9 38: Difference between revisions

Revision as of 21:09, 29 August 2005

9_37

9_39

Visit 9 38's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 38's page at Knotilus!

Visit 9 38's page at the original Knot Atlas!

9 38 Quick Notes

9 38 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₆₂₇ X_5,14,6,15 X_7,18,8,1 X_15,8,16,9 X_3,10,4,11 X_9,4,10,5 X_17,12,18,13 X_11,16,12,17 X_13,2,14,3
Gauss code	-1, 9, -5, 6, -2, 1, -3, 4, -6, 5, -8, 7, -9, 2, -4, 8, -7, 3
Dowker-Thistlethwaite code	6 10 14 18 4 16 2 8 12
Conway Notation	[.2.2.2]

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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{2,3\}}
3-genus	2
Bridge index	3
Super bridge index	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{4,7\}}
Nakanishi index	2
Maximal Thurston-Bennequin number	[-14][3]
Hyperbolic Volume	12.9329
A-Polynomial	See Data:9 38/A-polynomial

[edit Notes for 9 38's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	$2$
Topological 4 genus	$2$
Concordance genus	$2$
Rasmussen s-Invariant	-4

[edit Notes for 9 38's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$5t^{2}-14t+19-14t^{-1}+5t^{-2}$
Conway polynomial	$5z^{4}+6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 57, -4 }
Jones polynomial	$q^{-2}-3q^{-3}+7q^{-4}-8q^{-5}+10q^{-6}-10q^{-7}+8q^{-8}-6q^{-9}+3q^{-10}-q^{-11}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{10}+z^{4}a^{8}-z^{2}a^{8}-3a^{8}+3z^{4}a^{6}+7z^{2}a^{6}+4a^{6}+z^{4}a^{4}+z^{2}a^{4}$
Kauffman polynomial (db, data sources)	$z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-6z^{4}a^{12}+3z^{2}a^{12}+4z^{7}a^{11}-7z^{5}a^{11}+3z^{3}a^{11}-za^{11}+2z^{8}a^{10}+3z^{6}a^{10}-10z^{4}a^{10}+3z^{2}a^{10}+9z^{7}a^{9}-15z^{5}a^{9}+5z^{3}a^{9}+za^{9}+2z^{8}a^{8}+6z^{6}a^{8}-15z^{4}a^{8}+10z^{2}a^{8}-3a^{8}+5z^{7}a^{7}-4z^{5}a^{7}-2z^{3}a^{7}+3za^{7}+6z^{6}a^{6}-10z^{4}a^{6}+9z^{2}a^{6}-4a^{6}+3z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-z^{2}a^{4}$
The A2 invariant	$-q^{34}+q^{32}+q^{30}-3q^{28}-2q^{24}-q^{22}+2q^{20}+4q^{16}+q^{12}+2q^{10}-2q^{8}+q^{6}$
The G2 invariant	$q^{176}-2q^{174}+5q^{172}-8q^{170}+8q^{168}-6q^{166}-2q^{164}+18q^{162}-33q^{160}+47q^{158}-46q^{156}+21q^{154}+16q^{152}-65q^{150}+101q^{148}-104q^{146}+70q^{144}-6q^{142}-63q^{140}+112q^{138}-116q^{136}+77q^{134}-8q^{132}-60q^{130}+87q^{128}-70q^{126}+15q^{124}+52q^{122}-95q^{120}+98q^{118}-56q^{116}-20q^{114}+93q^{112}-152q^{110}+153q^{108}-105q^{106}+19q^{104}+69q^{102}-137q^{100}+159q^{98}-125q^{96}+52q^{94}+26q^{92}-90q^{90}+105q^{88}-66q^{86}+q^{84}+64q^{82}-84q^{80}+67q^{78}-9q^{76}-58q^{74}+106q^{72}-112q^{70}+82q^{68}-20q^{66}-43q^{64}+89q^{62}-94q^{60}+77q^{58}-36q^{56}+25q^{52}-40q^{50}+37q^{48}-25q^{46}+13q^{44}-5q^{40}+6q^{38}-6q^{36}+4q^{34}-2q^{32}+q^{30}$

Further Quantum Invariants

Further quantum knot invariants for 9_38.

