9 7

From Knot Atlas
Revision as of 21:52, 27 August 2005 by ScottTestRobot (talk | contribs)
Jump to navigationJump to search


9 6.gif

9_6

9 8.gif

9_8

9 7.gif Visit 9 7's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 7's page at Knotilus!

Visit 9 7's page at the original Knot Atlas!

9 7 Quick Notes


9 7 Further Notes and Views

Knot presentations

Planar diagram presentation X1425 X3,12,4,13 X5,16,6,17 X7,18,8,1 X17,6,18,7 X9,14,10,15 X13,10,14,11 X15,8,16,9 X11,2,12,3
Gauss code -1, 9, -2, 1, -3, 5, -4, 8, -6, 7, -9, 2, -7, 6, -8, 3, -5, 4
Dowker-Thistlethwaite code 4 12 16 18 14 2 10 8 6
Conway Notation [342]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index [math]\displaystyle{ \{4,6\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [-14][3]
Hyperbolic Volume 8.01486
A-Polynomial See Data:9 7/A-polynomial

[edit Notes for 9 7'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 7's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 3 t^2-7 t+9-7 t^{-1} +3 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ 3 z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 29, -4 }
Jones polynomial [math]\displaystyle{ q^{-2} - q^{-3} +3 q^{-4} -4 q^{-5} +5 q^{-6} -5 q^{-7} +4 q^{-8} -3 q^{-9} +2 q^{-10} - q^{-11} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^2 a^{10}-a^{10}+z^4 a^8+2 z^2 a^8+a^8+z^4 a^6+z^2 a^6-a^6+z^4 a^4+3 z^2 a^4+2 a^4 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^5 a^{13}-3 z^3 a^{13}+z a^{13}+2 z^6 a^{12}-6 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-6 z^5 a^{11}+5 z^3 a^{11}-2 z a^{11}+z^8 a^{10}-2 z^6 a^{10}+2 z^4 a^{10}-2 z^2 a^{10}+a^{10}+3 z^7 a^9-9 z^5 a^9+11 z^3 a^9-3 z a^9+z^8 a^8-3 z^6 a^8+7 z^4 a^8-4 z^2 a^8+a^8+z^7 a^7-z^5 a^7+2 z^3 a^7-z a^7+z^6 a^6-2 z^2 a^6+a^6+z^5 a^5-z^3 a^5-z a^5+z^4 a^4-3 z^2 a^4+2 a^4 }[/math]
The A2 invariant [math]\displaystyle{ -q^{34}-q^{28}+q^{26}-q^{18}+q^{16}+q^{12}+2 q^{10}+q^6 }[/math]
The G2 invariant [math]\displaystyle{ q^{176}-q^{174}+2 q^{172}-3 q^{170}+2 q^{168}-q^{166}-2 q^{164}+7 q^{162}-8 q^{160}+9 q^{158}-7 q^{156}+6 q^{152}-12 q^{150}+13 q^{148}-11 q^{146}+4 q^{144}+4 q^{142}-10 q^{140}+10 q^{138}-6 q^{136}+5 q^{132}-9 q^{130}+5 q^{128}-7 q^{124}+11 q^{122}-12 q^{120}+8 q^{118}-q^{116}-8 q^{114}+13 q^{112}-16 q^{110}+15 q^{108}-7 q^{106}-q^{104}+9 q^{102}-13 q^{100}+14 q^{98}-7 q^{96}+q^{94}+5 q^{92}-7 q^{90}+5 q^{88}+q^{86}-6 q^{84}+8 q^{82}-7 q^{80}+4 q^{76}-9 q^{74}+10 q^{72}-8 q^{70}+4 q^{68}-q^{66}-4 q^{64}+6 q^{62}-6 q^{60}+7 q^{58}-3 q^{56}+3 q^{54}+q^{52}-q^{50}+4 q^{48}-3 q^{46}+4 q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+2 q^{34}+q^{30} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_7.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{23}+q^{21}-q^{19}+q^{17}-q^{15}+q^{11}-q^9+2 q^7+q^3 }[/math]
