10 66

10_65

10_67

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10 66 Quick Notes

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]

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]=

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

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

Computer Talk

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

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 66]]
Out[2]=	10
In[3]:=	PD[Knot[10, 66]]
Out[3]=	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[4]:=	GaussCode[Knot[10, 66]]
Out[4]=	GaussCode[-1, 10, -2, 1, -3, 5, -4, 9, -10, 2, -7, 8, -9, 3, -5, 4, -6, 7, -8, 6]
In[5]:=	BR[Knot[10, 66]]
Out[5]=	BR[4, {-1, -1, -1, 2, -1, -3, -2, -2, -2, -3, -3}]
In[6]:=	alex = Alexander[Knot[10, 66]][t]
Out[6]=	3 9 16 2 3 -19 + -- - -- + -- + 16 t - 9 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[10, 66]][z]
Out[7]=	2 4 6 1 + 7 z + 9 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 66], Knot[11, Alternating, 245]}
In[9]:=	{KnotDet[Knot[10, 66]], KnotSignature[Knot[10, 66]]}
Out[9]=	{75, -6}
In[10]:=	J=Jones[Knot[10, 66]][q]
Out[10]=	-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[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 66]}
In[12]:=	A2Invariant[Knot[10, 66]][q]
Out[12]=	-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[13]:=	Kauffman[Knot[10, 66]][a, z]
Out[13]=	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[14]:=	{Vassiliev[2][Knot[10, 66]], Vassiliev[3][Knot[10, 66]]}
Out[14]=	{0, -17}
In[15]:=	Kh[Knot[10, 66]][q, t]
Out[15]=	-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