10 66

10_65

10_67

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10 66 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X5,14,6,15 X7,16,8,17 X15,6,16,7 X17,20,18,1 X11,18,12,19 X19,12,20,13 X13,8,14,9 X9,2,10,3
Gauss code	-1, 10, -2, 1, -3, 5, -4, 9, -10, 2, -7, 8, -9, 3, -5, 4, -6, 7, -8, 6
Dowker-Thistlethwaite code	4 10 14 16 2 18 8 6 20 12
Conway Notation	[31,21,21]

Minimum Braid Representative: Length is 11, width is 4. Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-17][5] Hyperbolic Volume 13.0293 A-Polynomial See Data:10 66/A-polynomial

[edit Notes for 10 66's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $3$ Topological 4 genus $3$ Concordance genus $3$ Rasmussen s-Invariant -6

[edit Notes for 10 66's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{3}-9t^{2}+16t-19+16t^{-1}-9t^{-2}+3t^{-3}$
Conway polynomial	$3z^{6}+9z^{4}+7z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 75, -6 }
Jones polynomial	$q^{-3}-2q^{-4}+6q^{-5}-8q^{-6}+11q^{-7}-13q^{-8}+12q^{-9}-10q^{-10}+7q^{-11}-4q^{-12}+q^{-13}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{12}+a^{12}-3z^{4}a^{10}-8z^{2}a^{10}-4a^{10}+2z^{6}a^{8}+8z^{4}a^{8}+9z^{2}a^{8}+2a^{8}+z^{6}a^{6}+4z^{4}a^{6}+5z^{2}a^{6}+2a^{6}$
Kauffman polynomial (db, data sources)	$z^{4}a^{16}+4z^{5}a^{15}-3z^{3}a^{15}+7z^{6}a^{14}-8z^{4}a^{14}+2z^{2}a^{14}+7z^{7}a^{13}-7z^{5}a^{13}+z^{3}a^{13}+4z^{8}a^{12}+3z^{6}a^{12}-13z^{4}a^{12}+5z^{2}a^{12}+a^{12}+z^{9}a^{11}+11z^{7}a^{11}-28z^{5}a^{11}+22z^{3}a^{11}-6za^{11}+7z^{8}a^{10}-13z^{6}a^{10}+8z^{4}a^{10}-8z^{2}a^{10}+4a^{10}+z^{9}a^{9}+6z^{7}a^{9}-22z^{5}a^{9}+20z^{3}a^{9}-5za^{9}+3z^{8}a^{8}-8z^{6}a^{8}+8z^{4}a^{8}-6z^{2}a^{8}+2a^{8}+2z^{7}a^{7}-5z^{5}a^{7}+2z^{3}a^{7}+za^{7}+z^{6}a^{6}-4z^{4}a^{6}+5z^{2}a^{6}-2a^{6}$
The A2 invariant	$q^{40}-2q^{36}+q^{34}-2q^{32}-3q^{26}+2q^{24}-2q^{22}+3q^{20}+2q^{18}+3q^{14}-q^{12}+q^{10}$
The G2 invariant	$q^{210}-3q^{208}+6q^{206}-10q^{204}+9q^{202}-6q^{200}-2q^{198}+19q^{196}-34q^{194}+50q^{192}-53q^{190}+33q^{188}-2q^{186}-44q^{184}+90q^{182}-117q^{180}+119q^{178}-79q^{176}+9q^{174}+73q^{172}-134q^{170}+158q^{168}-136q^{166}+64q^{164}+21q^{162}-95q^{160}+126q^{158}-92q^{156}+21q^{154}+64q^{152}-114q^{150}+100q^{148}-35q^{146}-74q^{144}+168q^{142}-211q^{140}+179q^{138}-68q^{136}-76q^{134}+198q^{132}-256q^{130}+228q^{128}-135q^{126}-5q^{124}+115q^{122}-177q^{120}+178q^{118}-106q^{116}+q^{114}+82q^{112}-117q^{110}+87q^{108}-17q^{106}-76q^{104}+136q^{102}-139q^{100}+90q^{98}+6q^{96}-107q^{94}+178q^{92}-175q^{90}+118q^{88}-30q^{86}-60q^{84}+118q^{82}-125q^{80}+102q^{78}-47q^{76}-2q^{74}+38q^{72}-49q^{70}+42q^{68}-25q^{66}+11q^{64}+2q^{62}-6q^{60}+7q^{58}-5q^{56}+4q^{54}-q^{52}+q^{50}$

Further Quantum Invariants

Further quantum knot invariants for 10_66.

A1 Invariants.

