9 6

9_5

9_7

Visit 9 6's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 6's page at Knotilus! Visit 9 6's page at the original Knot Atlas! 9 6 Quick Notes

9 6 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,12,4,13 X5,14,6,15 X7,16,8,17 X9,18,10,1 X15,6,16,7 X17,8,18,9 X13,10,14,11 X11,2,12,3
Gauss code	-1, 9, -2, 1, -3, 6, -4, 7, -5, 8, -9, 2, -8, 3, -6, 4, -7, 5
Dowker-Thistlethwaite code	4 12 14 16 18 2 10 6 8
Conway Notation	[522]

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 7.2036 A-Polynomial See Data:9 6/A-polynomial

[edit Notes for 9 6'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 6's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-4t^{2}+5t-5+5t^{-1}-4t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+8z^{4}+7z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 27, -6 }
Jones polynomial	$q^{-3}-q^{-4}+3q^{-5}-3q^{-6}+4q^{-7}-5q^{-8}+4q^{-9}-3q^{-10}+2q^{-11}-q^{-12}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{10}-3z^{2}a^{10}-a^{10}+z^{6}a^{8}+4z^{4}a^{8}+3z^{2}a^{8}-a^{8}+z^{6}a^{6}+5z^{4}a^{6}+7z^{2}a^{6}+3a^{6}$
Kauffman polynomial (db, data sources)	$z^{3}a^{15}-za^{15}+2z^{4}a^{14}-2z^{2}a^{14}+2z^{5}a^{13}-z^{3}a^{13}+2z^{6}a^{12}-2z^{4}a^{12}+z^{2}a^{12}+2z^{7}a^{11}-5z^{5}a^{11}+6z^{3}a^{11}-2za^{11}+z^{8}a^{10}-2z^{6}a^{10}+2z^{4}a^{10}-3z^{2}a^{10}+a^{10}+3z^{7}a^{9}-10z^{5}a^{9}+8z^{3}a^{9}-za^{9}+z^{8}a^{8}-3z^{6}a^{8}+z^{4}a^{8}+z^{2}a^{8}-a^{8}+z^{7}a^{7}-3z^{5}a^{7}+2za^{7}+z^{6}a^{6}-5z^{4}a^{6}+7z^{2}a^{6}-3a^{6}$
The A2 invariant	$-q^{36}-2q^{26}-q^{22}+q^{20}+2q^{18}+q^{16}+2q^{14}+q^{10}$
The G2 invariant	$q^{196}-q^{194}+2q^{192}-2q^{190}+q^{186}-2q^{184}+4q^{182}-4q^{180}+4q^{178}-2q^{176}-q^{174}+3q^{172}-4q^{170}+4q^{168}-5q^{166}+3q^{164}-3q^{162}-q^{160}+3q^{158}-4q^{156}+5q^{154}-4q^{152}+2q^{150}-4q^{146}+4q^{144}-2q^{142}-q^{140}+6q^{138}-5q^{136}+2q^{134}+3q^{132}-6q^{130}+9q^{128}-10q^{126}+3q^{124}-5q^{120}+10q^{118}-12q^{116}+6q^{114}-3q^{112}-2q^{110}+3q^{108}-9q^{106}+6q^{104}-4q^{102}+4q^{98}-6q^{96}+4q^{94}+3q^{92}-5q^{90}+7q^{88}-6q^{86}+2q^{84}+5q^{82}-7q^{80}+11q^{78}-7q^{76}+5q^{74}+2q^{72}-4q^{70}+7q^{68}-5q^{66}+5q^{64}+2q^{58}-q^{56}+2q^{54}+q^{50}$

Further Quantum Invariants

Further quantum knot invariants for 9_6.

A1 Invariants.

