9}+2z^{8}a^{8}+6z^{6}a^{8}-15z^{4}a^{8}+10z^{2}a^{8}-3a^{8}+5z^{7}a^{7}-4z^{5}a^{7}-2z^{3}a^{7}+3za^{7}+6z^{6}a^{6}-10z^{4}a^{6}+9z^{2}a^{6}-4a^{6}+3z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-z^{2}a^{4}$
The A2 invariant	$-q^{34}+q^{32}+q^{30}-3q^{28}-2q^{24}-q^{22}+2q^{20}+4q^{16}+q^{12}+2q^{10}-2q^{8}+q^{6}$
The G2 invariant	$q^{176}-2q^{174}+5q^{172}-8q^{170}+8q^{168}-6q^{166}-2q^{164}+18q^{162}-33q^{160}+47q^{158}-46q^{156}+21q^{154}+16q^{152}-65q^{150}+101q^{148}-104q^{146}+70q^{144}-6q^{142}-63q^{140}+112q^{138}-116q^{136}+77q^{134}-8q^{132}-60q^{130}+87q^{128}-70q^{126}+15q^{124}+52q^{122}-95q^{120}+98q^{118}-56q^{116}-20q^{114}+93q^{112}-152q^{110}+153q^{108}-105q^{106}+19q^{104}+69q^{102}-137q^{100}+159q^{98}-125q^{96}+52q^{94}+26q^{92}-90q^{90}+105q^{88}-66q^{86}+q^{84}+64q^{82}-84q^{80}+67q^{78}-9q^{76}-58q^{74}+106q^{72}-112q^{70}+82q^{68}-20q^{66}-43q^{64}+89q^{62}-94q^{60}+77q^{58}-36q^{56}+25q^{52}-40q^{50}+37q^{48}-25q^{46}+13q^{44}-5q^{40}+6q^{38}-6q^{36}+4q^{34}-2q^{32}+q^{30}$

Further Quantum Invariants

Further quantum knot invariants for 9_38.

A1 Invariants.