Weight	Invariant
1	$q^{27}-3q^{25}+3q^{23}-3q^{21}+2q^{19}-q^{17}-2q^{15}+3q^{13}-2q^{11}+4q^{9}-q^{7}+q^{5}$
2	$q^{74}-3q^{72}+9q^{68}-11q^{66}-4q^{64}+22q^{62}-16q^{60}-11q^{58}+30q^{56}-12q^{54}-19q^{52}+20q^{50}+2q^{48}-16q^{46}-q^{44}+16q^{42}-2q^{40}-20q^{38}+18q^{36}+10q^{34}-30q^{32}+10q^{30}+18q^{28}-22q^{26}+q^{24}+17q^{22}-8q^{20}-3q^{18}+7q^{16}-q^{12}+q^{10}$
3	$q^{141}-3q^{139}+6q^{135}+q^{133}-11q^{131}-7q^{129}+26q^{127}+10q^{125}-39q^{123}-22q^{121}+57q^{119}+39q^{117}-79q^{115}-64q^{113}+95q^{111}+96q^{109}-99q^{107}-127q^{105}+86q^{103}+151q^{101}-55q^{99}-158q^{97}+13q^{95}+145q^{93}+33q^{91}-112q^{89}-72q^{87}+69q^{85}+105q^{83}-23q^{81}-123q^{79}-21q^{77}+125q^{75}+64q^{73}-130q^{71}-97q^{69}+110q^{67}+131q^{65}-89q^{63}-147q^{61}+53q^{59}+157q^{57}-14q^{55}-149q^{53}-24q^{51}+122q^{49}+50q^{47}-88q^{45}-63q^{43}+47q^{41}+62q^{39}-20q^{37}-42q^{35}-q^{33}+28q^{31}+8q^{29}-10q^{27}-5q^{25}+5q^{23}+3q^{21}-q^{17}+q^{15}$
4	$q^{228}-3q^{226}+6q^{222}-2q^{220}+q^{218}-14q^{216}+3q^{214}+26q^{212}-8q^{210}-3q^{208}-50q^{206}+10q^{204}+91q^{202}-2q^{200}-41q^{198}-150q^{196}+25q^{194}+248q^{192}+71q^{190}-129q^{188}-395q^{186}-34q^{184}+511q^{182}+344q^{180}-157q^{178}-786q^{176}-321q^{174}+682q^{172}+793q^{170}+94q^{168}-1036q^{166}-799q^{164}+458q^{162}+1063q^{160}+586q^{158}-789q^{156}-1070q^{154}-114q^{152}+821q^{150}+919q^{148}-153q^{146}-859q^{144}-609q^{142}+227q^{140}+840q^{138}+449q^{136}-360q^{134}-811q^{132}-319q^{130}+547q^{128}+814q^{126}+94q^{124}-820q^{122}-673q^{120}+235q^{118}+1007q^{116}+467q^{114}-726q^{112}-929q^{110}-129q^{108}+1031q^{106}+823q^{104}-424q^{102}-1031q^{100}-588q^{98}+754q^{96}+1044q^{94}+104q^{92}-793q^{90}-931q^{88}+188q^{86}+882q^{84}+557q^{82}-245q^{80}-856q^{78}-307q^{76}+376q^{74}+579q^{72}+223q^{70}-416q^{68}-390q^{66}-62q^{64}+267q^{62}+299q^{60}-42q^{58}-175q^{56}-151q^{54}+15q^{52}+134q^{50}+52q^{48}-10q^{46}-60q^{44}-30q^{42}+24q^{40}+17q^{38}+13q^{36}-7q^{34}-8q^{32}+3q^{30}+q^{28}+3q^{26}-q^{22}+q^{20}$
5	$q^{335}-3q^{333}+6q^{329}-2q^{327}-2q^{325}-2q^{323}-4q^{321}+3q^{319}+14q^{317}+2q^{315}-21q^{313}-17q^{311}+8q^{309}+37q^{307}+32q^{305}-9q^{303}-87q^{301}-97q^{299}+54q^{297}+199q^{295}+169q^{293}-70q^{291}-379q^{289}-405q^{287}+82q^{285}+715q^{283}+779q^{281}+q^{279}-1114q^{277}-1451q^{275}-317q^{273}+1579q^{271}+2425q^{269}+966q^{267}-1920q^{265}-3624q^{263}-2100q^{261}+1908q^{259}+4915q^{257}+3688q^{255}-1357q^{253}-5925q^{251}-5542q^{249}+107q^{247}+6325q^{245}+7336q^{243}+1717q^{241}-5860q^{239}-8593q^{237}-3810q^{235}+4458q^{233}+8949q^{231}+5739q^{229}-2360q^{227}-8267q^{225}-7033q^{223}-12q^{221}+6636q^{219}+7459q^{217}+2214q^{215}-4427q^{213}-7041q^{211}-3883q^{209}+2090q^{207}+5971q^{205}+4893q^{203}+39q^{201}-4630q^{199}-5384q^{197}-1700q^{195}+3328q^{193}+5523q^{191}+2937q^{189}-2252q^{187}-5555q^{185}-3895q^{183}+1452q^{181}+5661q^{179}+4693q^{177}-776q^{175}-5782q^{173}-5573q^{171}+22q^{169}+5936q^{167}+6471q^{165}+923q^{163}-5746q^{161}-7404q^{159}-2238q^{157}+5203q^{155}+8090q^{153}+3751q^{151}-3987q^{149}-8307q^{147}-5358q^{145}+2246q^{143}+7823q^{141}+6653q^{139}-86q^{137}-6545q^{135}-7336q^{133}-2094q^{131}+4562q^{129}+7163q^{127}+3876q^{125}-2222q^{123}-6102q^{121}-4890q^{119}-44q^{117}+4350q^{115}+4975q^{113}+1791q^{111}-2353q^{109}-4220q^{107}-2702q^{105}+546q^{103}+2900q^{101}+2826q^{99}+715q^{97}-1541q^{95}-2271q^{93}-1272q^{91}+387q^{89}+1461q^{87}+1294q^{85}+247q^{83}-693q^{81}-936q^{79}-497q^{77}+164q^{75}+536q^{73}+433q^{71}+81q^{69}-214q^{67}-278q^{65}-130q^{63}+47q^{61}+125q^{59}+102q^{57}+14q^{55}-47q^{53}-46q^{51}-13q^{49}+6q^{47}+21q^{45}+14q^{43}-3q^{41}-5q^{39}-q^{35}+q^{33}+3q^{31}-q^{27}+q^{25}$