Weight	Invariant
1	$-q^{25}+q^{23}-q^{21}+q^{19}-q^{17}-q^{15}+q^{13}+2q^{9}+q^{5}$
2	$q^{68}-q^{66}-q^{64}+2q^{62}-q^{60}+3q^{56}-3q^{54}-q^{52}+3q^{50}-2q^{48}-2q^{46}+q^{44}+q^{42}-q^{40}-q^{38}+2q^{36}-q^{34}-3q^{32}+2q^{30}-3q^{26}+2q^{24}+3q^{22}-2q^{20}+q^{18}+3q^{16}+q^{10}$
3	$-q^{129}+q^{127}+q^{125}-2q^{121}-q^{119}+2q^{117}+q^{115}-q^{111}-q^{109}-q^{107}+2q^{105}+3q^{103}-3q^{101}-5q^{99}+2q^{97}+7q^{95}+q^{93}-5q^{91}-2q^{89}+4q^{87}+3q^{85}-4q^{81}-2q^{79}+4q^{77}+q^{75}-4q^{73}-5q^{71}+4q^{69}+3q^{67}-3q^{65}-5q^{63}+4q^{61}+5q^{59}-q^{57}-6q^{55}-2q^{53}+5q^{51}+3q^{49}-5q^{47}-6q^{45}+q^{43}+6q^{41}+2q^{39}-6q^{37}-2q^{35}+4q^{33}+5q^{31}-q^{29}-2q^{27}+q^{25}+3q^{23}+q^{21}+q^{15}$
4	$q^{208}-q^{206}-q^{204}+4q^{198}-q^{196}-2q^{194}-3q^{192}-3q^{190}+8q^{188}+3q^{186}+q^{184}-6q^{182}-10q^{180}+6q^{178}+8q^{176}+9q^{174}-6q^{172}-20q^{170}-3q^{168}+10q^{166}+22q^{164}+5q^{162}-24q^{160}-17q^{158}+q^{156}+25q^{154}+16q^{152}-13q^{150}-17q^{148}-12q^{146}+11q^{144}+17q^{142}+2q^{140}-5q^{138}-12q^{136}-3q^{134}+5q^{132}+8q^{130}+7q^{128}-5q^{126}-11q^{124}-q^{122}+12q^{120}+10q^{118}-3q^{116}-15q^{114}-4q^{112}+15q^{110}+10q^{108}-6q^{106}-20q^{104}-8q^{102}+16q^{100}+14q^{98}-19q^{94}-14q^{92}+9q^{90}+14q^{88}+10q^{86}-9q^{84}-14q^{82}-3q^{80}+2q^{78}+15q^{76}+6q^{74}-6q^{72}-9q^{70}-13q^{68}+5q^{66}+10q^{64}+7q^{62}-16q^{58}-6q^{56}+3q^{54}+10q^{52}+9q^{50}-6q^{48}-6q^{46}-4q^{44}+3q^{42}+8q^{40}+q^{38}-2q^{34}+3q^{30}+q^{28}+q^{26}+q^{20}$
5	$-q^{305}+q^{303}+q^{301}-2q^{295}-2q^{293}+q^{291}+4q^{289}+2q^{287}+q^{285}-4q^{283}-8q^{281}-3q^{279}+5q^{277}+11q^{275}+7q^{273}-3q^{271}-14q^{269}-14q^{267}+3q^{265}+21q^{263}+19q^{261}-2q^{259}-25q^{257}-31q^{255}-5q^{253}+35q^{251}+44q^{249}+11q^{247}-40q^{245}-59q^{243}-24q^{241}+40q^{239}+78q^{237}+42q^{235}-35q^{233}-86q^{231}-58q^{229}+20q^{227}+82q^{225}+73q^{223}-q^{221}-69q^{219}-75q^{217}-20q^{215}+44q^{213}+63q^{211}+33q^{209}-17q^{207}-47q^{205}-38q^{203}-2q^{201}+25q^{199}+31q^{197}+19q^{195}-7q^{193}-23q^{191}-24q^{189}-6q^{187}+17q^{185}+26q^{183}+13q^{181}-14q^{179}-31q^{177}-15q^{175}+19q^{173}+32q^{171}+15q^{169}-21q^{167}