A1 Invariants.

Weight	Invariant
1	$-q^{23}+2q^{21}-3q^{19}+2q^{17}-2q^{15}+2q^{11}-q^{9}+4q^{7}-2q^{5}+q^{3}$
2	$q^{64}-2q^{62}-q^{60}+8q^{58}-5q^{56}-11q^{54}+17q^{52}+q^{50}-21q^{48}+15q^{46}+10q^{44}-20q^{42}+4q^{40}+12q^{38}-8q^{36}-9q^{34}+6q^{32}+9q^{30}-17q^{28}-q^{26}+23q^{24}-14q^{22}-9q^{20}+21q^{18}-5q^{16}-9q^{14}+8q^{12}+q^{10}-2q^{8}+q^{6}$
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{123}+2 q^{121}+q^{119}-4 q^{117}-5 q^{115}+8 q^{113}+17 q^{111}-11 q^{109}-36 q^{107}+q^{105}+57 q^{103}+25 q^{101}-73 q^{99}-57 q^{97}+71 q^{95}+95 q^{93}-52 q^{91}-124 q^{89}+22 q^{87}+134 q^{85}+11 q^{83}-128 q^{81}-41 q^{79}+112 q^{77}+61 q^{75}-84 q^{73}-69 q^{71}+50 q^{69}+79 q^{67}-19 q^{65}-79 q^{63}-23 q^{61}+79 q^{59}+57 q^{57}-70 q^{55}-100 q^{53}+53 q^{51}+126 q^{49}-27 q^{47}-139 q^{45}-6 q^{43}+131 q^{41}+37 q^{39}-103 q^{37}-54 q^{35}+72 q^{33}+50 q^{31}-33 q^{29}-42 q^{27}+15 q^{25}+27 q^{23}-3 q^{21}-11 q^{19}+5 q^{15}+q^{13}-2 q^{11}+q^9}
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{200}-2 q^{198}-q^{196}+4 q^{194}+q^{192}+2 q^{190}-14 q^{188}-11 q^{186}+19 q^{184}+27 q^{182}+32 q^{180}-51 q^{178}-95 q^{176}-18 q^{174}+88 q^{172}+199 q^{170}+20 q^{168}-224 q^{166}-259 q^{164}-40 q^{162}+414 q^{160}+367 q^{158}-92 q^{156}-533 q^{154}-498 q^{152}+294 q^{150}+718 q^{148}+409 q^{146}-425 q^{144}-921 q^{142}-199 q^{140}+661 q^{138}+851 q^{136}+23 q^{134}-917 q^{132}-621 q^{130}+282 q^{128}+892 q^{126}+390 q^{124}-593 q^{122}-701 q^{120}-62 q^{118}+649 q^{116}+505 q^{114}-231 q^{112}-584 q^{110}-273 q^{108}+361 q^{106}+518 q^{104}+106 q^{102}-442 q^{100}-483 q^{98}+30 q^{96}+537 q^{94}+518 q^{92}-228 q^{90}-717 q^{88}-423 q^{86}+427 q^{84}+917 q^{82}+183 q^{80}-711 q^{78}-860 q^{76}+29 q^{74}+975 q^{72}+609 q^{70}-310 q^{68}-903 q^{66}-415 q^{64}+561 q^{62}+657 q^{60}+153 q^{58}-502 q^{56}-485 q^{54}+93 q^{52}+331 q^{50}+261 q^{48}-104 q^{46}-245 q^{44}-59 q^{42}+63 q^{40}+127 q^{38}+14 q^{36}-60 q^{34}-25 q^{32}-6 q^{30}+30 q^{28}+9 q^{26}-9 q^{24}-2 q^{22}-3 q^{20}+5 q^{18}+q^{16}-2 q^{14}+q^{12}}
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{295}+2 q^{293}+q^{291}-4 q^{289}-q^{287}+2 q^{285}+4 q^{283}+8 q^{281}+3 q^{279}-22 q^{277}-33 q^{275}-7 q^{273}+38 q^{271}+85 q^{269}+73 q^{267}-33 q^{265}-188 q^{263}-231 q^{261}-58 q^{259}+258 q^{257}+497 q^{255}+370 q^{253}-174 q^{251}-802 q^{249}-921 q^{247}-237 q^{245}+898 q^{243}+1612 q^{241}+1111 q^{239}-536 q^{237}-2185 q^{235}-2311 q^{233}-455 q^{231}+2207 q^{229}+3536 q^{227}+2067 q^{225}-1463 q^{223}-4335 q^{221}-3908 q^{219}-91 q^{217}+4298 q^{215}+5538 q^{213}+2172 q^{211}-3367 q^{209}-6484 q^{207}-4264 q^{205}+1709 q^{203}+6512 q^{201}+5917 q^{199}+248 q^{197}-5715 q^{195}-6815 q^{193}-2036 q^{191}+4383 q^{189}+6870 q^{187}+3358 q^{185}-2886 q^{183}-6293 q^{181}-4057 q^{179}+1545 q^{177}+5352 q^{175}+4193 q^{173}-499 q^{171}-4314 q^{169}-4003 q^{167}-233 q^{165}+3404 q^{163}+3692 q^{161}+729 q^{159}-2619 q^{157}-3475 q^{155}-1256 q^{153}+2009 q^{151}+3453 q^{149}+1874 q^{147}-1345 q^{145}-3561 q^{143}-2812 q^{141}+530 q^{139}+3729 q^{137}+3936 q^{135}+651 q^{133}-3654 q^{131}-5182 q^{129}-2181 q^{127}+3138 q^{125}+6194 q^{123}+3972 q^{121}-2022 q^{119}-6668 q^{117}-5687 q^{115}+350 q^{113}+6292 q^{111}+6943 q^{109}+1597 q^{107}-5076 q^{105}-7334 q^{103}-3374 q^{101}+3203 q^{99}+6731 q^{97}+4536 q^{95}-1149 q^{93}-5334 q^{91}-4783 q^{89}-550 q^{87}+3477 q^{85}+4201 q^{83}+1609 q^{81}-1770 q^{79}-3114 q^{77}-1837 q^{75}+474 q^{73}+1900 q^{71}+1586 q^{69}+198 q^{67}-956 q^{65}-1062 q^{63}-387 q^{61}+342 q^{59}+591 q^{57}+332 q^{55}-63 q^{53}-275 q^{51}-199 q^{49}-12 q^{47}+100 q^{45}+96 q^{43}+26 q^{41}-29 q^{39}-45 q^{37}-10 q^{35}+15 q^{33}+12 q^{31}+3 q^{29}-5 q^{25}-3 q^{23}+5 q^{21}+q^{19}-2 q^{17}+q^{15}}

