9 9

9_8

9_10

Visit 9 9's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 9's page at Knotilus! Visit 9 9's page at the original Knot Atlas! 9 9 Quick Notes

9 9 Further Notes and Views

Knot presentations

Planar diagram presentation	X1627 X3,12,4,13 X7,16,8,17 X9,18,10,1 X17,8,18,9 X15,10,16,11 X5,14,6,15 X11,2,12,3 X13,4,14,5
Gauss code	-1, 8, -2, 9, -7, 1, -3, 5, -4, 6, -8, 2, -9, 7, -6, 3, -5, 4
Dowker-Thistlethwaite code	6 12 14 16 18 2 4 10 8
Conway Notation	[423]

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 2 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-16][5] Hyperbolic Volume 8.01682 A-Polynomial See Data:9 9/A-polynomial

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

Polynomial invariants

Alexander polynomial	$2t^{3}-4t^{2}+6t-7+6t^{-1}-4t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+8z^{4}+8z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 31, -6 }
Jones polynomial	$q^{-3}-q^{-4}+3q^{-5}-4q^{-6}+5q^{-7}-5q^{-8}+5q^{-9}-4q^{-10}+2q^{-11}-q^{-12}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{10}-3z^{2}a^{10}-2a^{10}+z^{6}a^{8}+4z^{4}a^{8}+4z^{2}a^{8}+a^{8}+z^{6}a^{6}+5z^{4}a^{6}+7z^{2}a^{6}+2a^{6}$
Kauffman polynomial (db, data sources)	$z^{3}a^{15}-za^{15}+2z^{4}a^{14}-z^{2}a^{14}+3z^{5}a^{13}-3z^{3}a^{13}+2za^{13}+3z^{6}a^{12}-4z^{4}a^{12}+3z^{2}a^{12}+2z^{7}a^{11}-2z^{5}a^{11}+z^{8}a^{10}-z^{6}a^{10}+2z^{4}a^{10}-6z^{2}a^{10}+2a^{10}+3z^{7}a^{9}-8z^{5}a^{9}+5z^{3}a^{9}-2za^{9}+z^{8}a^{8}-3z^{6}a^{8}+3z^{4}a^{8}-3z^{2}a^{8}+a^{8}+z^{7}a^{7}-3z^{5}a^{7}+z^{3}a^{7}+za^{7}+z^{6}a^{6}-5z^{4}a^{6}+7z^{2}a^{6}-2a^{6}$
The A2 invariant	$-q^{36}-q^{32}-q^{30}-q^{26}+2q^{24}+q^{20}+q^{18}+2q^{14}+q^{10}$
The G2 invariant	$q^{196}-q^{194}+2q^{192}-2q^{190}+q^{188}-2q^{184}+5q^{182}-6q^{180}+6q^{178}-6q^{176}+2q^{174}+3q^{172}-7q^{170}+12q^{168}-11q^{166}+9q^{164}-5q^{162}-2q^{160}+6q^{158}-11q^{156}+11q^{154}-7q^{152}+3q^{148}-7q^{146}+5q^{144}-q^{142}-8q^{140}+9q^{138}-12q^{136}+5q^{134}+5q^{132}-15q^{130}+20q^{128}-18q^{126}+10q^{124}+q^{122}-12q^{120}+18q^{118}-18q^{116}+14q^{114}-4q^{112}-4q^{110}+11q^{108}-10q^{106}+7q^{104}-2q^{102}-5q^{100}+8q^{98}-8q^{96}+2q^{94}+6q^{92}-11q^{90}+16q^{88}-11q^{86}+q^{84}+7q^{82}-11q^{80}+16q^{78}-12q^{76}+6q^{74}+2q^{72}-5q^{70}+10q^{68}-7q^{66}+5q^{64}+2q^{58}-2q^{56}+2q^{54}+q^{50}$

Further Quantum Invariants

Further quantum knot invariants for 9_9.

A1 Invariants.