A2 Invariants.

Weight	Invariant
1,0	$q^{40}-2q^{36}+q^{34}-2q^{32}-3q^{26}+2q^{24}-2q^{22}+3q^{20}+2q^{18}+3q^{14}-q^{12}+q^{10}$
1,1	$q^{108}-6q^{106}+18q^{104}-38q^{102}+71q^{100}-128q^{98}+206q^{96}-296q^{94}+403q^{92}-522q^{90}+628q^{88}-696q^{86}+717q^{84}-676q^{82}+550q^{80}-328q^{78}+36q^{76}+310q^{74}-676q^{72}+1016q^{70}-1301q^{68}+1474q^{66}-1528q^{64}+1452q^{62}-1242q^{60}+964q^{58}-608q^{56}+228q^{54}+119q^{52}-424q^{50}+620q^{48}-762q^{46}+786q^{44}-738q^{42}+632q^{40}-496q^{38}+368q^{36}-236q^{34}+152q^{32}-76q^{30}+43q^{28}-16q^{26}+8q^{24}-2q^{22}+q^{20}$
2,0	$q^{100}-2q^{96}-q^{94}+q^{92}+q^{90}-5q^{88}+q^{86}+7q^{84}-3q^{80}+6q^{78}+10q^{76}-7q^{74}-10q^{72}+6q^{70}+2q^{68}-12q^{66}-q^{64}+10q^{62}-4q^{60}-6q^{58}+6q^{56}+q^{54}-13q^{52}-2q^{50}+8q^{48}-9q^{46}-8q^{44}+11q^{42}+7q^{40}-7q^{38}+2q^{36}+11q^{34}+3q^{32}-5q^{30}+3q^{28}+5q^{26}-q^{24}-q^{22}+q^{20}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{88}-3q^{86}+8q^{82}-9q^{80}-2q^{78}+17q^{76}-15q^{74}-8q^{72}+23q^{70}-13q^{68}-10q^{66}+22q^{64}-3q^{62}-8q^{60}+7q^{58}+3q^{56}-6q^{54}-15q^{52}+8q^{50}+2q^{48}-24q^{46}+10q^{44}+11q^{42}-20q^{40}+10q^{38}+14q^{36}-12q^{34}+9q^{32}+7q^{30}-4q^{28}+4q^{26}+2q^{24}-q^{22}+q^{20}$
1,0,0	$q^{53}+q^{49}-2q^{47}+q^{45}-4q^{43}+q^{41}-3q^{39}-2q^{35}+q^{31}+4q^{27}+4q^{23}-q^{21}+3q^{19}-q^{17}+q^{15}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{114}-3q^{110}-2q^{108}+4q^{106}+4q^{104}-6q^{102}-2q^{100}+11q^{98}-q^{96}-16q^{94}+3q^{92}+12q^{90}-8q^{88}-3q^{86}+20q^{84}+11q^{82}-9q^{80}+7q^{78}+12q^{76}-19q^{74}-18q^{72}+7q^{70}-13q^{68}-30q^{66}+3q^{64}+9q^{62}-13q^{60}-6q^{58}+17q^{56}+7q^{54}-6q^{52}+7q^{50}+13q^{48}+10q^{42}+2q^{40}+3q^{36}+2q^{34}-q^{32}+q^{30}$
1,0,0,0	$q^{66}+q^{62}+q^{60}-2q^{58}+q^{56}-4q^{54}-q^{52}-2q^{50}-3q^{48}-2q^{44}+q^{42}-q^{40}+3q^{38}+4q^{34}+q^{32}+2q^{30}+3q^{28}-q^{26}+3q^{24}-q^{22}+q^{20}$

