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]

Minimum Braid Representative: Length is 10, width is 3. Braid index is 3.

A Morse Link Presentation:

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]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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

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)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{-6}-q^{-7}+4q^{-9}-3q^{-10}-4q^{-11}+10q^{-12}-4q^{-13}-11q^{-14}+17q^{-15}-2q^{-16}-19q^{-17}+21q^{-18}+2q^{-19}-25q^{-20}+20q^{-21}+6q^{-22}-25q^{-23}+17q^{-24}+6q^{-25}-18q^{-26}+10q^{-27}+3q^{-28}-8q^{-29}+4q^{-30}+q^{-31}-2q^{-32}+q^{-33}$
3	$q^{-9}-q^{-10}+q^{-12}+3q^{-13}-3q^{-14}-3q^{-15}+q^{-16}+10q^{-17}-3q^{-18}-11q^{-19}-5q^{-20}+20q^{-21}+6q^{-22}-18q^{-23}-18q^{-24}+24q^{-25}+22q^{-26}-19q^{-27}-33q^{-28}+19q^{-29}+36q^{-30}-11q^{-31}-45q^{-32}+8q^{-33}+46q^{-34}+q^{-35}-51q^{-36}-7q^{-37}+51q^{-38}+15q^{-39}-53q^{-40}-18q^{-41}+48q^{-42}+22q^{-43}-44q^{-44}-21q^{-45}+37q^{-46}+18q^{-47}-28q^{-48}-15q^{-49}+23q^{-50}+7q^{-51}-13q^{-52}-6q^{-53}+11q^{-54}+q^{-55}-6q^{-56}-q^{-57}+5q^{-58}-q^{-59}-q^{-60}-q^{-61}+2q^{-62}-q^{-63}$
4	$q^{-12}-q^{-13}+q^{-15}+3q^{-17}-4q^{-18}-2q^{-19}+3q^{-20}+11q^{-22}-8q^{-23}-10q^{-24}-q^{-25}-2q^{-26}+30q^{-27}-3q^{-28}-16q^{-29}-16q^{-30}-21q^{-31}+51q^{-32}+15q^{-33}-3q^{-34}-28q^{-35}-61q^{-36}+54q^{-37}+28q^{-38}+31q^{-39}-17q^{-40}-102q^{-41}+38q^{-42}+17q^{-43}+65q^{-44}+19q^{-45}-121q^{-46}+15q^{-47}-17q^{-48}+85q^{-49}+63q^{-50}-119q^{-51}-q^{-52}-61q^{-53}+91q^{-54}+103q^{-55}-103q^{-56}-16q^{-57}-104q^{-58}+94q^{-59}+137q^{-60}-82q^{-61}-30q^{-62}-138q^{-63}+87q^{-64}+158q^{-65}-54q^{-66}-32q^{-67}-157q^{-68}+65q^{-69}+152q^{-70}-27q^{-71}-12q^{-72}-144q^{-73}+33q^{-74}+113q^{-75}-14q^{-76}+14q^{-77}-100q^{-78}+12q^{-79}+60q^{-80}-17q^{-81}+27q^{-82}-50q^{-83}+6q^{-84}+24q^{-85}-20q^{-86}+21q^{-87}-19q^{-88}+7q^{-89}+8q^{-90}-16q^{-91}+11q^{-92}-6q^{-93}+4q^{-94}+3q^{-95}-7q^{-96}+4q^{-97}-2q^{-98}+q^{-99}+q^{-100}-2q^{-101}+q^{-102}$
5	$q^{-15}-q^{-16}+q^{-18}+2q^{-21}-3q^{-22}-2q^{-23}+3q^{-24}+3q^{-25}+5q^{-27}-7q^{-28}-10q^{-29}-q^{-30}+7q^{-31}+8q^{-32}+18q^{-33}-4q^{-34}-23q^{-35}-20q^{-36}-7q^{-37}+12q^{-38}+46q^{-39}+25q^{-40}-15q^{-41}-40q^{-42}-49q^{-43}-22q^{-44}+55q^{-45}+70q^{-46}+33q^{-47}-17q^{-48}-79q^{-49}-89q^{-50}+4q^{-51}+77q^{-52}+90q^{-53}+56q^{-54}-45q^{-55}-125q^{-56}-80q^{-57}+10q^{-58}+92q^{-59}+126q^{-60}+47q^{-61}-80q^{-62}-128q^{-63}-95q^{-64}+15q^{-65}+139q^{-66}+143q^{-67}+26q^{-68