A2 Invariants.

Weight	Invariant
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{34}+q^{32}+q^{30}-3 q^{28}-2 q^{24}-q^{22}+2 q^{20}+4 q^{16}+q^{12}+2 q^{10}-2 q^8+q^6}
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{92}-4 q^{90}+12 q^{88}-28 q^{86}+58 q^{84}-106 q^{82}+176 q^{80}-260 q^{78}+349 q^{76}-430 q^{74}+474 q^{72}-466 q^{70}+394 q^{68}-256 q^{66}+58 q^{64}+176 q^{62}-408 q^{60}+628 q^{58}-794 q^{56}+896 q^{54}-910 q^{52}+836 q^{50}-700 q^{48}+484 q^{46}-260 q^{44}+10 q^{42}+192 q^{40}-348 q^{38}+441 q^{36}-456 q^{34}+438 q^{32}-366 q^{30}+290 q^{28}-200 q^{26}+136 q^{24}-82 q^{22}+46 q^{20}-20 q^{18}+10 q^{16}-4 q^{14}+q^{12}}
2,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{86}-q^{84}-q^{82}+q^{80}+5 q^{78}-9 q^{74}-3 q^{72}+8 q^{70}+5 q^{68}-7 q^{66}+12 q^{62}+4 q^{60}-11 q^{58}-4 q^{56}+4 q^{54}-7 q^{52}-8 q^{50}-2 q^{48}-q^{46}-5 q^{44}+4 q^{42}+7 q^{40}-6 q^{38}+q^{36}+14 q^{34}+4 q^{32}-11 q^{30}+3 q^{28}+12 q^{26}-9 q^{22}+q^{20}+7 q^{18}-q^{16}-2 q^{14}+q^{12}}

A3 Invariants.