B2 Invariants.

Weight	Invariant
0,1	$q^{88}-3q^{86}+6q^{84}-10q^{82}+15q^{80}-20q^{78}+23q^{76}-25q^{74}+24q^{72}-21q^{70}+13q^{68}-2q^{66}-10q^{64}+23q^{62}-34q^{60}+43q^{58}-49q^{56}+48q^{54}-45q^{52}+34q^{50}-26q^{48}+12q^{46}-11q^{42}+20q^{40}-22q^{38}+26q^{36}-22q^{34}+21q^{32}-15q^{30}+12q^{28}-6q^{26}+4q^{24}-q^{22}+q^{20}$
1,0	$q^{142}-3q^{138}-3q^{136}+3q^{134}+9q^{132}+2q^{130}-12q^{128}-11q^{126}+9q^{124}+20q^{122}+2q^{120}-23q^{118}-16q^{116}+15q^{114}+25q^{112}-3q^{110}-25q^{108}-7q^{106}+21q^{104}+16q^{102}-13q^{100}-17q^{98}+8q^{96}+18q^{94}-3q^{92}-19q^{90}-2q^{88}+15q^{86}+3q^{84}-18q^{82}-8q^{80}+15q^{78}+12q^{76}-16q^{74}-23q^{72}+5q^{70}+26q^{68}+6q^{66}-25q^{64}-18q^{62}+15q^{60}+26q^{58}-18q^{54}-8q^{52}+14q^{50}+13q^{48}-2q^{46}-8q^{44}+5q^{40}+3q^{38}-q^{36}-q^{34}+q^{30}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{122}-3q^{120}+3q^{118}-4q^{116}+9q^{114}-12q^{112}+12q^{110}-14q^{108}+20q^{106}-20q^{104}+16q^{102}-19q^{100}+19q^{98}-12q^{96}+5q^{94}-3q^{92}-2q^{90}+17q^{88}-17q^{86}+26q^{84}-30q^{82}+38q^{80}-37q^{78}+33q^{76}-43q^{74}+27q^{72}-31q^{70}+16q^{68}-21q^{66}+5q^{64}-4q^{60}+11q^{58}-14q^{56}+22q^{54}-16q^{52}+21q^{50}-16q^{48}+20q^{46}-11q^{44}+14q^{42}-7q^{40}+8q^{38}-2q^{36}+3q^{34}-q^{32}+q^{30}$

G2 Invariants.

Weight

Invariant

1,0

$q^{210}-3q^{208}+6q^{206}-10q^{204}+9q^{202}-6q^{200}-2q^{198}+19q^{196}-34q^{194}+50q^{192}-53q^{190}+33q^{188}-2q^{186}-44q^{184}+90q^{182}-117q^{180}+119q^{178}-79q^{176}+9q^{174}+73q^{172}-134q^{170}+158q^{168}-136q^{166}+64q^{164}+21q^{162}-95q^{160}+126q^{158}-92q^{156}+21q^{154}+64q^{152}-114q^{150}+100q^{148}-35q^{146}-74q^{144}+168q^{142}-211q^{140}+179q^{138}-68q^{136}-76q^{134}+198q^{132}-256q^{130}+228q^{128}-135q^{126}-5q^{124}+115q^{122}-177q^{120}+178q^{118}-106q^{116}+q^{114}+82q^{112}-117q^{110}+87q^{108}-17q^{106}-76q^{104}+136q^{102}-139q^{100}+90q^{98}+6q^{96}-107q^{94}+178q^{92}-175q^{90}+118q^{88}-30q^{86}-60q^{84}+118q^{82}-125q^{80}+102q^{78}-47q^{76}-2q^{74}+38q^{72}-49q^{70}+42q^{68}-25q^{66}+11q^{64}+2q^{62}-6q^{60}+7q^{58}-5q^{56}+4q^{54}-q^{52}+q^{50}$

.