Weight	Invariant
1	$-q^{23}+2q^{21}-3q^{19}+2q^{17}-2q^{15}+2q^{11}-q^{9}+4q^{7}-2q^{5}+q^{3}$
2	$q^{64}-2q^{62}-q^{60}+8q^{58}-5q^{56}-11q^{54}+17q^{52}+q^{50}-21q^{48}+15q^{46}+10q^{44}-20q^{42}+4q^{40}+12q^{38}-8q^{36}-9q^{34}+6q^{32}+9q^{30}-17q^{28}-q^{26}+23q^{24}-14q^{22}-9q^{20}+21q^{18}-5q^{16}-9q^{14}+8q^{12}+q^{10}-2q^{8}+q^{6}$
3	$-q^{123}+2q^{121}+q^{119}-4q^{117}-5q^{115}+8q^{113}+17q^{111}-11q^{109}-36q^{107}+q^{105}+57q^{103}+25q^{101}-73q^{99}-57q^{97}+71q^{95}+95q^{93}-52q^{91}-124q^{89}+22q^{87}+134q^{85}+11q^{83}-128q^{81}-41q^{79}+112q^{77}+61q^{75}-84q^{73}-69q^{71}+50q^{69}+79q^{67}-19q^{65}-79q^{63}-23q^{61}+79q^{59}+57q^{57}-70q^{55}-100q^{53}+53q^{51}+126q^{49}-27q^{47}-139q^{45}-6q^{43}+131q^{41}+37q^{39}-103q^{37}-54q^{35}+72q^{33}+50q^{31}-33q^{29}-42q^{27}+15q^{25}+27q^{23}-3q^{21}-11q^{19}+5q^{15}+q^{13}-2q^{11}+q^{9}$
4	$q^{200}-2q^{198}-q^{196}+4q^{194}+q^{192}+2q^{190}-14q^{188}-11q^{186}+19q^{184}+27q^{182}+32q^{180}-51q^{178}-95q^{176}-18q^{174}+88q^{172}+199q^{170}+20q^{168}-224q^{166}-259q^{164}-40q^{162}+414q^{160}+367q^{158}-92q^{156}-533q^{154}-498q^{152}+294q^{150}+718q^{148}+409q^{146}-425q^{144}-921q^{142}-199q^{140}+661q^{138}+851q^{136}+23q^{134}-917q^{132}-621q^{130}+282q^{128}+892q^{126}+390q^{124}-593q^{122}-701q^{120}-62q^{118}+649q^{116}+505q^{114}-231q^{112}-584q^{110}-273q^{108}+361q^{106}+518q^{104}+106q^{102}-442q^{100}-483q^{98}+30q^{96}+537q^{94}+518q^{92}-228q^{90}-717q^{88}-423q^{86}+427q^{84}+917q^{82}+183q^{80}-711q^{78}-860q^{76}+29q^{74}+975q^{72}+609q^{70}-310q^{68}-903q^{66}-415q^{64}+561q^{62}+657q^{60}+153q^{58}-502q^{56}-485q^{54}+93q^{52}+331q^{50}+261q^{48}-104q^{46}-245q^{44}-59q^{42}+63q^{40}+127q^{38}+14q^{36}-60q^{34}-25q^{32}-6q^{30}+30q^{28}+9q^{26}-9q^{24}-2q^{22}-3q^{20}+5q^{18}+q^{16}-2q^{14}+q^{12}$
5	$-q^{295}+2q^{293}+q^{291}-4q^{289}-q^{287}+2q^{285}+4q^{283}+8q^{281}+3q^{279}-22q^{277}-33q^{275}-7q^{273}+38q^{271}+85q^{269}+73q^{267}-33q^{265}-188q^{263}-231q^{261}-58q^{259}+258q^{257}+497q^{255}+370q^{253}-174q^{251}-802q^{249}-921q^{247}-237q^{245}+898q^{243}+1612q^{241}+1111q^{239}-536q^{237}-2185q^{235}-2311q^{233}-455q^{231}+2207q^{229}+3536q^{227}+2067q^{225}-1463q^{223}-4335q^{221}-3908q^{219}-91q^{217}+4298q^{215}+5538q^{213}+2172q^{211}-3367q^{209}-6484q^{207}-4264q^{205}+1709q^{203}+6512q^{201}+5