2 [math]\displaystyle{ q^{64}-q^{62}-q^{60}+3 q^{58}-q^{56}-4 q^{54}+3 q^{52}+2 q^{50}-4 q^{48}+2 q^{46}+3 q^{44}-4 q^{42}+2 q^{38}-q^{36}-2 q^{34}+3 q^{30}-4 q^{28}-2 q^{26}+5 q^{24}-2 q^{22}-2 q^{20}+4 q^{18}+2 q^{12}+q^6 }[/math]
3 [math]\displaystyle{ -q^{123}+q^{121}+q^{119}-q^{117}-2 q^{115}+q^{113}+5 q^{111}-7 q^{107}-3 q^{105}+6 q^{103}+7 q^{101}-4 q^{99}-10 q^{97}+q^{95}+10 q^{93}+4 q^{91}-10 q^{89}-7 q^{87}+9 q^{85}+9 q^{83}-6 q^{81}-9 q^{79}+5 q^{77}+8 q^{75}-3 q^{73}-8 q^{71}+6 q^{67}+2 q^{65}-3 q^{63}-6 q^{61}+3 q^{59}+9 q^{57}-12 q^{53}-4 q^{51}+10 q^{49}+6 q^{47}-11 q^{45}-7 q^{43}+5 q^{41}+7 q^{39}-2 q^{37}-5 q^{35}+q^{33}+2 q^{31}+q^{29}+q^{25}+q^{21}+q^{19}+2 q^{17}+q^9 }[/math]
4 [math]\displaystyle{ q^{200}-q^{198}-q^{196}+q^{194}+2 q^{190}-3 q^{188}-3 q^{186}+2 q^{184}+2 q^{182}+9 q^{180}-3 q^{178}-11 q^{176}-5 q^{174}-q^{172}+18 q^{170}+9 q^{168}-5 q^{166}-13 q^{164}-20 q^{162}+9 q^{160}+18 q^{158}+16 q^{156}-q^{154}-31 q^{152}-16 q^{150}+5 q^{148}+32 q^{146}+26 q^{144}-21 q^{142}-32 q^{140}-18 q^{138}+26 q^{136}+40 q^{134}-3 q^{132}-30 q^{130}-30 q^{128}+13 q^{126}+36 q^{124}+6 q^{122}-19 q^{120}-25 q^{118}+4 q^{116}+24 q^{114}+10 q^{112}-10 q^{110}-17 q^{108}-3 q^{106}+11 q^{104}+16 q^{102}+q^{100}-11 q^{98}-18 q^{96}-4 q^{94}+27 q^{92}+18 q^{90}-4 q^{88}-33 q^{86}-27 q^{84}+25 q^{82}+35 q^{80}+17 q^{78}-31 q^{76}-44 q^{74}+6 q^{72}+28 q^{70}+31 q^{68}-8 q^{66}-35 q^{64}-12 q^{62}+5 q^{60}+24 q^{58}+9 q^{56}-13 q^{54}-9 q^{52}-8 q^{50}+8 q^{48}+9 q^{46}-q^{42}-7 q^{40}+q^{38}+4 q^{36}+2 q^{34}+3 q^{32}-3 q^{30}+q^{26}+q^{24}+3 q^{22}+q^{12} }[/math]
5 [math]\displaystyle{ -q^{295}+q^{293}+q^{291}-q^{289}+q^{281}+2 q^{279}-2 q^{277}-5 q^{275}-2 q^{273}+q^{271}+6 q^{269}+9 q^{267}+5 q^{265}-9 q^{263}-17 q^{261}-11 q^{259}+q^{257}+19 q^{255}+24 q^{253}+13 q^{251}-12 q^{249}-30 q^{247}-27 q^{245}-9 q^{243}+20 q^{241}+40 q^{239}+35 q^{237}+2 q^{235}-35 q^{233}-54 q^{231}-39 q^{229}+11 q^{227}+61 q^{225}+75 q^{223}+27 q^{221}-51 q^{219}-96 q^{217}-73 q^{215}+18 q^{213}+105 q^{211}+112 q^{209}+21 q^{207}-92 q^{205}-132 q^{203}-62 q^{201}+67 q^{199}+140 q^{197}+92 q^{195}-40 q^{193}-132 q^{191}-105 q^{189}+11 q^{187}+112 q^{185}+107 q^{183}+6 q^{181}-91 q^{179}-95 q^{177}-15 q^{175}+67 q^{173}+81 q^{171}+19 q^{169}-50 q^{167}-63 