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["10 66"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{3}-9t^{2}+16t-19+16t^{-1}-9t^{-2}+3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{6}+9z^{4}+7z^{2}+1$

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]=

{ 75, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{4}a^{16}+4z^{5}a^{15}-3z^{3}a^{15}+7z^{6}a^{14}-8z^{4}a^{14}+2z^{2}a^{14}+7z^{7}a^{13}-7z^{5}a^{13}+z^{3}a^{13}+4z^{8}a^{12}+3z^{6}a^{12}-13z^{4}a^{12}+5z^{2}a^{12}+a^{12}+z^{9}a^{11}+11z^{7}a^{11}-28z^{5}a^{11}+22z^{3}a^{11}-6za^{11}+7z^{8}a^{10}-13z^{6}a^{10}+8z^{4}a^{10}-8z^{2}a^{10}+4a^{10}+z^{9}a^{9}+6z^{7}a^{9}-22z^{5}a^{9}+20z^{3}a^{9}-5za^{9}+3z^{8}a^{8}-8z^{6}a^{8}+8z^{4}a^{8}-6z^{2}a^{8}+2a^{8}+2z^{7}a^{7}-5z^{5}a^{7}+2z^{3}a^{7}+za^{7}+z^{6}a^{6}-4z^{4}a^{6}+5z^{2}a^{6}-2a^{6}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a245, ...}

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

Vassiliev invariants

V2 and V3:

(7, -17)

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 $28$ $-136$ $392$ ${\frac {2594}{3}}$ ${\frac {358}{3}}$ $-3808$ $-{\frac {18448}{3}}$ $-{\frac {3232}{3}}$ $-680$ ${\frac {10976}{3}}$ $9248$ ${\frac {72632}{3}}$ ${\frac {10024}{3}}$ ${\frac {1349977}{30}}$ ${\frac {41206}{15}}$ ${\frac {677354}{45}}$ ${\frac {3431}{18}}$ ${\frac {55417}{30}}$