Weight	Invariant
0,1,0	$q^{74}-2q^{72}+q^{70}+4q^{68}-8q^{66}+4q^{64}+9q^{62}-15q^{60}+7q^{58}+10q^{56}-17q^{54}+4q^{52}+10q^{50}-10q^{48}-3q^{46}+2q^{44}-2q^{42}-8q^{40}-7q^{38}+12q^{36}-3q^{34}-8q^{32}+21q^{30}+q^{28}-10q^{26}+16q^{24}-q^{22}-7q^{20}+6q^{18}-2q^{14}+q^{12}$
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{45}+q^{43}+q^{39}-3 q^{37}-4 q^{33}-q^{31}-q^{29}+2 q^{27}+2 q^{25}+2 q^{23}+4 q^{21}+2 q^{17}-q^{15}+2 q^{13}-2 q^{11}+q^9}

A4 Invariants.

Weight

Invariant

0,1,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{96}-q^{94}-3 q^{92}+3 q^{90}+5 q^{88}-4 q^{86}-4 q^{84}+9 q^{82}+6 q^{80}-11 q^{78}-4 q^{76}+14 q^{74}+q^{72}-14 q^{70}+6 q^{68}+11 q^{66}-11 q^{64}-11 q^{62}+4 q^{60}-9 q^{58}-20 q^{56}-q^{54}+6 q^{52}-11 q^{50}-q^{48}+23 q^{46}+7 q^{44}-7 q^{42}+10 q^{40}+16 q^{38}-3 q^{36}-6 q^{34}+7 q^{32}+6 q^{30}-5 q^{28}-2 q^{26}+4 q^{24}-2 q^{20}+q^{18}}

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{56}+q^{54}+q^{48}-3 q^{46}-4 q^{42}-3 q^{40}-q^{38}-q^{36}+2 q^{34}+2 q^{32}+4 q^{30}+2 q^{28}+4 q^{26}+2 q^{22}-q^{18}+2 q^{16}-2 q^{14}+q^{12}}

B2 Invariants.

Weight

Invariant

0,1

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{74}+2 q^{72}-5 q^{70}+8 q^{68}-12 q^{66}+16 q^{64}-17 q^{62}+17 q^{60}-15 q^{58}+10 q^{56}-3 q^{54}-6 q^{52}+14 q^{50}-24 q^{48}+29 q^{46}-34 q^{44}+32 q^{42}-30 q^{40}+23 q^{38}-14 q^{36}+7 q^{34}+4 q^{32}-9 q^{30}+17 q^{28}-16 q^{26}+18 q^{24}-15 q^{22}+13 q^{20}-8 q^{18}+4 q^{16}-2 q^{14}+q^{12}}

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{120}-2 q^{116}-2 q^{114}+3 q^{112}+6 q^{110}-q^{108}-10 q^{106}-5 q^{104}+12 q^{102}+13 q^{100}-7 q^{98}-18 q^{96}-2 q^{94}+19 q^{92}+10 q^{90}-14 q^{88}-15 q^{86}+6 q^{84}+15 q^{82}-14 q^{78}-3 q^{76}+11 q^{74}+4 q^{72}-13 q^{70}-9 q^{68}+7 q^{66}+9 q^{64}-8 q^{62}-14 q^{60}+4 q^{58}+15 q^{56}+2 q^{54}-16 q^{52}-5 q^{50}+17 q^{48}+17 q^{46}-8 q^{44}-16 q^{42}+q^{40}+17 q^{38}+8 q^{36}-8 q^{34}-10 q^{32}+2 q^{30}+7 q^{28}+2 q^{26}-2 q^{24}-2 q^{22}+q^{18}}

D4 Invariants.