917q^{199}+248q^{197}-5715q^{195}-6815q^{193}-2036q^{191}+4383q^{189}+6870q^{187}+3358q^{185}-2886q^{183}-6293q^{181}-4057q^{179}+1545q^{177}+5352q^{175}+4193q^{173}-499q^{171}-4314q^{169}-4003q^{167}-233q^{165}+3404q^{163}+3692q^{161}+729q^{159}-2619q^{157}-3475q^{155}-1256q^{153}+2009q^{151}+3453q^{149}+1874q^{147}-1345q^{145}-3561q^{143}-2812q^{141}+530q^{139}+3729q^{137}+3936q^{135}+651q^{133}-3654q^{131}-5182q^{129}-2181q^{127}+3138q^{125}+6194q^{123}+3972q^{121}-2022q^{119}-6668q^{117}-5687q^{115}+350q^{113}+6292q^{111}+6943q^{109}+1597q^{107}-5076q^{105}-7334q^{103}-3374q^{101}+3203q^{99}+6731q^{97}+4536q^{95}-1149q^{93}-5334q^{91}-4783q^{89}-550q^{87}+3477q^{85}+4201q^{83}+1609q^{81}-1770q^{79}-3114q^{77}-1837q^{75}+474q^{73}+1900q^{71}+1586q^{69}+198q^{67}-956q^{65}-1062q^{63}-387q^{61}+342q^{59}+591q^{57}+332q^{55}-63q^{53}-275q^{51}-199q^{49}-12q^{47}+100q^{45}+96q^{43}+26q^{41}-29q^{39}-45q^{37}-10q^{35}+15q^{33}+12q^{31}+3q^{29}-5q^{25}-3q^{23}+5q^{21}+q^{19}-2q^{17}+q^{15}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{34}+q^{32}+q^{30}-3q^{28}-2q^{24}-q^{22}+2q^{20}+4q^{16}+q^{12}+2q^{10}-2q^{8}+q^{6}$
1,1	$q^{92}-4q^{90}+12q^{88}-28q^{86}+58q^{84}-106q^{82}+176q^{80}-260q^{78}+349q^{76}-430q^{74}+474q^{72}-466q^{70}+394q^{68}-256q^{66}+58q^{64}+176q^{62}-408q^{60}+628q^{58}-794q^{56}+896q^{54}-910q^{52}+836q^{50}-700q^{48}+484q^{46}-260q^{44}+10q^{42}+192q^{40}-348q^{38}+441q^{36}-456q^{34}+438q^{32}-366q^{30}+290q^{28}-200q^{26}+136q^{24}-82q^{22}+46q^{20}-20q^{18}+10q^{16}-4q^{14}+q^{12}$
2,0	$q^{86}-q^{84}-q^{82}+q^{80}+5q^{78}-9q^{74}-3q^{72}+8q^{70}+5q^{68}-7q^{66}+12q^{62}+4q^{60}-11q^{58}-4q^{56}+4q^{54}-7q^{52}-8q^{50}-2q^{48}-q^{46}-5q^{44}+4q^{42}+7q^{40}-6q^{38}+q^{36}+14q^{34}+4q^{32}-11q^{30}+3q^{28}+12q^{26}-9q^{22}+q^{20}+7q^{18}-q^{16}-2q^{14}+q^{12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{74}-2q^{72}+q^{70}+4q^{68}-8q^{66}+4q^{64}+9q^{62}-15q^{60}+7q^{58}+10q^{56}-17q^{54}+4q^{52}+10q^{50}-10q^{48}-3q^{46}+2q^{44}-2q^{42}-8q^{40}-7q^{38}+12q^{36}-3q^{34}-8q^{32}+21q^{30}+q^{28}-10q^{26}+16q^{24}-q^{22}-7q^{20}+6q^{18}-2q^{14}+q^{12}$