q^{165}-18 q^{163}+35 q^{161}+48 q^{159}+20 q^{157}-20 q^{155}-44 q^{153}-26 q^{151}+8 q^{149}+35 q^{147}+43 q^{145}+12 q^{143}-36 q^{141}-62 q^{139}-36 q^{137}+31 q^{135}+86 q^{133}+72 q^{131}-15 q^{129}-102 q^{127}-111 q^{125}-14 q^{123}+111 q^{121}+142 q^{119}+50 q^{117}-93 q^{115}-160 q^{113}-92 q^{111}+65 q^{109}+157 q^{107}+110 q^{105}-18 q^{103}-127 q^{101}-124 q^{99}-19 q^{97}+87 q^{95}+109 q^{93}+40 q^{91}-44 q^{89}-84 q^{87}-53 q^{85}+11 q^{83}+55 q^{81}+45 q^{79}+5 q^{77}-27 q^{75}-35 q^{73}-14 q^{71}+13 q^{69}+22 q^{67}+13 q^{65}-3 q^{63}-15 q^{61}-11 q^{59}+q^{57}+7 q^{55}+8 q^{53}+4 q^{51}-6 q^{49}-5 q^{47}+2 q^{43}+4 q^{41}+5 q^{39}-q^{37}-2 q^{35}+q^{29}+3 q^{27}+q^{25}+q^{15} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{34}-q^{28}+q^{26}-q^{18}+q^{16}+q^{12}+2 q^{10}+q^6 }[/math]
1,1 [math]\displaystyle{ q^{92}-2 q^{90}+4 q^{88}-8 q^{86}+15 q^{84}-20 q^{82}+26 q^{80}-34 q^{78}+35 q^{76}-34 q^{74}+28 q^{72}-16 q^{70}+3 q^{68}+14 q^{66}-30 q^{64}+42 q^{62}-53 q^{60}+58 q^{58}-58 q^{56}+56 q^{54}-46 q^{52}+36 q^{50}-22 q^{48}+6 q^{46}+q^{44}-18 q^{42}+18 q^{40}-24 q^{38}+21 q^{36}-20 q^{34}+20 q^{32}-14 q^{30}+16 q^{28}-10 q^{26}+12 q^{24}-6 q^{22}+8 q^{20}-2 q^{18}+4 q^{16}+q^{12} }[/math]
2,0 [math]\displaystyle{ q^{86}+q^{78}-3 q^{74}-q^{72}+q^{70}+q^{68}-q^{66}+3 q^{62}+q^{60}-2 q^{58}-q^{56}+q^{54}-q^{52}-q^{50}-q^{46}-3 q^{44}-2 q^{42}-3 q^{38}-q^{36}+4 q^{34}+2 q^{32}-q^{30}+2 q^{28}+4 q^{26}+2 q^{24}-q^{22}+2 q^{20}+2 q^{18}+q^{12} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{74}-q^{72}+q^{68}-2 q^{66}+2 q^{64}+2 q^{62}-3 q^{60}+2 q^{58}+q^{56}-5 q^{54}-q^{52}+q^{50}-3 q^{48}-q^{46}+q^{44}+q^{42}+4 q^{36}-2 q^{34}-3 q^{32}+3 q^{30}-2 q^{28}-3 q^{26}+4 q^{24}+2 q^{22}+q^{20}+3 q^{18}+2 q^{16}+q^{12} }[/math]
1,0,0 [math]\displaystyle{ -q^{45}-q^{41}-q^{37}+q^{35}+q^{31}-q^{25}-q^{23}+q^{21}+2 q^{17}+q^{15}+2 q^{13}+q^9 }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{96}-q^{92}+q^{90}+q^{88}-2 q^{86}+4 q^{82}+2 q^{80}-2 q^{78}+2 q^{74}-3 q^{72}-6 q^{70}-q^{68}-2 q^{66}-5 q^{64}-q^{62}+q^{60}-q^{58}+4 q^{54}+3 q^{52}+q^{48}+3 q^{46}-2 q^{44}-5 q^{42}-q^{40}-q^{36}+3 q^{32}+5 q^{30}+3 q^{28}+3 q^{26}+3 q^{24}+2 q^{22}+q^{18} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{56}-q^{52}-q^{50}-q^{46}+q^{44}+q^{40}+q^{38}-q^{32}-q^{30}-q^{28}+q^{26}+2 q^{22}+2 q^{20}+q^{18}+2 q^{16}+q^{12} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{74}+q^{72}-2 q^{70}+3 q^{68}-4 q^{66}+4 q^{64}-4 q^{62}+3 q^{60}-2 q^{58}+q^{56}+q^{54}-3 q^{52}+5 q^{50}-7 q^{48}+7 q^{46}-7 q^{44}+7 q^{42}-6 q^{40}+4 q^{38}-2 q^{36}+q^{32}-3 q^{30}+4 q^{28}-3 q^{26}+4 q^{24}-2 q^{22}+3 q^{20}-q^{18}+2 q^{16}+q^{12} }[/math]