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 $s=$-6 is the signature of 10 66. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                     1 1 -7                   2 1 -1 -9                 4     4 -11               4 2     -2 -13             7 4       3 -15           6 4         -2 -17         6 7           -1 -19       4 6             2 -21     3 6               -3 -23   1 4                 3 -25   3                   -3 -27 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-7$ $i=-5$ $r=-10$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-7$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-6$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{-6}-2q^{-7}+q^{-8}+8q^{-9}-12q^{-10}-4q^{-11}+33q^{-12}-28q^{-13}-27q^{-14}+73q^{-15}-36q^{-16}-67q^{-17}+113q^{-18}-28q^{-19}-105q^{-20}+131q^{-21}-10q^{-22}-122q^{-23}+116q^{-24}+8q^{-25}-104q^{-26}+77q^{-27}+15q^{-28}-62q^{-29}+36q^{-30}+10q^{-31}-24q^{-32}+10q^{-33}+3q^{-34}-4q^{-35}+q^{-36}$
3	$q^{-9}-2q^{-10}+q^{-11}+3q^{-12}+3q^{-13}-12q^{-14}-4q^{-15}+21q^{-16}+23q^{-17}-41q^{-18}-45q^{-19}+43q^{-20}+105q^{-21}-56q^{-22}-155q^{-23}+18q^{-24}+243q^{-25}+16q^{-26}-301q^{-27}-107q^{-28}+378q^{-29}+187q^{-30}-405q^{-31}-307q^{-32}+436q^{-33}+407q^{-34}-426q^{-35}-514q^{-36}+403q^{-37}+601q^{-38}-365q^{-39}-660q^{-40}+301q^{-41}+701q^{-42}-237q^{-43}-696q^{-44}+160q^{-45}+661q^{-46}-92q^{-47}-584q^{-48}+28q^{-49}+490q^{-50}+11q^{-51}-378q^{-52}-37q^{-53}+277q^{-54}+39q^{-55}-183q^{-56}-38q^{-57}+118q^{-58}+24q^{-59}-65q^{-60}-20q^{-61}+39q^{-62}+7q^{-63}-16q^{-64}-4q^{-65}+6q^{-66}+3q^{-67}-4q^{-68}+q^{-69}$
4	$q^{-12}-2q^{-13}+q^{-14}+3q^{-15}-2q^{-16}+3q^{-17}-13q^{-18}+2q^{-19}+23q^{-20}+2q^{-21}+10q^{-22}-67q^{-23}-28q^{-24}+73q^{-25}+64q^{-26}+92q^{-27}-186q^{-28}-194q^{-29}+49q^{-30}+197q^{-31}+433q^{-32}-218q^{-33}-523q^{-34}-279q^{-35}+171q^{-36}+1072q^{-37}+138q^{-38}-726q^{-39}-962q^{-40}-378q^{-41}+1683q^{-42}+940q^{-43}-401q^{-44}-1656q^{-45}-1497q^{-46}+1821q^{-47}+1837q^{-48}+539q^{-49}-1946q^{-50}-2839q^{-51}+1378q^{-52}+2444q^{-53}+1786q^{-54}-1738q^{-55}-3999q^{-56}+578q^{-57}+2647q^{-58}+2979q^{-59}-1198q^{-60}-4771q^{-61}-330q^{-62}+2500q^{-63}+3893q^{-64}-478q^{-65}-5038q^{-66}-1196q^{-67}+2008q^{-68}+4344q^{-69}+331q^{-70}-4647q^{-71}-1809q^{-72}+1172q^{-73}+4094q^{-74}+1037q^{-75}-3575q^{-76}-1907q^{-77}+237q^{-78}+3138q^{-79}+1318q^{-80}-2200q^{-81}-1430q^{-82}-368q^{-83}+1881q^{-84}+1081q^{-85}-1070q^{-86}-731q^{-87}-479q^{-88}+878q^{-89}+616q^{-90}-441q^{-91}-230q^{-92}-312q^{-93}+333q^{-94}+255q^{-95}-175q^{-96}-30q^{-97}-135q^{-98}+110q^{-99}+80q^{-100}-66q^{-101}+9q^{-102}-42q^{-103}+29q^{-104}+20q^{-105}-19q^{-106}+4q^{-107}-8q^{-108}+6q^{-109}+3q^{-110}-4q^{-111}+q^{-112}$
5	$q^{-15}-2q^{-16}+q^{-17}+3q^{-18}-2q^{-19}-2q^{-20}+2q^{-21}-7q^{-22}+3q^{-23}+20q^{-24}+5q^{-25}-17q^{-26}-17q^{-27}-40q^{-28}+2q^{-29}+81q^{-30}+93q^{-31}+6q^{-32}-95q^{-33}-217q^{-34}-146q^{-35}+145q^{-36}+388q^{-37}+358q^{-38}+8q^{-39}-589q^{-40}-807q^{-41}-294q^{-42}+631q^{-43}+1298q^{-44}+1055q^{-45}-422q^{-46}-1881q^{-47}-1953q^{-48}-368q^{-49}+2028q^{-50}+3311q^{-51}+1689q^{-52}-1807q^{-53}-4307q^{-54}-3616q^{-55}+510q^{-56}+5178q^{-57}+5833q^{-58}+1377q^{-59}-4932q^{-60}-8010q^{-61}-4336q^{-62}+3966q^{-63}+9713q^{-64}+7475q^{-65}-1645q^{-66}-10611q^{-67}-10992q^{-68}-1276q^{-69}+10504q^{-70}+13934q^{-71}+5094q^{-72}-9441q^{-73}-16569q^{-74}-8880q^{-75}+7555q^{-76}+18254q^{-77}+12832q^{-78}-5102q^{-79}-19456q^{-80}-16321q^{-81}+2389q^{-82}+19912q^{-83}+19501q^{-84}+446q^{-85}-19991q^{-86}-22235q^{-87}-3206q^{-88}+19703q^{-89}+24507q^{-90