Weight	Invariant
1	$-q^{25}+q^{23}-2q^{21}+q^{19}+q^{13}-q^{11}+2q^{9}+q^{5}$
2	$q^{68}-q^{66}+3q^{62}-3q^{60}-q^{58}+5q^{56}-5q^{54}-2q^{52}+5q^{50}-2q^{48}-2q^{46}+q^{44}+q^{42}-3q^{40}-2q^{38}+4q^{36}-4q^{32}+4q^{30}+2q^{28}-5q^{26}+2q^{24}+3q^{22}-3q^{20}+q^{18}+3q^{16}+q^{10}$
3	$-q^{129}+q^{127}-q^{123}-q^{121}+2q^{119}+2q^{117}-3q^{115}-q^{113}+5q^{111}-7q^{107}-q^{105}+11q^{103}+2q^{101}-13q^{99}-2q^{97}+12q^{95}+6q^{93}-10q^{91}-6q^{89}+5q^{87}+8q^{85}-q^{83}-8q^{81}-5q^{79}+6q^{77}+8q^{75}-6q^{73}-11q^{71}+4q^{69}+10q^{67}-2q^{65}-12q^{63}-q^{61}+11q^{59}+3q^{57}-11q^{55}-6q^{53}+9q^{51}+10q^{49}-6q^{47}-10q^{45}+3q^{43}+10q^{41}+q^{39}-9q^{37}-3q^{35}+5q^{33}+5q^{31}-2q^{29}-2q^{27}+q^{25}+3q^{23}+q^{21}+q^{15}$
4	$q^{208}-q^{206}+q^{202}-q^{200}+2q^{198}-3q^{196}-q^{194}+2q^{192}-2q^{190}+5q^{188}-4q^{186}+q^{184}+4q^{182}-9q^{180}+q^{178}-3q^{176}+13q^{174}+12q^{172}-19q^{170}-13q^{168}-10q^{166}+26q^{164}+32q^{162}-18q^{160}-31q^{158}-28q^{156}+25q^{154}+46q^{152}+2q^{150}-24q^{148}-37q^{146}+2q^{144}+34q^{142}+21q^{140}+q^{138}-26q^{136}-20q^{134}+2q^{132}+21q^{130}+23q^{128}-5q^{126}-26q^{124}-19q^{122}+15q^{120}+29q^{118}+8q^{116}-26q^{114}-29q^{112}+14q^{110}+29q^{108}+13q^{106}-27q^{104}-33q^{102}+11q^{100}+27q^{98}+25q^{96}-19q^{94}-39q^{92}-5q^{90}+18q^{88}+37q^{86}+q^{84}-33q^{82}-22q^{80}-6q^{78}+35q^{76}+24q^{74}-10q^{72}-23q^{70}-26q^{68}+14q^{66}+26q^{64}+13q^{62}-5q^{60}-26q^{58}-7q^{56}+8q^{54}+14q^{52}+9q^{50}-10q^{48}-8q^{46}-4q^{44}+4q^{42}+8q^{40}-2q^{34}+3q^{30}+q^{28}+q^{26}+q^{20}$
5	$-q^{305}+q^{303}-q^{299}+q^{297}-q^{293}+2q^{291}+2q^{289}-2q^{287}-q^{285}-q^{283}-3q^{281}+2q^{279}+4q^{277}+4q^{275}+q^{273}-3q^{271}-8q^{269}-12q^{267}+16q^{263}+24q^{261}+9q^{259}-22q^{257}-43q^{255}-27q^{253}+27q^{251}+72q^{249}+52q^{247}-26q^{245}-94q^{243}-86q^{241}+6q^{239}+113q^{237}+124q^{235}+21q^{233}-114q^{231}-154q^{229}-57q^{227}+88q^{225}+165