-42q^{165}-16q^{163}+33q^{161}+48q^{159}+22q^{157}-27q^{155}-58q^{153}-30q^{151}+30q^{149}+62q^{147}+39q^{145}-18q^{143}-62q^{141}-51q^{139}+6q^{137}+55q^{135}+56q^{133}+9q^{131}-45q^{129}-58q^{127}-26q^{125}+25q^{123}+49q^{121}+35q^{119}-7q^{117}-35q^{115}-34q^{113}-12q^{111}+13q^{109}+26q^{107}+23q^{105}+6q^{103}-11q^{101}-18q^{99}-21q^{97}-8q^{95}+10q^{93}+22q^{91}+18q^{89}+5q^{87}-14q^{85}-28q^{83}-17q^{81}+3q^{79}+19q^{77}+22q^{75}+9q^{73}-11q^{71}-20q^{69}-14q^{67}+13q^{63}+14q^{61}+5q^{59}-3q^{57}-9q^{55}-6q^{53}+q^{51}+6q^{49}+4q^{47}+3q^{45}-2q^{41}+2q^{37}+q^{35}+q^{33}+q^{31}+q^{25}$
6	$q^{420}-q^{418}-q^{416}+2q^{410}+2q^{406}-3q^{404}-3q^{402}-q^{398}+4q^{396}+4q^{394}+8q^{392}-5q^{390}-8q^{388}-7q^{386}-8q^{384}+3q^{382}+11q^{380}+21q^{378}-8q^{374}-15q^{372}-19q^{370}-2q^{368}+18q^{366}+33q^{364}+2q^{362}-13q^{360}-25q^{358}-26q^{356}+5q^{354}+37q^{352}+46q^{350}-5q^{348}-42q^{346}-58q^{344}-38q^{342}+32q^{340}+96q^{338}+96q^{336}-3q^{334}-97q^{332}-141q^{330}-96q^{328}+43q^{326}+172q^{324}+194q^{322}+59q^{320}-113q^{318}-223q^{316}-201q^{314}-26q^{312}+176q^{310}+263q^{308}+163q^{306}-27q^{304}-195q^{302}-248q^{300}-132q^{298}+61q^{296}+204q^{294}+197q^{292}+88q^{290}-59q^{288}-167q^{286}-155q^{284}-58q^{282}+60q^{280}+114q^{278}+108q^{276}+48q^{274}-36q^{272}-85q^{270}-84q^{268}-36q^{266}+14q^{264}+60q^{262}+69q^{260}+36q^{258}-19q^{256}-65q^{254}-56q^{252}-15q^{250}+38q^{248}+65q^{246}+45q^{244}-19q^{242}-77q^{240}-65q^{238}+q^{236}+71q^{234}+92q^{232}+48q^{230}-49q^{228}-124q^{226}-98q^{224}+6q^{222}+109q^{220}+140q^{218}+79q^{216}-55q^{214}-163q^{212}-152q^{210}-27q^{208}+113q^{206}+179q^{204}+137q^{202}-10q^{200}-157q^{198}-196q^{196}-98q^{194}+56q^{192}+175q^{190}+196q^{188}+79q^{186}-87q^{184}-194q^{182}-169q^{180}-52q^{178}+95q^{176}+196q^{174}+158q^{172}+30q^{170}-112q^{168}-168q^{166}-135q^{164}-28q^{162}+103q^{160}+149q^{158}+108q^{156}+6q^{154}-71q^{152}-115q^{150}-92q^{148}-13q^{146}+49q^{144}+79q^{142}+53q^{140}+32q^{138}-15q^{136}-46q^{134}-42q^{132}-33q^{130}-9q^{128}+36q^{124}+40q^{122}+31q^{120}+14q^{118}-15q^{116}-35q^{114}-57q^{112}-24q^{110}+4q^{108}+34q^{106}+48q^{104