1,0 [math]\displaystyle{ q^{120}-q^{116}-q^{114}+q^{112}+2 q^{110}-q^{108}-3 q^{106}+4 q^{102}+3 q^{100}-3 q^{98}-4 q^{96}+q^{94}+4 q^{92}+q^{90}-4 q^{88}-3 q^{86}+q^{84}+2 q^{82}-q^{80}-3 q^{78}+2 q^{74}-3 q^{70}-q^{68}+3 q^{66}+2 q^{64}-2 q^{62}-2 q^{60}+2 q^{58}+3 q^{56}-q^{54}-4 q^{52}-q^{50}+4 q^{48}+2 q^{46}-3 q^{44}-3 q^{42}+4 q^{38}+2 q^{36}-q^{32}+q^{30}+2 q^{28}+2 q^{26}+q^{18} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{102}-q^{100}+q^{98}-2 q^{96}+3 q^{94}-3 q^{92}+3 q^{90}-3 q^{88}+4 q^{86}-2 q^{84}+2 q^{82}-q^{80}-q^{76}-4 q^{74}+q^{72}-5 q^{70}+3 q^{68}-7 q^{66}+5 q^{64}-5 q^{62}+7 q^{60}-4 q^{58}+5 q^{56}-2 q^{54}+4 q^{52}-q^{46}-3 q^{44}+2 q^{42}-4 q^{40}+q^{38}-3 q^{36}+4 q^{34}+4 q^{30}+q^{28}+4 q^{26}+q^{24}+2 q^{22}+q^{18} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{176}-q^{174}+2 q^{172}-3 q^{170}+2 q^{168}-q^{166}-2 q^{164}+7 q^{162}-8 q^{160}+9 q^{158}-7 q^{156}+6 q^{152}-12 q^{150}+13 q^{148}-11 q^{146}+4 q^{144}+4 q^{142}-10 q^{140}+10 q^{138}-6 q^{136}+5 q^{132}-9 q^{130}+5 q^{128}-7 q^{124}+11 q^{122}-12 q^{120}+8 q^{118}-q^{116}-8 q^{114}+13 q^{112}-16 q^{110}+15 q^{108}-7 q^{106}-q^{104}+9 q^{102}-13 q^{100}+14 q^{98}-7 q^{96}+q^{94}+5 q^{92}-7 q^{90}+5 q^{88}+q^{86}-6 q^{84}+8 q^{82}-7 q^{80}+4 q^{76}-9 q^{74}+10 q^{72}-8 q^{70}+4 q^{68}-q^{66}-4 q^{64}+6 q^{62}-6 q^{60}+7 q^{58}-3 q^{56}+3 q^{54}+q^{52}-q^{50}+4 q^{48}-3 q^{46}+4 q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+2 q^{34}+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 7"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 3 t^2-7 t+9-7 t^{-1} +3 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 3 z^4+5 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]= { 29, -4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^{-2} - q^{-3} +3 q^{-4} -4 q^{-5} +5 q^{-6} -5 q^{-7} +4 q^{-8} -3 q^{-9} +2 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}-a^{10}+z^4 a^8+2 z^2 a^8+a^8+z^4 a^6+z^2 a^6-a^6+z^4 a^4+3 z^2 a^4+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}-3 z^3 a^{13}+z a^{13}+2 z^6 a^{12}-6 z^4 a^{12}+3 z^2 a^{12}+2 z^7 a^{11}-6 z^5 a^{11}+5 z^3 a^{11}-2 z a^{11}+z^8 a^{10}-2 z^6 a^{10}+2 z^4 a^{10}-2 z^2 a^{10}+a^{10}+3 z^7 a^9-9 z^5 a^9+11 z^3 a^9-3 z a^9+z^8 a^8-3 z^6 a^8+7 z^4 a^8-4 z^2 a^8+a^8+z^7 a^7-z^5 a^7+2 z^3 a^7-z a^7+z^6 a^6-2 z^2 a^6+a^6+z^5 a^5-z^3 a^5-z a^5+z^4 a^4-3 z^2 a^4+2 a^4 }[/math]