q^{223}+98q^{221}-56q^{219}-151q^{217}-117q^{215}+2q^{213}+112q^{211}+125q^{209}+39q^{207}-63q^{205}-102q^{203}-68q^{201}+7q^{199}+75q^{197}+80q^{195}+32q^{193}-36q^{191}-78q^{189}-60q^{187}+8q^{185}+73q^{183}+75q^{181}+9q^{179}-67q^{177}-78q^{175}-13q^{173}+67q^{171}+82q^{169}+14q^{167}-73q^{165}-88q^{163}-11q^{161}+82q^{159}+95q^{157}+19q^{155}-88q^{153}-112q^{151}-28q^{149}+84q^{147}+124q^{145}+49q^{143}-71q^{141}-129q^{139}-77q^{137}+41q^{135}+124q^{133}+100q^{131}-6q^{129}-102q^{127}-115q^{125}-38q^{123}+67q^{121}+115q^{119}+70q^{117}-22q^{115}-92q^{113}-94q^{111}-27q^{109}+57q^{107}+93q^{105}+59q^{103}-14q^{101}-71q^{99}-78q^{97}-27q^{95}+40q^{93}+70q^{91}+47q^{89}-q^{87}-46q^{85}-55q^{83}-21q^{81}+21q^{79}+41q^{77}+33q^{75}+4q^{73}-24q^{71}-28q^{69}-14q^{67}+5q^{65}+18q^{63}+15q^{61}+2q^{59}-6q^{57}-10q^{55}-6q^{53}+2q^{51}+6q^{49}+3q^{47}+3q^{45}-2q^{41}+2q^{37}+q^{35}+q^{33}+q^{31}+q^{25}$
6	$q^{420}-q^{418}+q^{414}-q^{412}-q^{408}+2q^{406}-3q^{404}-2q^{402}+5q^{400}+q^{396}+3q^{392}-8q^{390}-8q^{388}+6q^{386}+q^{384}+6q^{382}+9q^{380}+14q^{378}-14q^{376}-24q^{374}-8q^{372}-13q^{370}+13q^{368}+37q^{366}+50q^{364}-6q^{362}-56q^{360}-58q^{358}-59q^{356}+18q^{354}+106q^{352}+143q^{350}+38q^{348}-103q^{346}-176q^{344}-183q^{342}-13q^{340}+207q^{338}+326q^{336}+176q^{334}-108q^{332}-327q^{330}-405q^{328}-157q^{326}+232q^{324}+516q^{322}+422q^{320}+45q^{318}-352q^{316}-602q^{314}-417q^{312}+50q^{310}+508q^{308}+601q^{306}+322q^{304}-124q^{302}-538q^{300}-570q^{298}-253q^{296}+219q^{294}+491q^{292}+462q^{290}+190q^{288}-207q^{286}-421q^{284}-384q^{282}-117q^{280}+152q^{278}+318q^{276}+312q^{274}+117q^{272}-106q^{270}-264q^{268}-250q^{266}-127q^{264}+75q^{262}+229q^{260}+240q^{258}+105q^{256}-102q^{254}-222q^{252}-213q^{250}-36q^{248}+147q^{246}+239q^{244}+141q^{242}-73q^{240}-216q^{238}-204q^{236}-6q^{234}+181q^{232}+262q^{230}+118q^{228}-139q^{226}-297q^{224}-236q^{222