1,0,0	$-q^{45}+q^{43}+q^{39}-3q^{37}-4q^{33}-q^{31}-q^{29}+2q^{27}+2q^{25}+2q^{23}+4q^{21}+2q^{17}-q^{15}+2q^{13}-2q^{11}+q^{9}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{96}-q^{94}-3q^{92}+3q^{90}+5q^{88}-4q^{86}-4q^{84}+9q^{82}+6q^{80}-11q^{78}-4q^{76}+14q^{74}+q^{72}-14q^{70}+6q^{68}+11q^{66}-11q^{64}-11q^{62}+4q^{60}-9q^{58}-20q^{56}-q^{54}+6q^{52}-11q^{50}-q^{48}+23q^{46}+7q^{44}-7q^{42}+10q^{40}+16q^{38}-3q^{36}-6q^{34}+7q^{32}+6q^{30}-5q^{28}-2q^{26}+4q^{24}-2q^{20}+q^{18}$
1,0,0,0	$-q^{56}+q^{54}+q^{48}-3q^{46}-4q^{42}-3q^{40}-q^{38}-q^{36}+2q^{34}+2q^{32}+4q^{30}+2q^{28}+4q^{26}+2q^{22}-q^{18}+2q^{16}-2q^{14}+q^{12}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{74}+2q^{72}-5q^{70}+8q^{68}-12q^{66}+16q^{64}-17q^{62}+17q^{60}-15q^{58}+10q^{56}-3q^{54}-6q^{52}+14q^{50}-24q^{48}+29q^{46}-34q^{44}+32q^{42}-30q^{40}+23q^{38}-14q^{36}+7q^{34}+4q^{32}-9q^{30}+17q^{28}-16q^{26}+18q^{24}-15q^{22}+13q^{20}-8q^{18}+4q^{16}-2q^{14}+q^{12}$
1,0	$q^{120}-2q^{116}-2q^{114}+3q^{112}+6q^{110}-q^{108}-10q^{106}-5q^{104}+12q^{102}+13q^{100}-7q^{98}-18q^{96}-2q^{94}+19q^{92}+10q^{90}-14q^{88}-15q^{86}+6q^{84}+15q^{82}-14q^{78}-3q^{76}+11q^{74}+4q^{72}-13q^{70}-9q^{68}+7q^{66}+9q^{64}-8q^{62}-14q^{60}+4q^{58}+15q^{56}+2q^{54}-16q^{52}-5q^{50}+17q^{48}+17q^{46}-8q^{44}-16q^{42}+q^{40}+17q^{38}+8q^{36}-8q^{34}-10q^{32}+2q^{30}+7q^{28}+2q^{26}-2q^{24}-2q^{22}+q^{18}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{102}-2q^{100}+3q^{98}-4q^{96}+7q^{94}-10q^{92}+11q^{90}-12q^{88}+15q^{86}-14q^{84}+12q^{82}-10q^{80}+9q^{78}-4q^{76}-4q^{74}+6q^{72}-9q^{70}+16q^{68}-23q^{66}+21q^{64}-26q^{62}+24q^{60}-29q^{58}+16q^{56}-23q^{54}+15q^{52}-9q^{50}+5q^{48}+q^{46}+q^{44}+16q^{42}-7q^{40}+15q^{38}-12q^{36}+17q^{34}-10q^{32}+10q^{30}-10q^{28}+7q^{26}-3q^{24}+2q^{22}-2q^{20}+q^{18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{176}-2q^{174}+5q^{172}-8q^{170}+8q^{168}-6q^{166}-2q^{164}+18q^{162}-33q^{160}+47q^{158}-46q^{156}+21q^{154}+16q^{152}-65q^{150}+101q^{148}-104q^{146}+70q^{144}-6q^{142}-63q^{140}+112q^{138}-116q^{136}+77q^{134}-8q^{132}-60q^{130}+87q^{128}-70q^{126}+15q^{124}+52q^{122}-95q^{120}+98q^{118}-56q^{116}-20q^{114}+93q^{112}-152q^{110}+153q^{108}-105q^{106}+19q^{104}+69q^{102}-137q^{100}+159q^{98}-125q^{96}+52q^{94}+26q^{92}-90q^{90}+105q^{88}-66q^{86}+q^{84}+64q^{82}-84q^{80}+67q^{78}-9q^{76}-58q^{74}+106q^{72}-112q^{70}+82q^{68}-20q^{66}-43q^{64}+89q^{62}-94q^{60}+77q^{58}-36q^{56}+25q^{52}-40q^{50}+37q^{48}-25q^{46}+13q^{44}-5q^{40}+6q^{38}-6q^{36}+4q^{34}-2q^{32}+q^{30}$