Vassiliev invariants

V2 and V3: (5, -12)
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{ 20 }[/math] [math]\displaystyle{ -96 }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ \frac{1654}{3} }[/math] [math]\displaystyle{ \frac{218}{3} }[/math] [math]\displaystyle{ -1920 }[/math] [math]\displaystyle{ -3552 }[/math] [math]\displaystyle{ -576 }[/math] [math]\displaystyle{ -448 }[/math] [math]\displaystyle{ \frac{4000}{3} }[/math] [math]\displaystyle{ 4608 }[/math] [math]\displaystyle{ \frac{33080}{3} }[/math] [math]\displaystyle{ \frac{4360}{3} }[/math] [math]\displaystyle{ \frac{140719}{6} }[/math] [math]\displaystyle{ \frac{1954}{3} }[/math] [math]\displaystyle{ \frac{77486}{9} }[/math] [math]\displaystyle{ \frac{4789}{18} }[/math] [math]\displaystyle{ \frac{6415}{6} }[/math]

V2,1 through V6,9 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 7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 -9-8-7-6-5-4-3-2-10χ
-3         11
-5        110
-7       2  2
-9      21  -1
-11     32   1
-13    22    0
-15   23     -1
-17  12      1
-19 12       -1
-21 1        1
-231         -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, 7]]
Out[2]=  
9
In[3]:=
PD[Knot[9, 7]]
Out[3]=  
PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[5, 16, 6, 17], X[7, 18, 8, 1], 

 X[17, 6, 18, 7], X[9, 14, 10, 15], X[13, 10, 14, 11], 



  X[15, 8, 16, 9], X[11, 2, 12, 3]]
In[4]:=
GaussCode[Knot[9, 7]]
Out[4]=  
GaussCode[-1, 9, -2, 1, -3, 5, -4, 8, -6, 7, -9, 2, -7, 6, -8, 3, -5, 4]
In[5]:=
BR[Knot[9, 7]]
Out[5]=  
BR[4, {-1, -1, -1, -1, -2, 1, -2, -3, 2, -3, -3}]
In[6]:=
alex = Alexander[Knot[9, 7]][t]
Out[6]=  
    3    7            2

9 + -- - - - 7 t + 3 t



    2   t


    t
In[7]:=
Conway[Knot[9, 7]][z]
Out[7]=  
       2      4
1 + 5 z  + 3 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[9, 7]}
In[9]:=
{KnotDet[Knot[9, 7]], KnotSignature[Knot[9, 7]]}
Out[9]=  
{29, -4}
In[10]:=
J=Jones[Knot[9, 7]][q]
Out[10]=  
  -11    2    3    4    5    5    4    3     -3    -2

-q    + --- - -- + -- - -- + -- - -- + -- - q   + q



        10    9    8    7    6    5    4


        q     q    q    q    q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[9, 7]}
In[12]:=
A2Invariant[Knot[9, 7]][q]
Out[12]=  
  -34    -28    -26    -18    -16    -12    2     -6

-q    - q    + q    - q    + q    + q    + --- + q



                                           10


                                           q
In[13]:=
Kauffman[Knot[9, 7]][a, z]
Out[13]=  
   4    6    8    10    5      7        9        11      13

2 a  + a  + a  + a   - a  z - a  z - 3 a  z - 2 a   z + a   z - 



    4  2      6  2      8  2      10  2      12  2    5  3      7  3
 3 a  z  - 2 a  z  - 4 a  z  - 2 a   z  + 3 a   z  - a  z  + 2 a  z  + 

     9  3      11  3      13  3    4  4      8  4      10  4
 11 a  z  + 5 a   z  - 3 a   z  + a  z  + 7 a  z  + 2 a   z  - 

    12  4    5  5    7  5      9  5      11  5    13  5    6  6
 6 a   z  + a  z  - a  z  - 9 a  z  - 6 a   z  + a   z  + a  z  - 

    8  6      10  6      12  6    7  7      9  7      11  7    8  8
 3 a  z  - 2 a   z  + 2 a   z  + a  z  + 3 a  z  + 2 a   z  + a  z  + 

  10  8


  a   z
In[14]:=
{Vassiliev[2][Knot[9, 7]], Vassiliev[3][Knot[9, 7]]}
Out[14]=  
{0, -12}
In[15]:=
Kh[Knot[9, 7]][q, t]
Out[15]=  
 -5    -3     1        1        1        2        1        2

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

   2        3        2        2        3        2        2       1
 ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
  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

   2      1
 ----- + ----
  7  2    5


  q  t    q  t
Retrieved from "https://katlas.org/index.php?title=9_7&oldid=26351"