}+5915q^{-91}-19023q^{-92}-26444q^{-93}-8553q^{-94}+18043q^{-95}+27810q^{-96}+11104q^{-97}-16437q^{-98}-28639q^{-99}-13581q^{-100}+14359q^{-101}+28564q^{-102}+15773q^{-103}-11583q^{-104}-27561q^{-105}-17462q^{-106}+8386q^{-107}+25406q^{-108}+18387q^{-109}-4942q^{-110}-22316q^{-111}-18285q^{-112}+1738q^{-113}+18385q^{-114}+17153q^{-115}+965q^{-116}-14217q^{-117}-15075q^{-118}-2753q^{-119}+10117q^{-120}+12370q^{-121}+3698q^{-122}-6640q^{-123}-9456q^{-124}-3764q^{-125}+3899q^{-126}+6721q^{-127}+3315q^{-128}-2072q^{-129}-4411q^{-130}-2537q^{-131}+892q^{-132}+2713q^{-133}+1791q^{-134}-368q^{-135}-1525q^{-136}-1078q^{-137}+46q^{-138}+817q^{-139}+657q^{-140}-31q^{-141}-410q^{-142}-300q^{-143}-18q^{-144}+184q^{-145}+170q^{-146}-5q^{-147}-101q^{-148}-61q^{-149}+12q^{-150}+39q^{-151}+19q^{-152}+5q^{-153}-23q^{-154}-20q^{-155}+17q^{-156}+10q^{-157}-6q^{-158}+q^{-159}-8q^{-161}+6q^{-162}+3q^{-163}-4q^{-164}+q^{-165}$
6	$q^{-18}-2q^{-19}+q^{-20}+3q^{-21}-2q^{-22}-2q^{-23}-3q^{-24}+8q^{-25}-6q^{-26}+22q^{-28}-4q^{-29}-15q^{-30}-33q^{-31}+12q^{-32}-11q^{-33}+12q^{-34}+109q^{-35}+45q^{-36}-30q^{-37}-169q^{-38}-92q^{-39}-145q^{-40}-8q^{-41}+399q^{-42}+417q^{-43}+267q^{-44}-312q^{-45}-484q^{-46}-945q^{-47}-718q^{-48}+517q^{-49}+1363q^{-50}+1806q^{-51}+759q^{-52}-232q^{-53}-2577q^{-54}-3504q^{-55}-1710q^{-56}+1087q^{-57}+4331q^{-58}+4838q^{-59}+4099q^{-60}-1819q^{-61}-7223q^{-62}-8412q^{-63}-5186q^{-64}+2619q^{-65}+9264q^{-66}+14519q^{-67}+7614q^{-68}-4181q^{-69}-14862q^{-70}-18731q^{-71}-10973q^{-72}+3647q^{-73}+23531q^{-74}+25770q^{-75}+14209q^{-76}-8018q^{-77}-28920q^{-78}-34705q^{-79}-20922q^{-80}+15622q^{-81}+38976q^{-82}+44006q^{-83}+20800q^{-84}-18491q^{-85}-52186q^{-86}-58073q^{-87}-17154q^{-88}+29079q^{-89}+66699q^{-90}+63094q^{-91}+19104q^{-92}-45268q^{-93}-87954q^{-94}-64499q^{-95}-8764q^{-96}+64897q^{-97}+98738q^{-98}+72032q^{-99}-10940q^{-100}-94527q^{-101}-106650q^{-102}-61777q^{-103}+37275q^{-104}+113524q^{-105}+121253q^{-106}+37707q^{-107}-77814q^{-108}-131152q^{-109}-112182q^{-110}-3359q^{-111}+108395q^{-112}+155711q^{-113}+84670q^{-114}-49577q^{-115}-139215q^{-116}-150494q^{-117}-43178q^{-118}+93488q^{-119}+176291q^{-120}+122336q^{-121}-20779q^{-122}-138757q^{-123}-177616q^{-124}-76882q^{-125}+76464q^{-126}+188492q^{-127}+152178q^{-128}+6344q^{-129}-133783q^{-130}-197392q^{-131}-107215q^{-132}+56166q^{-133}+192865q^{-134}+177073q^{-135}+36403q^{-136}-119619q^{-137}-207712q^{-138}-136562q^{-139}+26139q^{-140}+181743q^{-141}+192656q^{-142}+71194q^{-143}-88400q^{-144}-198592q^{-145}-158247q^{-146}-14279q^{-147}+146792q^{-148}+186918q^{-149}+101207q^{-150}-41454q^{-151}-161908q^{-152}-158190q^{-153}-52386q^{-154}+92208q^{-155}+152282q^{-156}+110215q^{-157}+5020q^{-158}-104830q^{-159}-129424q^{-160}-70153q^{-161}+37579q^{-162}+98405q^{-163}+91783q^{-164}+31675q^{-165}-49065q^{-166}-83258q^{-167}-61888q^{-168}+2693q^{-169}+47459q^{-170}+57844q^{-171}+33653q^{-172}-13203q^{-173}-40753q^{-174}-39411q^{-175}-8703q^{-176}+15411q^{-177}+27114q^{-178}+22137q^{-179}+838q^{-180}-14669q^{-181}-18678q^{-182}-6914q^{-183}+2273q^{-184}+9133q^{-185}+10343q^{-186}+2700q^{-187}-3705q^{-188}-6736q^{-189}-2762q^{-190}-656q^{-191}+2006q^{-192}+3625q^{-193}+1353q^{-194}-613q^{-195}-1941q^{-196}-506q^{-197}-527q^{-198}+158q^{-199}+1014q^{-200}+376q^{-201}-74q^{-202}-504q^{-203}+89q^{-204}-169q^{-205}-79q^{-206}+248q^{-207}+64q^{-208}-20q^{-209}-140q^{-210}+100q^{-211}-35q^{-212}-41q^{-213}+57q^{-214}+2q^{-215}-4q^{-216}-42q^{-217}+39q^{-218}-2q^{-219}-16q^{-220}+14q^{-221}-3q^{-222}-8q^{-224}+6q^{-225}+3q^{-226}-4q^{-227}+q^{-228}$
7	Not Available