}+36q^{102}+9q^{100}-37q^{98}-41q^{96}-36q^{94}-11q^{92}+17q^{90}+36q^{88}+34q^{86}+5q^{84}-8q^{82}-24q^{80}-22q^{78}-12q^{76}+7q^{74}+18q^{72}+10q^{70}+9q^{68}-q^{66}-7q^{64}-9q^{62}-2q^{60}+4q^{58}+2q^{56}+5q^{54}+3q^{52}+q^{50}-2q^{48}+2q^{44}+q^{40}+q^{38}+q^{36}+q^{30}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{36}-2q^{26}-q^{22}+q^{20}+2q^{18}+q^{16}+2q^{14}+q^{10}$
1,1	$q^{100}-2q^{98}+4q^{96}-6q^{94}+7q^{92}-10q^{90}+12q^{88}-10q^{86}+11q^{84}-10q^{82}+10q^{80}-10q^{78}+7q^{76}-10q^{74}+8q^{72}-4q^{70}-q^{68}+10q^{66}-16q^{64}+26q^{62}-31q^{60}+38q^{58}-40q^{56}+36q^{54}-35q^{52}+24q^{50}-22q^{48}+2q^{46}+2q^{44}-18q^{42}+20q^{40}-24q^{38}+28q^{36}-18q^{34}+22q^{32}-8q^{30}+12q^{28}-2q^{26}+4q^{24}+q^{20}$
2,0	$q^{90}-q^{84}+q^{80}+q^{78}+q^{76}-q^{74}+q^{70}-2q^{66}+q^{62}-2q^{60}-2q^{58}-q^{56}-3q^{52}-2q^{50}-q^{48}-3q^{46}-q^{44}+q^{42}+q^{40}+4q^{36}+4q^{34}+2q^{32}+q^{30}+3q^{28}+2q^{26}+q^{20}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{82}-q^{80}+2q^{76}-2q^{74}-q^{72}+3q^{70}-q^{68}-2q^{66}+3q^{64}-2q^{60}+q^{58}-q^{56}-q^{54}-2q^{52}+q^{50}-4q^{46}-q^{44}-q^{42}-4q^{40}-q^{38}+3q^{36}+4q^{32}+4q^{30}+2q^{28}+3q^{26}+2q^{24}+q^{20}$
1,0,0	$-q^{47}-q^{43}+q^{41}-q^{39}-2q^{35}-q^{33}-q^{31}+2q^{27}+q^{25}+3q^{23}+q^{21}+2q^{19}+q^{15}$
1,0,1	$q^{132}-2q^{130}+3q^{128}-q^{126}-3q^{124}+6q^{122}-8q^{120}+4q^{118}+2q^{116}-5q^{114}+9q^{112}-5q^{110}+2q^{108}+2q^{106}-5q^{104}-q^{102}+4q^{100}-11q^{98}+9q^{96}-9q^{92}+17q^{90}-16q^{88}+15q^{86}-6q^{84}+3q^{82}+7q^{80}-6q^{78}+11q^{76}-7q^{74}+9q^{72}-12q^{70}+7q^{68}-17q^{66}-5q^{64}-10q^{62}-18q^{60}+2q^{58}-22q^{56}+11q^{54}-8q^{52}+9q^{50}+12q^{48}+2q^{46}+19q^{44}+3q^{42}+10q^{40}+6q^{38}+2q^{36}+4q^{34}+q^{30}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{104}-q^{100}+q^{98}+2q^{96}-q^{94}-q^{92}+2q^{90}+q^{88}-q^{86}+2q^{82}-q^{80}-2q^{78}+q^{76}-q^{74}-3q^{72}-5q^{66}-3q^{64}-2q^{62}-4q^{60}-6q^{58}-3q^{56}-q^{54}-q^{52}+q^{50}+4q^{48}+5q^{46}+5q^{44}+7q^{42}+4q^{40}+4q^{38}+3q^{36}+2q^{34}+q^{30}$
1,0,0,0	$-q^{58}-q^{54}-q^{48}-2q^{44}-q^{42}-2q^{40}+2q^{34}+2q^{32}+2q^{30}+3q^{28}+q^{26}+2q^{24}+q^{20}$