Weight

Invariant

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{102}-2 q^{100}+3 q^{98}-4 q^{96}+7 q^{94}-10 q^{92}+11 q^{90}-12 q^{88}+15 q^{86}-14 q^{84}+12 q^{82}-10 q^{80}+9 q^{78}-4 q^{76}-4 q^{74}+6 q^{72}-9 q^{70}+16 q^{68}-23 q^{66}+21 q^{64}-26 q^{62}+24 q^{60}-29 q^{58}+16 q^{56}-23 q^{54}+15 q^{52}-9 q^{50}+5 q^{48}+q^{46}+q^{44}+16 q^{42}-7 q^{40}+15 q^{38}-12 q^{36}+17 q^{34}-10 q^{32}+10 q^{30}-10 q^{28}+7 q^{26}-3 q^{24}+2 q^{22}-2 q^{20}+q^{18}}

G2 Invariants.

Weight

Invariant

1,0

q^{176}-2q^{174}+5q^{172}-8q^{170}+8q^{168}-6q^{166}-2q^{164}+18q^{162}-33q^{160}+47q^{158}-46q^{156}+21q^{154}+16q^{152}-65q^{150}+101q^{148}-104q^{146}+70q^{144}-6q^{142}-63q^{140}+112q^{138}-116q^{136}+77q^{134}-8q^{132}-60q^{130}+87q^{128}-70q^{126}+15q^{124}+52q^{122}-95q^{120}+98q^{118}-56q^{116}-20q^{114}+93q^{112}-152q^{110}+153q^{108}-105q^{106}+19q^{104}+69q^{102}-137q^{100}+159q^{98}-125q^{96}+52q^{94}+26q^{92}-90q^{90}+105q^{88}-66q^{86}+q^{84}+64q^{82}-84q^{80}+67q^{78}-9q^{76}-58q^{74}+106q^{72}-112q^{70}+82q^{68}-20q^{66}-43q^{64}+89q^{62}-94q^{60}+77q^{58}-36q^{56}+25q^{52}-40q^{50}+37q^{48}-25q^{46}+13q^{44}-5q^{40}+6q^{38}-6q^{36}+4q^{34}-2q^{32}+q^{30}

.

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.

In[3]:= K = Knot["9 38"];

In[4]:= Alexander[K][t]

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

Out[4]= $5t^{2}-14t+19-14t^{-1}+5t^{-2}$

In[5]:= Conway[K][z]

Out[5]= $5z^{4}+6z^{2}+1$

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]= { 57, -4 }

In[8]:= Jones[K][q]

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

Out[8]= $q^{-2}-3q^{-3}+7q^{-4}-8q^{-5}+10q^{-6}-10q^{-7}+8q^{-8}-6q^{-9}+3q^{-10}-q^{-11}$

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= $-z^{2}a^{10}+z^{4}a^{8}-z^{2}a^{8}-3a^{8}+3z^{4}a^{6}+7z^{2}a^{6}+4a^{6}+z^{4}a^{4}+z^{2}a^{4}$

In[10]:= Kauffman[K][a, z]

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

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_63, ...}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {...}

Vassiliev invariants

V₂ and V₃:

(6, -14)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 24}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -112}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 288}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 684}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 100}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2688}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{13888}{3}}

-{\frac {2368}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -592}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2304}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6272}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 16416}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2400}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{160231}{5}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{15308}{15}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{59748}{5}}

259

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{7671}{5}}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} -4 is the signature of 9 38. 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