.

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 38"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$5t^{2}-14t+19-14t^{-1}+5t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$5z^{4}+6z^{2}+1$

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]=

{ 57, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{2}a^{10}+z^{4}a^{8}-z^{2}a^{8}-3a^{8}+3z^{4}a^{6}+7z^{2}a^{6}+4a^{6}+z^{4}a^{4}+z^{2}a^{4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-6z^{4}a^{12}+3z^{2}a^{12}+4z^{7}a^{11}-7z^{5}a^{11}+3z^{3}a^{11}-za^{11}+2z^{8}a^{10}+3z^{6}a^{10}-10z^{4}a^{10}+3z^{2}a^{10}+9z^{7}a^{9}-15z^{5}a^{9}+5z^{3}a^{9}+za^{9}+2z^{8}a^{8}+6z^{6}a^{8}-15z^{4}a^{8}+10z^{2}a^{8}-3a^{8}+5z^{7}a^{7}-4z^{5}a^{7}-2z^{3}a^{7}+3za^{7}+6z^{6}a^{6}-10z^{4}a^{6}+9z^{2}a^{6}-4a^{6}+3z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-z^{2}a^{4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_63,}

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

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 38"];

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]=

{ $5t^{2}-14t+19-14t^{-1}+5t^{-2}$, $q^{-2}-3q^{-3}+7q^{-4}-8q^{-5}+10q^{-6}-10q^{-7}+8q^{-8}-6q^{-9}+3q^{-10}-q^{-11}$ }

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]=

{10_63,}

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:

(6, -14)

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 $24$ $-112$ $288$ $684$ $100$ $-2688$ $-{\frac {13888}{3}}$ $-{\frac {2368}{3}}$ $-592$ $2304$ $6272$ $16416$ $2400$ ${\frac {160231}{5}}$ ${\frac {15308}{15}}$ ${\frac {59748}{5}}$ $259$ ${\frac {7671}{5}}$