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{114}-q^{112}+q^{110}-q^{108}+2q^{106}-2q^{104}+q^{102}-2q^{100}+3q^{98}-2q^{96}+q^{94}-2q^{92}+2q^{90}+q^{84}-2q^{82}+3q^{80}-4q^{78}+3q^{76}-6q^{74}+3q^{72}-5q^{70}+3q^{68}-5q^{66}+2q^{64}-4q^{62}-3q^{58}-2q^{56}-2q^{52}+2q^{50}+6q^{46}+q^{44}+6q^{42}+q^{40}+5q^{38}+q^{36}+2q^{34}+q^{30}$

G2 Invariants.

Weight

Invariant

1,0

$q^{196}-q^{194}+2q^{192}-2q^{190}+q^{186}-2q^{184}+4q^{182}-4q^{180}+4q^{178}-2q^{176}-q^{174}+3q^{172}-4q^{170}+4q^{168}-5q^{166}+3q^{164}-3q^{162}-q^{160}+3q^{158}-4q^{156}+5q^{154}-4q^{152}+2q^{150}-4q^{146}+4q^{144}-2q^{142}-q^{140}+6q^{138}-5q^{136}+2q^{134}+3q^{132}-6q^{130}+9q^{128}-10q^{126}+3q^{124}-5q^{120}+10q^{118}-12q^{116}+6q^{114}-3q^{112}-2q^{110}+3q^{108}-9q^{106}+6q^{104}-4q^{102}+4q^{98}-6q^{96}+4q^{94}+3q^{92}-5q^{90}+7q^{88}-6q^{86}+2q^{84}+5q^{82}-7q^{80}+11q^{78}-7q^{76}+5q^{74}+2q^{72}-4q^{70}+7q^{68}-5q^{66}+5q^{64}+2q^{58}-q^{56}+2q^{54}+q^{50}$

.

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 6"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-4t^{2}+5t-5+5t^{-1}-4t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{6}+8z^{4}+7z^{2}+1$

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]=

{ 27, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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

Vassiliev invariants

V2 and V3:

(7, -18)

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 $28$ $-144$ $392$ ${\frac {2834}{3}}$ ${\frac {382}{3}}$ $-4032$ $-6912$ $-1184$ $-784$ ${\frac {10976}{3}}$ $10368$ ${\frac {79352}{3}}$ ${\frac {10696}{3}}$ ${\frac {1559017}{30}}$ ${\frac {13882}{5}}$ ${\frac {797474}{45}}$ ${\frac {5687}{18}}$ ${\frac {64777}{30}}$