}+27q^{220}+262q^{218}+360q^{216}+168q^{214}-165q^{212}-396q^{210}-344q^{208}-43q^{206}+271q^{204}+463q^{202}+310q^{200}-59q^{198}-400q^{196}-459q^{194}-225q^{192}+129q^{190}+456q^{188}+453q^{186}+166q^{184}-233q^{182}-447q^{180}-400q^{178}-137q^{176}+257q^{174}+447q^{172}+372q^{170}+62q^{168}-229q^{166}-396q^{164}-354q^{162}-64q^{160}+214q^{158}+365q^{156}+278q^{154}+94q^{152}-154q^{150}-324q^{148}-268q^{146}-99q^{144}+121q^{142}+229q^{140}+261q^{138}+127q^{136}-76q^{134}-194q^{132}-221q^{130}-122q^{128}+156q^{124}+192q^{122}+121q^{120}+14q^{118}-99q^{116}-142q^{114}-135q^{112}-24q^{110}+62q^{108}+106q^{106}+101q^{104}+43q^{102}-23q^{100}-86q^{98}-69q^{96}-38q^{94}+7q^{92}+44q^{90}+55q^{88}+36q^{86}-7q^{84}-19q^{82}-31q^{80}-24q^{78}-8q^{76}+13q^{74}+20q^{72}+8q^{70}+7q^{68}-3q^{66}-8q^{64}-9q^{62}-q^{60}+4q^{58}+q^{56}+5q^{54}+3q^{52}+q^{50}-2q^{48}+2q^{44}+q^{40}+q^{38}+q^{36}+q^{30}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{36}-q^{32}-q^{30}-q^{26}+2q^{24}+q^{20}+q^{18}+2q^{14}+q^{10}$
1,1	$q^{100}-2q^{98}+4q^{96}-6q^{94}+11q^{92}-14q^{90}+18q^{88}-24q^{86}+28q^{84}-30q^{82}+28q^{80}-30q^{78}+27q^{76}-16q^{74}+8q^{72}+4q^{70}-20q^{68}+36q^{66}-52q^{64}+64q^{62}-70q^{60}+72q^{58}-68q^{56}+54q^{54}-47q^{52}+22q^{50}-10q^{48}-10q^{46}+21q^{44}-34q^{42}+42q^{40}-38q^{38}+38q^{36}-28q^{34}+28q^{32}-12q^{30}+12q^{28}-4q^{26}+4q^{24}+q^{20}$
2,0	$q^{90}+q^{86}+q^{84}+q^{82}-2q^{74}-2q^{72}+q^{70}-3q^{66}+q^{62}-2q^{60}-4q^{58}-q^{56}-q^{54}-3q^{52}+3q^{48}+3q^{42}+q^{40}-q^{38}+2q^{36}+3q^{34}+q^{32}+3q^{28}+2q^{26}+q^{20}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{82}-q^{80}+3q^{76}-2q^{74}-2q^{72}+5q^{70}-2q^{68}-3q^{66}+4q^{64}-q^{62}-3q^{60}+q^{58}-q^{56}-3q^{54}-3q^{52}-5q^{46}+q^{44}+3q^{42}-3q^{40}+2q^{38}+5q^{36}-q^{34}+3q^{32}+3q^{30}+2q^{26}+2q^{24}+q^{20}$
1,0,0	$-q^{47}-2q^{43}-2q^{39}-q^{35}+q^{33}+q^{31}+q^{29}+2q^{27}+2q^{23}+2q^{19}+q^{15}$