}-104q^{-69}-178q^{-70}-103q^{-71}+87q^{-72}+201q^{-73}+146q^{-74}-29q^{-75}-218q^{-76}-215q^{-77}+3q^{-78}+225q^{-79}+253q^{-80}+47q^{-81}-231q^{-82}-308q^{-83}-74q^{-84}+240q^{-85}+340q^{-86}+115q^{-87}-246q^{-88}-388q^{-89}-139q^{-90}+251q^{-91}+416q^{-92}+181q^{-93}-248q^{-94}-453q^{-95}-207q^{-96}+233q^{-97}+458q^{-98}+249q^{-99}-200q^{-100}-458q^{-101}-275q^{-102}+158q^{-103}+424q^{-104}+288q^{-105}-98q^{-106}-372q^{-107}-288q^{-108}+48q^{-109}+305q^{-110}+254q^{-111}-3q^{-112}-218q^{-113}-221q^{-114}-29q^{-115}+160q^{-116}+157q^{-117}+37q^{-118}-83q^{-119}-118q^{-120}-40q^{-121}+53q^{-122}+65q^{-123}+29q^{-124}-15q^{-125}-40q^{-126}-20q^{-127}+8q^{-128}+11q^{-129}+13q^{-130}+6q^{-131}-9q^{-132}-5q^{-133}-5q^{-135}+q^{-136}+10q^{-137}-4q^{-138}-q^{-139}+3q^{-140}-5q^{-141}-q^{-142}+5q^{-143}-2q^{-144}-q^{-145}+2q^{-146}-q^{-147}-q^{-148}+2q^{-149}-q^{-150}$
6	$q^{-18}-q^{-19}+q^{-21}-q^{-24}+3q^{-25}-3q^{-26}-2q^{-27}+4q^{-28}+2q^{-29}+2q^{-30}-5q^{-31}+6q^{-32}-8q^{-33}-10q^{-34}+5q^{-35}+7q^{-36}+12q^{-37}-4q^{-38}+18q^{-39}-15q^{-40}-31q^{-41}-11q^{-42}+24q^{-44}+8q^{-45}+61q^{-46}+4q^{-47}-42q^{-48}-48q^{-49}-45q^{-50}-7q^{-51}-9q^{-52}+124q^{-53}+70q^{-54}+16q^{-55}-43q^{-56}-89q^{-57}-93q^{-58}-120q^{-59}+117q^{-60}+113q^{-61}+129q^{-62}+64q^{-63}-18q^{-64}-129q^{-65}-276q^{-66}-5q^{-67}+14q^{-68}+156q^{-69}+182q^{-70}+185q^{-71}+5q^{-72}-308q^{-73}-113q^{-74}-206q^{-75}-13q^{-76}+126q^{-77}+355q^{-78}+253q^{-79}-124q^{-80}-26q^{-81}-357q^{-82}-291q^{-83}-164q^{-84}+313q^{-85}+420q^{-86}+167q^{-87}+284q^{-88}-282q^{-89}-481q^{-90}-558q^{-91}+50q^{-92}+373q^{-93}+381q^{-94}+683q^{-95}+5q^{-96}-478q^{-97}-885q^{-98}-304q^{-99}+139q^{-100}+440q^{-101}+1024q^{-102}+374q^{-103}-325q^{-104}-1077q^{-105}-618q^{-106}-162q^{-107}+388q^{-108}+1255q^{-109}+707q^{-110}-133q^{-111}-1175q^{-112}-853q^{-113}-425q^{-114}+327q^{-115}+1413q^{-116}+964q^{-117}+11q^{-118}-1256q^{-119}-1040q^{-120}-623q^{-121}+315q^{-122}+1556q^{-123}+1178q^{-124}+109q^{-125}-1348q^{-126}-1223q^{-127}-800q^{-128}+306q^{-129}+1676q^{-130}+1385q^{-131}+244q^{-132}-1359q^{-133}-1377q^{-134}-1002q^{-135}+183q^{-136}+1662q^{-137}+1543q^{-138}+467q^{-139}-1164q^{-140}-1371q^{-141}-1168q^{-142}-86q^{-143}+1395q^{-144}+1506q^{-145}+681q^{-146}-767q^{-147}-1099q^{-148}-1139q^{-149}-358q^{-150}+923q^{-151}+1189q^{-152}+713q^{-153}-353q^{-154}-653q^{-155}-860q^{-156}-451q^{-157}+465q^{-158}+722q^{-159}+528q^{-160}-103q^{-161}-256q^{-162}-483q^{-163}-357q^{-164}+181q^{-165}+333q^{-166}+280q^{-167}-25q^{-168}-37q^{-169}-199q^{-170