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 38. 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                 3 1 -2 -7               4     4 -9             4 3     -1 -11           6 4       2 -13         4 4         0 -15       4 6           -2 -17     2 4             2 -19   1 4               -3 -21   2                 2 -23 1                   -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{-4}-3q^{-5}+3q^{-6}+8q^{-7}-20q^{-8}+7q^{-9}+34q^{-10}-50q^{-11}+2q^{-12}+71q^{-13}-74q^{-14}-14q^{-15}+97q^{-16}-77q^{-17}-29q^{-18}+98q^{-19}-57q^{-20}-37q^{-21}+74q^{-22}-27q^{-23}-32q^{-24}+38q^{-25}-5q^{-26}-16q^{-27}+10q^{-28}+q^{-29}-3q^{-30}+q^{-31}$
3	$q^{-6}-3q^{-7}+3q^{-8}+4q^{-9}-4q^{-10}-14q^{-11}+11q^{-12}+34q^{-13}-16q^{-14}-71q^{-15}+20q^{-16}+117q^{-17}+6q^{-18}-197q^{-19}-29q^{-20}+257q^{-21}+100q^{-22}-334q^{-23}-162q^{-24}+369q^{-25}+253q^{-26}-407q^{-27}-315q^{-28}+399q^{-29}+380q^{-30}-385q^{-31}-417q^{-32}+343q^{-33}+440q^{-34}-287q^{-35}-446q^{-36}+224q^{-37}+425q^{-38}-142q^{-39}-395q^{-40}+71q^{-41}+338q^{-42}-3q^{-43}-272q^{-44}-41q^{-45}+192q^{-46}+69q^{-47}-125q^{-48}-65q^{-49}+64q^{-50}+53q^{-51}-27q^{-52}-33q^{-53}+8q^{-54}+16q^{-55}-2q^{-56}-5q^{-57}-q^{-58}+3q^{-59}-q^{-60}$
4	$q^{-8}-3q^{-9}+3q^{-10}+4q^{-11}-8q^{-12}+2q^{-13}-10q^{-14}+21q^{-15}+25q^{-16}-44q^{-17}-17q^{-18}-45q^{-19}+95q^{-20}+138q^{-21}-108q^{-22}-139q^{-23}-231q^{-24}+236q^{-25}+503q^{-26}-38q^{-27}-377q^{-28}-809q^{-29}+219q^{-30}+1158q^{-31}+466q^{-32}-473q^{-33}-1785q^{-34}-269q^{-35}+1751q^{-36}+1385q^{-37}-107q^{-38}-2731q^{-39}-1158q^{-40}+1900q^{-41}+2279q^{-42}+627q^{-43}-3221q^{-44}-2008q^{-45}+1606q^{-46}+2768q^{-47}+1373q^{-48}-3202q^{-49}-2515q^{-50}+1093q^{-51}+2809q^{-52}+1921q^{-53}-2790q^{-54}-2672q^{-55}+459q^{-56}+2498q^{-57}+2274q^{-58}-2054q^{-59}-2528q^{-60}-252q^{-61}+1859q^{-62}+2382q^{-63}-1071q^{-64}-2026q^{-65}-862q^{-66}+956q^{-67}+2086q^{-68}-131q^{-69}-1198q^{-70}-1052q^{-71}+96q^{-72}+1364q^{-73}+365q^{-74}-364q^{-75}-743q^{-76}-328q^{-77}+572q^{-78}+330q^{-79}+77q^{-80}-284q^{-81}-281q^{-82}+118q^{-83}+111q^{-84}+112q^{-85}-40q^{-86}-102q^{-87}+7q^{-88}+5q^{-89}+35q^{-90}+4q^{-91}-19q^{-92}+2q^{-93}-3q^{-94}+5q^{-95}+q^{-96}-3q^{-97}+q^{-98}$
5	$q^{-10}-3q^{-11}+3q^{-12}+4q^{-13}-8q^{-14}-2q^{-15}+6q^{-16}+12q^{-18}+7q^{-19}-33q^{-20}-37q^{-21}+22q^{-22}+55q^{-23}+82q^{-24}+11q^{-25}-145q^{-26}-224q^{-27}-54q^{-28}+267q^{-29}+477q^{-30}+270q^{-31}-394q^{-32}-953q^{-33}-729q^{-34}+373q^{-35}+1631q^{-36}+1658q^{-37}-80q^{-38}-2379q^{-39}-3040q^{-40}-904q^{-41}+2975q^{-42}+5037q^{-43}+2512q^{-44}-3103q^{-45}-7067q^{-46}-5137q^{-47}+2424q^{-48}+9222q^{-49}+8197q^{-50}-908q^{-51}-10595q^{-52}-11714q^{-53}-1536q^{-54}+11480q^{-55}+14870q^{-56}+4438q^{-57}-11246q^{-58}-17656q^{-59}-7573q^{-60}+10499q^{-61}+19516q^{-62}+10432q^{-63}-9024q^{-64}-20712q^{-65}-12892q^{-66}+7498q^{-67}+21044q^{-68}+14737q^{-69}-5739q^{-70}-20919q^{-71}-16091q^{-72}+4156q^{-73}+20295q^{-74}+16953q^{-75}-2520q^{-76}-19340q^{-77}-17535q^{-78}+891q^{-79}+18076q^{-80}+17809q^{-81}+828q^{-82}-16377q^{-83}-17823q^{-84}-2746q^{-85}+14306q^{-86}+17498q^{-87}+4643q^{-88}-11685q^{-89}-16664q^{-90}-6553q^{-91}+8704q^{-92}+15262q^{-93}+8050q^{-94}-5441q^{-95}-13152q^{-96}-9040q^{-97}+2285q^{-98}+10483q^{-99}+9150q^{-100}+522q^{-101}-7483q^{-102}-8445q^{-103}-2518q^{-104}+4510q^{-105}+6930q^{-106}+3639q^{-107}-1944q^{-108}-5079q^{-109}-3758q^{-110}+121q^{-111}+3113q^{-112}+3212q^{-113}+928q^{-114}-1549q^{-115}-2289q^{-116}-1208q^{-117}+451q^{-118}+1356q^{-119}+1054q^{-120}+100q^{-121}-642q^{-122}-707q^{-123}-263q^{-124}+221q^{-125}+370q^{-126}+219q^{-127}-14q^{-128}-163q^{-129}-136q^{-130}-18q^{-131}+54q^{-132}+46q^{-133}+29q^{-134}-8q^{-135}-30q^{-136}-6q^{-137}+7q^{-138}+q^{-139}+3q^{-140}+3q^{-141}-5q^{-142}-q^{-143}+3q^{-144}-q^{-145}$

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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