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 9 6. 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             1 1     0 -13           3 2       1 -15         2 1         -1 -17       2 3           -1 -19     1 2             1 -21   1 2               -1 -23   1                 1 -25 1                   -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{-6}-q^{-7}+4q^{-9}-3q^{-10}-3q^{-11}+9q^{-12}-4q^{-13}-8q^{-14}+12q^{-15}-2q^{-16}-13q^{-17}+14q^{-18}+q^{-19}-16q^{-20}+14q^{-21}+3q^{-22}-16q^{-23}+11q^{-24}+3q^{-25}-11q^{-26}+7q^{-27}+q^{-28}-5q^{-29}+4q^{-30}-2q^{-32}+q^{-33}$
3	$q^{-9}-q^{-10}+q^{-12}+3q^{-13}-3q^{-14}-3q^{-15}+2q^{-16}+9q^{-17}-4q^{-18}-9q^{-19}-2q^{-20}+17q^{-21}-14q^{-23}-9q^{-24}+18q^{-25}+8q^{-26}-12q^{-27}-16q^{-28}+14q^{-29}+13q^{-30}-6q^{-31}-17q^{-32}+5q^{-33}+15q^{-34}-16q^{-36}-4q^{-37}+16q^{-38}+5q^{-39}-13q^{-40}-10q^{-41}+14q^{-42}+9q^{-43}-10q^{-44}-9q^{-45}+8q^{-46}+6q^{-47}-4q^{-48}-3q^{-49}+3q^{-50}-q^{-51}-2q^{-52}+3q^{-53}+2q^{-54}-4q^{-55}-2q^{-56}+3q^{-57}+3q^{-58}-3q^{-59}-q^{-60}+2q^{-62}-q^{-63}$
4	$q^{-12}-q^{-13}+q^{-15}+3q^{-17}-4q^{-18}-2q^{-19}+3q^{-20}+q^{-21}+10q^{-22}-9q^{-23}-9q^{-24}+q^{-25}+q^{-26}+25q^{-27}-8q^{-28}-16q^{-29}-8q^{-30}-9q^{-31}+41q^{-32}-q^{-33}-13q^{-34}-13q^{-35}-27q^{-36}+45q^{-37}+2q^{-38}-q^{-39}-4q^{-40}-40q^{-41}+40q^{-42}-9q^{-43}+4q^{-44}+15q^{-45}-36q^{-46}+35q^{-47}-32q^{-48}-q^{-49}+34q^{-50}-22q^{-51}+37q^{-52}-56q^{-53}-13q^{-54}+48q^{-55}-6q^{-56}+42q^{-57}-75q^{-58}-24q^{-59}+60q^{-60}+7q^{-61}+44q^{-62}-88q^{-63}-34q^{-64}+66q^{-65}+19q^{-66}+45q^{-67}-91q^{-68}-42q^{-69}+57q^{-70}+26q^{-71}+52q^{-72}-76q^{-73}-48q^{-74}+34q^{-75}+21q^{-76}+56q^{-77}-47q^{-78}-39q^{-79}+10q^{-80}+3q^{-81}+49q^{-82}-18q^{-83}-22q^{-84}-2q^{-85}-10q^{-86}+32q^{-87}-4q^{-88}-7q^{-89}-3q^{-90}-12q^{-91}+16q^{-92}-q^{-95}-7q^{-96}+5q^{-97}+q^{-99}-2q^{-101}+q^{-102}$
5	$q^{-15}-q^{-16}+q^{-18}+2q^{-21}-3q^{-22}-2q^{-23}+3q^{-24}+3q^{-25}+q^{-26}+4q^{-27}-8q^{-28}-9q^{-29}+9q^{-31}+9q^{-32}+13q^{-33}-9q^{-34}-22q^{-35}-14q^{-36}+3q^{-37}+18q^{-38}+33q^{-39}+4q^{-40}-25q^{-41}-30q^{-42}-17q^{-43}+7q^{-44}+47q^{-45}+23q^{-46}-12q^{-47}-26q^{-48}-29q^{-49}-11q^{-50}+34q^{-51}+26q^{-52}-5q^{-53}-9q^{-54}-12q^{-55}-8q^{-56}+21q^{-57}+q^{-58}-27q^{-59}-10q^{-60}+16q^{-61}+34q^{-62}+35q^{-63}-23q^{-64}-78q^{-65}-42q^{-66}+29q^{-67}+88q^{-68}+82q^{-69}-24q^{-70}-127q^{-71}-99q^{-72}+18q^{-73}+132q^{-74}+139q^{-75}-q^{-76}-159q^{-77}-159q^{-78}-10q^{-79}+163q^{-80}+188q^{-81}+25q^{-82}-174q^{-83}-208q^{-84}-36q^{-85}+184q^{-86}+224q^{-87}+42q^{-88}-187q^{-89}-242q^