-3

1

-5

3

1

-2

-7

4

-9

4

3

-1

-11

6

4

2

-13

4

0

-15

4

6

-2

-17

2

4

2

-19

1

4

-3

-21

2

-23

1

-1

Integral Khovanov Homology

(db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-5}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-3}
$r=-9$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-8}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2}	${\mathbb {Z} }$
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-7}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-6}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-5}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{6}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-4} -3 q^{-5} +3 q^{-6} +8 q^{-7} -20 q^{-8} +7 q^{-9} +34 q^{-10} -50 q^{-11} +2 q^{-12} +71 q^{-13} -74 q^{-14} -14 q^{-15} +97 q^{-16} -77 q^{-17} -29 q^{-18} +98 q^{-19} -57 q^{-20} -37 q^{-21} +74 q^{-22} -27 q^{-23} -32 q^{-24} +38 q^{-25} -5 q^{-26} -16 q^{-27} +10 q^{-28} + q^{-29} -3 q^{-30} + q^{-31} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-6} -3 q^{-7} +3 q^{-8} +4 q^{-9} -4 q^{-10} -14 q^{-11} +11 q^{-12} +34 q^{-13} -16 q^{-14} -71 q^{-15} +20 q^{-16} +117 q^{-17} +6 q^{-18} -197 q^{-19} -29 q^{-20} +257 q^{-21} +100 q^{-22} -334 q^{-23} -162 q^{-24} +369 q^{-25} +253 q^{-26} -407 q^{-27} -315 q^{-28} +399 q^{-29} +380 q^{-30} -385 q^{-31} -417 q^{-32} +343 q^{-33} +440 q^{-34} -287 q^{-35} -446 q^{-36} +224 q^{-37} +425 q^{-38} -142 q^{-39} -395 q^{-40} +71 q^{-41} +338 q^{-42} -3 q^{-43} -272 q^{-44} -41 q^{-45} +192 q^{-46} +69 q^{-47} -125 q^{-48} -65 q^{-49} +64 q^{-50} +53 q^{-51} -27 q^{-52} -33 q^{-53} +8 q^{-54} +16 q^{-55} -2 q^{-56} -5 q^{-57} - q^{-58} +3 q^{-59} - q^{-60} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-8} -3 q^{-9} +3 q^{-10} +4 q^{-11} -8 q^{-12} +2 q^{-13} -10 q^{-14} +21 q^{-15} +25 q^{-16} -44 q^{-17} -17 q^{-18} -45 q^{-19} +95 q^{-20} +138 q^{-21} -108 q^{-22} -139 q^{-23} -231 q^{-24} +236 q^{-25} +503 q^{-26} -38 q^{-27} -377 q^{-28} -809 q^{-29} +219 q^{-30} +1158 q^{-31} +466 q^{-32} -473 q^{-33} -1785 q^{-34} -269 q^{-35} +1751 q^{-36} +1385 q^{-37} -107 q^{-38} -2731 q^{-39} -1158 q^{-40} +1900 q^{-41} +2279 q^{-42} +627 q^{-43} -3221 q^{-44} -2008 q^{-45} +1606 q^{-46} +2768 q^{-47} +1373 q^{-48} -3202 q^{-49} -2515 q^{-50} +1093 q^{-51} +2809 q^{-52} +1921 q^{-53} -2790 q^{-54} -2672 q^{-55} +459 q^{-56} +2498 q^{-57} +2274 q^{-58} -2054 q^{-59} -2528 q^{-60} -252 q^{-61} +1859 q^{-62} +2382 q^{-63} -1071 q^{-64} -2026 q^{-65} -862 q^{-66} +956 q^{-67} +2086 q^{-68} -131 q^{-69} -1198 q^{-70} -1052 q^{-71} +96 q^{-72} +1364 q^{-73} +365 q^{-74} -364 q^{-75} -743 q^{-76} -328 q^{-77} +572 q^{-78} +330 q^{-79} +77 q^{-80} -284 q^{-81} -281 q^{-82} +118 q^{-83} +111 q^{-84} +112 