1,0,1	$q^{132}-2q^{130}+3q^{128}-q^{126}-3q^{124}+9q^{122}-9q^{120}+5q^{118}+5q^{116}-17q^{114}+19q^{112}-11q^{110}-6q^{108}+19q^{106}-26q^{104}+16q^{102}+5q^{100}-18q^{98}+31q^{96}-23q^{94}+3q^{92}+14q^{90}-31q^{88}+27q^{86}-8q^{84}+2q^{82}+17q^{80}-2q^{78}-4q^{76}+3q^{74}-14q^{72}-19q^{70}+10q^{68}-39q^{66}+21q^{64}-20q^{62}-8q^{60}+22q^{58}-30q^{56}+32q^{54}-11q^{52}+9q^{50}+16q^{48}-5q^{46}+20q^{44}-2q^{42}+6q^{40}+4q^{38}+4q^{34}+q^{30}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{104}-q^{100}+2q^{98}+3q^{96}-q^{94}+3q^{90}+q^{88}-2q^{86}+q^{84}+2q^{82}-3q^{80}-4q^{78}-q^{76}-5q^{74}-9q^{72}-2q^{70}-2q^{68}-6q^{66}-q^{64}+3q^{62}-2q^{58}+3q^{56}+3q^{54}+q^{52}+2q^{50}+5q^{48}+3q^{46}+2q^{44}+5q^{42}+2q^{40}+2q^{38}+2q^{36}+2q^{34}+q^{30}$
1,0,0,0	$-q^{58}-2q^{54}-q^{52}-q^{50}-2q^{48}-q^{44}+q^{42}+2q^{38}+q^{36}+2q^{34}+q^{32}+q^{30}+2q^{28}+2q^{24}+q^{20}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{82}+q^{80}-2q^{78}+3q^{76}-4q^{74}+4q^{72}-5q^{70}+4q^{68}-3q^{66}+2q^{64}+q^{62}-3q^{60}+5q^{58}-7q^{56}+7q^{54}-9q^{52}+8q^{50}-8q^{48}+5q^{46}-3q^{44}+q^{42}+q^{40}-2q^{38}+5q^{36}-3q^{34}+5q^{32}-3q^{30}+4q^{28}-2q^{26}+2q^{24}+q^{20}$
1,0	$q^{132}-q^{128}-q^{126}+q^{124}+3q^{122}+q^{120}-3q^{118}-3q^{116}+q^{114}+5q^{112}+q^{110}-4q^{108}-3q^{106}+2q^{104}+4q^{102}-q^{100}-4q^{98}-q^{96}+3q^{94}+q^{92}-4q^{90}-3q^{88}+q^{86}+2q^{84}-2q^{82}-3q^{80}+2q^{76}-q^{74}-4q^{72}-q^{70}+4q^{68}+3q^{66}-3q^{64}-4q^{62}+2q^{60}+6q^{58}+2q^{56}-2q^{54}-3q^{52}+3q^{50}+4q^{48}+q^{46}-2q^{44}+2q^{40}+2q^{38}+q^{30}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{114}-q^{112}+q^{110}-q^{108}+3q^{106}-3q^{104}+2q^{102}-3q^{100}+5q^{98}-3q^{96}+2q^{94}-3q^{92}+2q^{90}-q^{86}+q^{84}-4q^{82}+4q^{80}-6q^{78}+4q^{76}-9q^{74}+4q^{72}-8q^{70}+4q^{68}-7q^{66}+3q^{64}-3q^{62}+2q^{60}+4q^{54}-q^{52}+4q^{50}-q^{48}+6q^{46}-q^{44}+5q^{42}-q^{40}+4q^{38}+2q^{34}+q^{30}$