{-90}-52q^{-91}+201q^{-92}+251q^{-93}+55q^{-94}-196q^{-95}-265q^{-96}-70q^{-97}+202q^{-98}+267q^{-99}+81q^{-100}-184q^{-101}-271q^{-102}-97q^{-103}+166q^{-104}+258q^{-105}+111q^{-106}-134q^{-107}-241q^{-108}-116q^{-109}+102q^{-110}+203q^{-111}+117q^{-112}-66q^{-113}-167q^{-114}-107q^{-115}+40q^{-116}+125q^{-117}+89q^{-118}-15q^{-119}-90q^{-120}-71q^{-121}+2q^{-122}+61q^{-123}+54q^{-124}+4q^{-125}-39q^{-126}-38q^{-127}-7q^{-128}+21q^{-129}+28q^{-130}+10q^{-131}-16q^{-132}-17q^{-133}-5q^{-134}+3q^{-135}+11q^{-136}+10q^{-137}-5q^{-138}-7q^{-139}-q^{-140}-3q^{-141}+3q^{-142}+5q^{-143}-q^{-144}-2q^{-145}-q^{-147}+2q^{-149}-q^{-150}$
6	$q^{-18}-q^{-19}+q^{-21}-q^{-24}+3q^{-25}-3q^{-26}-2q^{-27}+4q^{-28}+2q^{-29}+2q^{-30}-4q^{-31}+5q^{-32}-9q^{-33}-9q^{-34}+6q^{-35}+8q^{-36}+12q^{-37}-3q^{-38}+13q^{-39}-20q^{-40}-28q^{-41}-4q^{-42}+6q^{-43}+28q^{-44}+10q^{-45}+42q^{-46}-18q^{-47}-47q^{-48}-32q^{-49}-19q^{-50}+23q^{-51}+14q^{-52}+88q^{-53}+9q^{-54}-35q^{-55}-46q^{-56}-49q^{-57}-5q^{-58}-19q^{-59}+110q^{-60}+29q^{-61}-6q^{-62}-29q^{-63}-40q^{-64}-9q^{-65}-55q^{-66}+101q^{-67}+5q^{-68}-15q^{-69}-33q^{-70}-9q^{-71}+38q^{-72}-34q^{-73}+127q^{-74}-25q^{-75}-77q^{-76}-112q^{-77}-32q^{-78}+82q^{-79}+43q^{-80}+229q^{-81}+16q^{-82}-123q^{-83}-243q^{-84}-139q^{-85}+49q^{-86}+99q^{-87}+371q^{-88}+144q^{-89}-85q^{-90}-344q^{-91}-286q^{-92}-68q^{-93}+74q^{-94}+478q^{-95}+310q^{-96}+32q^{-97}-365q^{-98}-405q^{-99}-222q^{-100}-24q^{-101}+517q^{-102}+457q^{-103}+179q^{-104}-323q^{-105}-471q^{-106}-362q^{-107}-149q^{-108}+506q^{-109}+565q^{-110}+313q^{-111}-262q^{-112}-502q^{-113}-465q^{-114}-253q^{-115}+480q^{-116}+640q^{-117}+410q^{-118}-218q^{-119}-523q^{-120}-535q^{-121}-319q^{-122}+468q^{-123}+698q^{-124}+474q^{-125}-198q^{-126}-550q^{-127}-588q^{-128}-360q^{-129}+459q^{-130}+744q^{-131}+529q^{-132}-165q^{-133}-559q^{-134}-634q^{-135}-410q^{-136}+411q^{-137}+743q^{-138}+578q^{-139}-81q^{-140}-499q^{-141}-628q^{-142}-464q^{-143}+293q^{-144}+646q^{-145}+566q^{-146}+27q^{-147}-352q^{-148}-519q^{-149}-457q^{-150}+150q^{-151}+453q^{-152}+450q^{-153}+80q^{-154}-184q^{-155}-329q^{-156}-357q^{-157}+63q^{-158}+251q^{-159}+275q^{-160}+58q^{-161}-74q^{-162}-157q^{-163}-222q^{-164}+41q^{-165}+122q^{-166}+136q^{-167}+13q^{-168}-30q^{-169}-63q^{-170}-123q^{-171}+41q^{-172}+58q^{-173}+66q^{-174}-7q^{-175}-14q^{-176}-26q^{-177}-72q^{-178}+32q^{-179}+26q^{-180}+35q^{-181}-6q^{-182}-2q^{-183}-11q^{-184}-41q^{-185}+17q^{-186}+6q^{-187}+18q^{-188}-2q^{-189}+5q^{-190}-3q^{-191}-20q^{-192}+7q^{-193}-2q^{-194}+7q^{-195}-q^{-196}+4q^{-197}-7q^{-199}+3q^{-200}-2q^{-201}+2q^{-202}+q^{-204}-2q^{-206}+q^{-207}$
