9 38

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 X5,14,6,15 X7,18,8,1 X15,8,16,9 X3,10,4,11 X9,4,10,5 X17,12,18,13 X11,16,12,17 X13,2,14,3
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]

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{2,3\} }[/math] 3-genus 2 Bridge index 3 Super bridge index [math]\displaystyle{ \{4,7\} }[/math] 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 [math]\displaystyle{ 2 }[/math] Topological 4 genus [math]\displaystyle{ 2 }[/math] Concordance genus [math]\displaystyle{ 2 }[/math] Rasmussen s-Invariant -4

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

Polynomial invariants

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

Further Quantum Invariants

Further quantum knot invariants for 9_38.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{23}+2 q^{21}-3 q^{19}+2 q^{17}-2 q^{15}+2 q^{11}-q^9+4 q^7-2 q^5+q^3 }[/math]
2	[math]\displaystyle{ q^{64}-2 q^{62}-q^{60}+8 q^{58}-5 q^{56}-11 q^{54}+17 q^{52}+q^{50}-21 q^{48}+15 q^{46}+10 q^{44}-20 q^{42}+4 q^{40}+12 q^{38}-8 q^{36}-9 q^{34}+6 q^{32}+9 q^{30}-17 q^{28}-q^{26}+23 q^{24}-14 q^{22}-9 q^{20}+21 q^{18}-5 q^{16}-9 q^{14}+8 q^{12}+q^{10}-2 q^8+q^6 }[/math]
3	[math]\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 }[/math]
4	[math]\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} }[/math]
5	[math]\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} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\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 }[/math]
1,1	[math]\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} }[/math]
2,0	[math]\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} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{74}-2 q^{72}+q^{70}+4 q^{68}-8 q^{66}+4 q^{64}+9 q^{62}-15 q^{60}+7 q^{58}+10 q^{56}-17 q^{54}+4 q^{52}+10 q^{50}-10 q^{48}-3 q^{46}+2 q^{44}-2 q^{42}-8 q^{40}-7 q^{38}+12 q^{36}-3 q^{34}-8 q^{32}+21 q^{30}+q^{28}-10 q^{26}+16 q^{24}-q^{22}-7 q^{20}+6 q^{18}-2 q^{14}+q^{12} }[/math]
1,0,0	[math]\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 }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\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} }[/math]
1,0,0,0	[math]\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} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\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} }[/math]
1,0	[math]\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} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\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} }[/math]

G2 Invariants.

Weight

Invariant

1,0

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

.

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 38"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 5 t^2-14 t+19-14 t^{-1} +5 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 5 z^4+6 z^2+1 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 57, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{-2} -3 q^{-3} +7 q^{-4} -8 q^{-5} +10 q^{-6} -10 q^{-7} +8 q^{-8} -6 q^{-9} +3 q^{-10} - q^{-11} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^2 a^{10}+z^4 a^8-z^2 a^8-3 a^8+3 z^4 a^6+7 z^2 a^6+4 a^6+z^4 a^4+z^2 a^4 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-6 z^4 a^{12}+3 z^2 a^{12}+4 z^7 a^{11}-7 z^5 a^{11}+3 z^3 a^{11}-z a^{11}+2 z^8 a^{10}+3 z^6 a^{10}-10 z^4 a^{10}+3 z^2 a^{10}+9 z^7 a^9-15 z^5 a^9+5 z^3 a^9+z a^9+2 z^8 a^8+6 z^6 a^8-15 z^4 a^8+10 z^2 a^8-3 a^8+5 z^7 a^7-4 z^5 a^7-2 z^3 a^7+3 z a^7+6 z^6 a^6-10 z^4 a^6+9 z^2 a^6-4 a^6+3 z^5 a^5-2 z^3 a^5+z^4 a^4-z^2 a^4 }[/math]

Vassiliev invariants

V2 and V3:

(6, -14)

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 [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ -112 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 684 }[/math] [math]\displaystyle{ 100 }[/math] [math]\displaystyle{ -2688 }[/math] [math]\displaystyle{ -\frac{13888}{3} }[/math] [math]\displaystyle{ -\frac{2368}{3} }[/math] [math]\displaystyle{ -592 }[/math] [math]\displaystyle{ 2304 }[/math] [math]\displaystyle{ 6272 }[/math] [math]\displaystyle{ 16416 }[/math] [math]\displaystyle{ 2400 }[/math] [math]\displaystyle{ \frac{160231}{5} }[/math] [math]\displaystyle{ \frac{15308}{15} }[/math] [math]\displaystyle{ \frac{59748}{5} }[/math] [math]\displaystyle{ 259 }[/math] [math]\displaystyle{ \frac{7671}{5} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]-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	1
-5									3	1	-2
-7								4			4
-9							4	3			-1
-11						6	4				2
-13					4	4					0
-15				4	6						-2
-17			2	4							2
-19		1	4								-3
-21		2									2
-23	1										-1

Computer Talk

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

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 38]]
Out[2]=	9
In[3]:=	PD[Knot[9, 38]]
Out[3]=	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[4]:=	GaussCode[Knot[9, 38]]
Out[4]=	GaussCode[-1, 9, -5, 6, -2, 1, -3, 4, -6, 5, -8, 7, -9, 2, -4, 8, -7, 3]
In[5]:=	BR[Knot[9, 38]]
Out[5]=	BR[4, {-1, -1, -2, -2, 3, -2, 1, -2, -3, -3, -2}]
In[6]:=	alex = Alexander[Knot[9, 38]][t]
Out[6]=	5 14 2 19 + -- - -- - 14 t + 5 t 2 t t
In[7]:=	Conway[Knot[9, 38]][z]
Out[7]=	2 4 1 + 6 z + 5 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 38], Knot[10, 63]}
In[9]:=	{KnotDet[Knot[9, 38]], KnotSignature[Knot[9, 38]]}
Out[9]=	{57, -4}
In[10]:=	J=Jones[Knot[9, 38]][q]
Out[10]=	-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[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 38]}
In[12]:=	A2Invariant[Knot[9, 38]][q]
Out[12]=	-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[13]:=	Kauffman[Knot[9, 38]][a, z]
Out[13]=	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[14]:=	{Vassiliev[2][Knot[9, 38]], Vassiliev[3][Knot[9, 38]]}
Out[14]=	{0, -14}
In[15]:=	Kh[Knot[9, 38]][q, t]
Out[15]=	-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