q^{-85} -40 q^{-86} -102 q^{-87} +7 q^{-88} +5 q^{-89} +35 q^{-90} +4 q^{-91} -19 q^{-92} +2 q^{-93} -3 q^{-94} +5 q^{-95} + q^{-96} -3 q^{-97} + q^{-98} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-10} -3 q^{-11} +3 q^{-12} +4 q^{-13} -8 q^{-14} -2 q^{-15} +6 q^{-16} +12 q^{-18} +7 q^{-19} -33 q^{-20} -37 q^{-21} +22 q^{-22} +55 q^{-23} +82 q^{-24} +11 q^{-25} -145 q^{-26} -224 q^{-27} -54 q^{-28} +267 q^{-29} +477 q^{-30} +270 q^{-31} -394 q^{-32} -953 q^{-33} -729 q^{-34} +373 q^{-35} +1631 q^{-36} +1658 q^{-37} -80 q^{-38} -2379 q^{-39} -3040 q^{-40} -904 q^{-41} +2975 q^{-42} +5037 q^{-43} +2512 q^{-44} -3103 q^{-45} -7067 q^{-46} -5137 q^{-47} +2424 q^{-48} +9222 q^{-49} +8197 q^{-50} -908 q^{-51} -10595 q^{-52} -11714 q^{-53} -1536 q^{-54} +11480 q^{-55} +14870 q^{-56} +4438 q^{-57} -11246 q^{-58} -17656 q^{-59} -7573 q^{-60} +10499 q^{-61} +19516 q^{-62} +10432 q^{-63} -9024 q^{-64} -20712 q^{-65} -12892 q^{-66} +7498 q^{-67} +21044 q^{-68} +14737 q^{-69} -5739 q^{-70} -20919 q^{-71} -16091 q^{-72} +4156 q^{-73} +20295 q^{-74} +16953 q^{-75} -2520 q^{-76} -19340 q^{-77} -17535 q^{-78} +891 q^{-79} +18076 q^{-80} +17809 q^{-81} +828 q^{-82} -16377 q^{-83} -17823 q^{-84} -2746 q^{-85} +14306 q^{-86} +17498 q^{-87} +4643 q^{-88} -11685 q^{-89} -16664 q^{-90} -6553 q^{-91} +8704 q^{-92} +15262 q^{-93} +8050 q^{-94} -5441 q^{-95} -13152 q^{-96} -9040 q^{-97} +2285 q^{-98} +10483 q^{-99} +9150 q^{-100} +522 q^{-101} -7483 q^{-102} -8445 q^{-103} -2518 q^{-104} +4510 q^{-105} +6930 q^{-106} +3639 q^{-107} -1944 q^{-108} -5079 q^{-109} -3758 q^{-110} +121 q^{-111} +3113 q^{-112} +3212 q^{-113} +928 q^{-114} -1549 q^{-115} -2289 q^{-116} -1208 q^{-117} +451 q^{-118} +1356 q^{-119} +1054 q^{-120} +100 q^{-121} -642 q^{-122} -707 q^{-123} -263 q^{-124} +221 q^{-125} +370 q^{-126} +219 q^{-127} -14 q^{-128} -163 q^{-129} -136 q^{-130} -18 q^{-131} +54 q^{-132} +46 q^{-133} +29 q^{-134} -8 q^{-135} -30 q^{-136} -6 q^{-137} +7 q^{-138} + q^{-139} +3 q^{-140} +3 q^{-141} -5 q^{-142} - q^{-143} +3 q^{-144} - q^{-145} }
6	Not Available
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, 38]]
Out[2]=	PD[X[1, 6, 2, 7], X[5, 14, 6, 15], X[7, 18, 8, 1], X[15, 8, 16, 9], X[3, 10, 4, 11], X[9, 4, 10, 5], X[17, 12, 18, 13], X[11, 16, 12, 17], X[13, 2, 14, 3]]
In[3]:=	GaussCode[Knot[9, 38]]
Out[3]=	GaussCode[-1, 9, -5, 6, -2, 1, -3, 4, -6, 5, -8, 7, -9, 2, -4, 8, -7, 3]
In[4]:=	DTCode[Knot[9, 38]]
Out[4]=	DTCode[6, 10, 14, 18, 4, 16, 2, 8, 12]
In[5]:=	br = BR[Knot[9, 38]]
Out[5]=	BR[4, {-1, -1, -2, -2, 3, -2, 1, -2, -3, -3, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[9, 38]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 38]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[9, 38]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, {2, 3}, 2, 3, {4, 7}, 2}