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, 6]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[5, 14, 6, 15], X[7, 16, 8, 17], X[9, 18, 10, 1], X[15, 6, 16, 7], X[17, 8, 18, 9], X[13, 10, 14, 11], X[11, 2, 12, 3]]
In[3]:=	GaussCode[Knot[9, 6]]
Out[3]=	GaussCode[-1, 9, -2, 1, -3, 6, -4, 7, -5, 8, -9, 2, -8, 3, -6, 4, -7, 5]
In[4]:=	DTCode[Knot[9, 6]]
Out[4]=	DTCode[4, 12, 14, 16, 18, 2, 10, 6, 8]
In[5]:=	br = BR[Knot[9, 6]]
Out[5]=	BR[3, {-1, -1, -1, -1, -1, -1, -2, 1, -2, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[9, 6]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[9, 6]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 6]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 3, 3, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 6]][t]
Out[10]=	2 4 5 2 3 -5 + -- - -- + - + 5 t - 4 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[9, 6]][z]
Out[11]=	2 4 6 1 + 7 z + 8 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 6]}
In[13]:=	{KnotDet[Knot[9, 6]], KnotSignature[Knot[9, 6]]}
Out[13]=	{27, -6}
In[14]:=	Jones[Knot[9, 6]][q]
Out[14]=	-12 2 3 4 5 4 3 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, 6]}
In[16]:=	A2Invariant[Knot[9, 6]][q]
Out[16]=	-36 2 -22 -20 2 -16 2 -10 -q - --- - q + q + --- + q + --- + q 26 18 14 q q q
In[17]:=	HOMFLYPT[Knot[9, 6]][a, z]
Out[17]=	6 8 10 6 2 8 2 10 2 6 4 8 4 3 a - a - a + 7 a z + 3 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, 6]][a, z]
Out[18]=	6 8 10 7 9 11 15 6 2 8 2 -3 a - a + a + 2 a z - a z - 2 a z - a z + 7 a z + a z - 10 2 12 2 14 2 9 3 11 3 13 3 15 3 3 a z + a z - 2 a z + 8 a z + 6 a z - a z + a z - 6 4 8 4 10 4 12 4 14 4 7 5 5 a z + a z + 2 a z - 2 a z + 2 a z - 3 a z - 9 5 11 5 13 5 6 6 8 6 10 6 10 a z - 5 a z + 2 a z + a z - 3 a z - 2 a z + 12 6 7 7 9 7 11 7 8 8 10 8 2 a z + a z + 3 a z + 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[9, 6]], Vassiliev[3][Knot[9, 6]]}
Out[19]=	{7, -18}
In[20]:=	Kh[Knot[9, 6]][q, t]
Out[20]=	-7 -5 1 1 1 2 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 2 3 2 1 3 2 1 ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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, 6], 2][q]
Out[21]=	-33 2 4 5 -28 7 11 3 11 16 3 q - --- + --- - --- + q + --- - --- + --- + --- - --- + --- + 32 30 29 27 26 25 24 23 22 q q q q q q q q q 14 16 -19 14 13 2 12 8 4 9 3 --- - --- + q + --- - --- - --- + --- - --- - --- + --- - --- - 21 20 18 17 16 15 14 13 12 11 q q q q q q q q q q 3 4 -7 -6 --- + -- - q + q 10 9 q q

See/edit the Rolfsen_Splice_Template. Back to the top.

9_5 9_7