G2 Invariants.

Weight

Invariant

1,0

$q^{196}-q^{194}+2q^{192}-2q^{190}+q^{188}-2q^{184}+5q^{182}-6q^{180}+6q^{178}-6q^{176}+2q^{174}+3q^{172}-7q^{170}+12q^{168}-11q^{166}+9q^{164}-5q^{162}-2q^{160}+6q^{158}-11q^{156}+11q^{154}-7q^{152}+3q^{148}-7q^{146}+5q^{144}-q^{142}-8q^{140}+9q^{138}-12q^{136}+5q^{134}+5q^{132}-15q^{130}+20q^{128}-18q^{126}+10q^{124}+q^{122}-12q^{120}+18q^{118}-18q^{116}+14q^{114}-4q^{112}-4q^{110}+11q^{108}-10q^{106}+7q^{104}-2q^{102}-5q^{100}+8q^{98}-8q^{96}+2q^{94}+6q^{92}-11q^{90}+16q^{88}-11q^{86}+q^{84}+7q^{82}-11q^{80}+16q^{78}-12q^{76}+6q^{74}+2q^{72}-5q^{70}+10q^{68}-7q^{66}+5q^{64}+2q^{58}-2q^{56}+2q^{54}+q^{50}$

.

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["9 9"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-4t^{2}+6t-7+6t^{-1}-4t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{6}+8z^{4}+8z^{2}+1$

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]=

{ 31, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}a^{10}-3z^{2}a^{10}-2a^{10}+z^{6}a^{8}+4z^{4}a^{8}+4z^{2}a^{8}+a^{8}+z^{6}a^{6}+5z^{4}a^{6}+7z^{2}a^{6}+2a^{6}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

Vassiliev invariants

V2 and V3:

(8, -22)

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 $32$ $-176$ $512$ ${\frac {3760}{3}}$ ${\frac {560}{3}}$ $-5632$ $-{\frac {29696}{3}}$ $-{\frac {5216}{3}}$ $-1264$ ${\frac {16384}{3}}$ $15488$ ${\frac {120320}{3}}$ ${\frac {17920}{3}}$ ${\frac {1196284}{15}}$ ${\frac {40544}{15}}$ ${\frac {1354096}{45}}$ ${\frac {4628}{9}}$ ${\frac {57964}{15}}$

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 9 9. 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 -5                   1 1 -7                 1 1 0 -9               2     2 -11             2 1     -1 -13           3 2       1 -15         2 2         0 -17       3 3           0 -19     1 2             1 -21   1 3               -2 -23   1                 1 -25 1                   -1

Integral Khovanov Homology (db, data source)

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

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[9, 9]]
Out[2]=	9
In[3]:=	PD[Knot[9, 9]]
Out[3]=	PD[X[1, 6, 2, 7], X[3, 12, 4, 13], X[7, 16, 8, 17], X[9, 18, 10, 1], X[17, 8, 18, 9], X[15, 10, 16, 11], X[5, 14, 6, 15], X[11, 2, 12, 3], X[13, 4, 14, 5]]
In[4]:=	GaussCode[Knot[9, 9]]
Out[4]=	GaussCode[-1, 8, -2, 9, -7, 1, -3, 5, -4, 6, -8, 2, -9, 7, -6, 3, -5, 4]
In[5]:=	BR[Knot[9, 9]]
Out[5]=	BR[3, {-1, -1, -1, -1, -1, -2, 1, -2, -2, -2}]
In[6]:=	alex = Alexander[Knot[9, 9]][t]
Out[6]=	2 4 6 2 3 -7 + -- - -- + - + 6 t - 4 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[9, 9]][z]
Out[7]=	2 4 6 1 + 8 z + 8 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 9]}
In[9]:=	{KnotDet[Knot[9, 9]], KnotSignature[Knot[9, 9]]}
Out[9]=	{31, -6}
In[10]:=	J=Jones[Knot[9, 9]][q]
Out[10]=	-12 2 4 5 5 5 4 3 -4 -3 -q + --- - --- + -- - -- + -- - -- + -- - q + q 11 10 9 8 7 6 5 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 9]}
In[12]:=	A2Invariant[Knot[9, 9]][q]
Out[12]=	-36 -32 -30 -26 2 -20 -18 2 -10 -q - q - q - q + --- + q + q + --- + q 24 14 q q
In[13]:=	Kauffman[Knot[9, 9]][a, z]
Out[13]=	6 8 10 7 9 13 15 6 2 -2 a + a + 2 a + a z - 2 a z + 2 a z - a z + 7 a z - 8 2 10 2 12 2 14 2 7 3 9 3 13 3 3 a z - 6 a z + 3 a z - a z + a z + 5 a z - 3 a z + 15 3 6 4 8 4 10 4 12 4 14 4 a z - 5 a z + 3 a z + 2 a z - 4 a z + 2 a z - 7 5 9 5 11 5 13 5 6 6 8 6 10 6 3 a z - 8 a z - 2 a z + 3 a z + a z - 3 a z - a z + 12 6 7 7 9 7 11 7 8 8 10 8 3 a z + a z + 3 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[9, 9]], Vassiliev[3][Knot[9, 9]]}
Out[14]=	{0, -22}
In[15]:=	Kh[Knot[9, 9]][q, t]
Out[15]=	-7 -5 1 1 1 3 1 2 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 25 9 23 8 21 8 21 7 19 7 19 6 q t q t q t q t q t q t 3 3 2 2 3 2 2 ------ + ------ + ------ + ------ + ------ + ------ + ------ + 17 6 17 5 15 5 15 4 13 4 13 3 11 3 q t q t q t q t q t q t q t 1 2 1 ------ + ----- + ---- 11 2 9 2 7 q t q t q t