In[10]:=	alex = Alexander[Knot[9, 38]][t]
Out[10]=	5 14 2 19 + -- - -- - 14 t + 5 t 2 t t
In[11]:=	Conway[Knot[9, 38]][z]
Out[11]=	2 4 1 + 6 z + 5 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 38], Knot[10, 63]}
In[13]:=	{KnotDet[Knot[9, 38]], KnotSignature[Knot[9, 38]]}
Out[13]=	{57, -4}
In[14]:=	Jones[Knot[9, 38]][q]
Out[14]=	-11 3 6 8 10 10 8 7 3 -2 -q + --- - -- + -- - -- + -- - -- + -- - -- + q 10 9 8 7 6 5 4 3 q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 38]}
In[16]:=	A2Invariant[Knot[9, 38]][q]
Out[16]=	-34 -32 -30 3 2 -22 2 4 -12 2 2 -q + q + q - --- - --- - q + --- + --- + q + --- - -- + 28 24 20 16 10 8 q q q q q q -6 q
In[17]:=	HOMFLYPT[Knot[9, 38]][a, z]
Out[17]=	6 8 4 2 6 2 8 2 10 2 4 4 6 4 8 4 4 a - 3 a + a z + 7 a z - a z - a z + a z + 3 a z + a z
In[18]:=	Kauffman[Knot[9, 38]][a, z]
Out[18]=	6 8 7 9 11 13 4 2 6 2 -4 a - 3 a + 3 a z + a z - a z + a z - a z + 9 a z + 8 2 10 2 12 2 5 3 7 3 9 3 10 a z + 3 a z + 3 a z - 2 a z - 2 a z + 5 a z + 11 3 13 3 4 4 6 4 8 4 10 4 3 a z - 2 a z + a z - 10 a z - 15 a z - 10 a z - 12 4 5 5 7 5 9 5 11 5 13 5 6 a z + 3 a z - 4 a z - 15 a z - 7 a z + a z + 6 6 8 6 10 6 12 6 7 7 9 7 6 a z + 6 a z + 3 a z + 3 a z + 5 a z + 9 a z + 11 7 8 8 10 8 4 a z + 2 a z + 2 a z
In[19]:=	{Vassiliev[2][Knot[9, 38]], Vassiliev[3][Knot[9, 38]]}
Out[19]=	{6, -14}
In[20]:=	Kh[Knot[9, 38]][q, t]
Out[20]=	-5 -3 1 2 1 4 2 4 q + q + ------ + ------ + ------ + ------ + ------ + ------ + 23 9 21 8 19 8 19 7 17 7 17 6 q t q t q t q t q t q t 4 6 4 4 6 4 4 3 ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 15 6 15 5 13 5 13 4 11 4 11 3 9 3 9 2 q t q t q t q t q t q t q t q t 4 3 ----- + ---- 7 2 5 q t q t
In[21]:=	ColouredJones[Knot[9, 38], 2][q]
Out[21]=	-31 3 -29 10 16 5 38 32 27 74 37 q - --- + q + --- - --- - --- + --- - --- - --- + --- - --- - 30 28 27 26 25 24 23 22 21 q q q q q q q q q 57 98 29 77 97 14 74 71 2 50 34 --- + --- - --- - --- + --- - --- - --- + --- + --- - --- + --- + 20 19 18 17 16 15 14 13 12 11 10 q q q q q q q q q q q 7 20 8 3 3 -4 -- - -- + -- + -- - -- + q 9 8 7 6 5 q q q q q

See/edit the Rolfsen_Splice_Template.

Back to the top.

9_37

9_39

@@ Line 1: / Line 1: @@
+<!-- This page was generated from the splice template "Rolfsen_Splice_Template". Please do not edit! -->
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 </table>
+{| width=100%
-See/edit the [[Rolfsen_Splice_Template]].
+|align=left|See/edit the [[Rolfsen_Splice_Template]].
+Back to the [[#top|top]].
+|align=right|{{Knot Navigation Links|ext=gif}}
+|}
  [[Category:Knot Page]]

9 38: Difference between revisions

Revision as of 21:09, 29 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

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