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[10, 66]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 14, 6, 15], X[7, 16, 8, 17], X[15, 6, 16, 7], X[17, 20, 18, 1], X[11, 18, 12, 19], X[19, 12, 20, 13], X[13, 8, 14, 9], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[10, 66]]
Out[3]=	GaussCode[-1, 10, -2, 1, -3, 5, -4, 9, -10, 2, -7, 8, -9, 3, -5, 4, -6, 7, -8, 6]
In[4]:=	DTCode[Knot[10, 66]]
Out[4]=	DTCode[4, 10, 14, 16, 2, 18, 8, 6, 20, 12]
In[5]:=	br = BR[Knot[10, 66]]
Out[5]=	BR[4, {-1, -1, -1, 2, -1, -3, -2, -2, -2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 66]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 66]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 66]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 3, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 66]][t]
Out[10]=	3 9 16 2 3 -19 + -- - -- + -- + 16 t - 9 t + 3 t 3 2 t t t
In[11]:=	Conway[Knot[10, 66]][z]
Out[11]=	2 4 6 1 + 7 z + 9 z + 3 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 66], Knot[11, Alternating, 245]}
In[13]:=	{KnotDet[Knot[10, 66]], KnotSignature[Knot[10, 66]]}
Out[13]=	{75, -6}
In[14]:=	Jones[Knot[10, 66]][q]
Out[14]=	-13 4 7 10 12 13 11 8 6 2 -3 q - --- + --- - --- + -- - -- + -- - -- + -- - -- + q 12 11 10 9 8 7 6 5 4 q q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 66]}
In[16]:=	A2Invariant[Knot[10, 66]][q]
Out[16]=	-40 2 -34 2 3 2 2 3 2 3 -12 q - --- + q - --- - --- + --- - --- + --- + --- + --- - q + 36 32 26 24 22 20 18 14 q q q q q q q q -10 q
In[17]:=	HOMFLYPT[Knot[10, 66]][a, z]
Out[17]=	6 8 10 12 6 2 8 2 10 2 12 2 2 a + 2 a - 4 a + a + 5 a z + 9 a z - 8 a z + a z + 6 4 8 4 10 4 6 6 8 6 4 a z + 8 a z - 3 a z + a z + 2 a z
In[18]:=	Kauffman[Knot[10, 66]][a, z]
Out[18]=	6 8 10 12 7 9 11 6 2 -2 a + 2 a + 4 a + a + a z - 5 a z - 6 a z + 5 a z - 8 2 10 2 12 2 14 2 7 3 9 3 6 a z - 8 a z + 5 a z + 2 a z + 2 a z + 20 a z + 11 3 13 3 15 3 6 4 8 4 10 4 22 a z + a z - 3 a z - 4 a z + 8 a z + 8 a z - 12 4 14 4 16 4 7 5 9 5 11 5 13 a z - 8 a z + a z - 5 a z - 22 a z - 28 a z - 13 5 15 5 6 6 8 6 10 6 12 6 7 a z + 4 a z + a z - 8 a z - 13 a z + 3 a z + 14 6 7 7 9 7 11 7 13 7 8 8 7 a z + 2 a z + 6 a z + 11 a z + 7 a z + 3 a z + 10 8 12 8 9 9 11 9 7 a z + 4 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 66]], Vassiliev[3][Knot[10, 66]]}
Out[19]=	{7, -17}
In[20]:=	Kh[Knot[10, 66]][q, t]
Out[20]=	-7 -5 1 3 1 4 3 6 q + q + ------- + ------ + ------ + ------ + ------ + ------ + 27 10 25 9 23 9 23 8 21 8 21 7 q t q t q t q t q t q t 4 6 6 7 6 4 7 ------ + ------ + ------ + ------ + ------ + ------ + ------ + 19 7 19 6 17 6 17 5 15 5 15 4 13 4 q t q t q t q t q t q t q t 4 4 2 4 2 ------ + ------ + ------ + ----- + ---- 13 3 11 3 11 2 9 2 7 q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 66], 2][q]
Out[21]=	-36 4 3 10 24 10 36 62 15 77 104 q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + 35 34 33 32 31 30 29 28 27 26 q q q q q q q q q q 8 116 122 10 131 105 28 113 67 36 73 --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 25 24 23 22 21 20 19 18 17 16 15 q q q q q q q q q q q 27 28 33 4 12 8 -8 2 -6 --- - --- + --- - --- - --- + -- + q - -- + q 14 13 12 11 10 9 7 q q q q q q q

See/edit the Rolfsen_Splice_Template.