}-207q^{-171}+62q^{-172}+113q^{-173}+110q^{-174}-18q^{-175}+36q^{-176}-58q^{-177}-102q^{-178}+25q^{-179}+25q^{-180}+33q^{-181}-17q^{-182}+38q^{-183}-8q^{-184}-46q^{-185}+12q^{-186}+q^{-187}+7q^{-188}-12q^{-189}+22q^{-190}+2q^{-191}-18q^{-192}+7q^{-193}-2q^{-194}+2q^{-195}-7q^{-196}+8q^{-197}+2q^{-198}-7q^{-199}+4q^{-200}-q^{-201}+q^{-202}-2q^{-203}+q^{-204}+q^{-205}-2q^{-206}+q^{-207}$
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[9, 9]]
Out[2]=	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[3]:=	GaussCode[Knot[9, 9]]
Out[3]=	GaussCode[-1, 8, -2, 9, -7, 1, -3, 5, -4, 6, -8, 2, -9, 7, -6, 3, -5, 4]
In[4]:=	DTCode[Knot[9, 9]]
Out[4]=	DTCode[6, 12, 14, 16, 18, 2, 4, 10, 8]
In[5]:=	br = BR[Knot[9, 9]]
Out[5]=	BR[3, {-1, -1, -1, -1, -1, -2, 1, -2, -2, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[9, 9]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[9, 9]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 9]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 3, 3, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 9]][t]
Out[10]=	2 4 6 2 3 -7 + -- - -- + - + 6 t - 4 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[9, 9]][z]
Out[11]=	2 4 6 1 + 8 z + 8 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 9]}
In[13]:=	{KnotDet[Knot[9, 9]], KnotSignature[Knot[9, 9]]}
Out[13]=	{31, -6}
In[14]:=	Jones[Knot[9, 9]][q]
Out[14]=	-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[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 9]}
In[16]:=	A2Invariant[Knot[9, 9]][q]
Out[16]=	-36 -32 -30 -26 2 -20 -18 2 -10 -q - q - q - q + --- + q + q + --- + q 24 14 q q
In[17]:=	HOMFLYPT[Knot[9, 9]][a, z]
Out[17]=	6 8 10 6 2 8 2 10 2 6 4 8 4 2 a + a - 2 a + 7 a z + 4 a z - 3 a z + 5 a z + 4 a z - 10 4 6 6 8 6 a z + a z + a z
In[18]:=	Kauffman[Knot[9, 9]][a, z]
Out[18]=	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[19]:=	{Vassiliev[2][Knot[9, 9]], Vassiliev[3][Knot[9, 9]]}
Out[19]=	{8, -22}
In[20]:=	Kh[Knot[9, 9]][q, t]
Out[20]=	-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
In[21]:=	ColouredJones[Knot[9, 9], 2][q]
Out[21]=	-33 2 -31 4 8 3 10 18 6 17 25 q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- + 32 30 29 28 27 26 25 24 23 q q q q q q q q q 6 20 25 2 21 19 2 17 11 4 10 --- + --- - --- + --- + --- - --- - --- + --- - --- - --- + --- - 22 21 20 19 18 17 16 15 14 13 12 q q q q q q q q q q q 4 3 4 -7 -6 --- - --- + -- - q + q 11 10 9 q q q

See/edit the Rolfsen_Splice_Template. Back to the top.

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