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Visit 5 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)
Visit 5 1's page at Knotilus!
Visit 5 1's page at the original Knot Atlas!
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An interlaced pentagram, this is known variously as the "Cinquefoil Knot", after certain herbs and shrubs of the rose family which have 5-lobed leaves and 5-petaled flowers (see e.g. [4]),
as the "Pentafoil Knot" (visit Bert Jagers' pentafoil page),
as the "Double Overhand Knot", as 5_1, or finally as the torus knot T(5,2).
When taken off the post the strangle knot (hitch) of practical knot tying deforms to 5_1
|
 A kolam of a 2x3 dot array |
 The VISA Interlink Logo [1] |
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 A pentagonal table by Bob Mackay [2] |
 The Utah State Parks logo |
 As impossible object ("Penrose" pentagram) |
 Folded ribbon which is single-sided (more complex version of Möbius Strip). |
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 Alternate pentagram of intersecting circles. |
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 Partial view of US bicentennial logo on a shirt seen in Lisboa [3] |
 Non-prime knot with two 5_1 configurations on a closed loop. |
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 Sum of two 5_1s, Vienna, orthodox church |
This sentence was last edited by Dror.
Sometime later, Scott added this sentence.
Knot presentations
| Planar diagram presentation
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X1627 X3849 X5,10,6,1 X7283 X9,4,10,5
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| Gauss code
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-1, 4, -2, 5, -3, 1, -4, 2, -5, 3
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| Dowker-Thistlethwaite code
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6 8 10 2 4
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| Conway Notation
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[5]
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Four dimensional invariants
| Smooth 4 genus
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[math]\displaystyle{ 2 }[/math]
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| Topological 4 genus
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[math]\displaystyle{ 2 }[/math]
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| Concordance genus
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[math]\displaystyle{ \textrm{ConcordanceGenus}(\textrm{Knot}(5,1)) }[/math]
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| Rasmussen s-Invariant
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-4
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[edit Notes for 5 1's four dimensional invariants]
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Polynomial invariants
Further Quantum Invariants
Further quantum knot invariants for 5_1.
A1 Invariants.
| Weight
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Invariant
|
| 1
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[math]\displaystyle{ -q^{15}+q^7+q^5+q^3 }[/math]
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| 2
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[math]\displaystyle{ q^{40}-q^{32}-q^{30}-q^{28}+q^{14}+q^{12}+q^{10}+q^8+q^6 }[/math]
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| 3
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[math]\displaystyle{ -q^{75}+q^{67}+q^{65}+q^{63}-q^{49}-q^{47}-q^{45}-q^{43}-q^{41}+q^{21}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^9 }[/math]
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| 4
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[math]\displaystyle{ q^{120}-q^{112}-q^{110}-q^{108}+q^{94}+q^{92}+q^{90}+q^{88}+q^{86}-q^{66}-q^{64}-q^{62}-q^{60}-q^{58}-q^{56}-q^{54}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{16}+q^{14}+q^{12} }[/math]
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| 5
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[math]\displaystyle{ -q^{175}+q^{167}+q^{165}+q^{163}-q^{149}-q^{147}-q^{145}-q^{143}-q^{141}+q^{121}+q^{119}+q^{117}+q^{115}+q^{113}+q^{111}+q^{109}-q^{83}-q^{81}-q^{79}-q^{77}-q^{75}-q^{73}-q^{71}-q^{69}-q^{67}+q^{35}+q^{33}+q^{31}+q^{29}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15} }[/math]
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| 6
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[math]\displaystyle{ q^{240}-q^{232}-q^{230}-q^{228}+q^{214}+q^{212}+q^{210}+q^{208}+q^{206}-q^{186}-q^{184}-q^{182}-q^{180}-q^{178}-q^{176}-q^{174}+q^{148}+q^{146}+q^{144}+q^{142}+q^{140}+q^{138}+q^{136}+q^{134}+q^{132}-q^{100}-q^{98}-q^{96}-q^{94}-q^{92}-q^{90}-q^{88}-q^{86}-q^{84}-q^{82}-q^{80}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{18} }[/math]
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| 8
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[math]\displaystyle{ q^{400}-q^{392}-q^{390}-q^{388}+q^{374}+q^{372}+q^{370}+q^{368}+q^{366}-q^{346}-q^{344}-q^{342}-q^{340}-q^{338}-q^{336}-q^{334}+q^{308}+q^{306}+q^{304}+q^{302}+q^{300}+q^{298}+q^{296}+q^{294}+q^{292}-q^{260}-q^{258}-q^{256}-q^{254}-q^{252}-q^{250}-q^{248}-q^{246}-q^{244}-q^{242}-q^{240}+q^{202}+q^{200}+q^{198}+q^{196}+q^{194}+q^{192}+q^{190}+q^{188}+q^{186}+q^{184}+q^{182}+q^{180}+q^{178}-q^{134}-q^{132}-q^{130}-q^{128}-q^{126}-q^{124}-q^{122}-q^{120}-q^{118}-q^{116}-q^{114}-q^{112}-q^{110}-q^{108}-q^{106}+q^{56}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24} }[/math]
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A2 Invariants.
| Weight
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Invariant
|
| 1,0
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[math]\displaystyle{ -q^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2 q^{10}+q^8+q^6 }[/math]
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| 1,1
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[math]\displaystyle{ q^{60}-2 q^{36}-2 q^{34}-4 q^{32}-4 q^{30}-3 q^{28}+2 q^{24}+4 q^{22}+5 q^{20}+4 q^{18}+4 q^{16}+2 q^{14}+q^{12} }[/math]
|
| 2,0
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[math]\displaystyle{ q^{54}+q^{52}+2 q^{50}+q^{48}-2 q^{44}-3 q^{42}-3 q^{40}-3 q^{38}-2 q^{36}-q^{34}+q^{28}+q^{26}+2 q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+2 q^{16}+q^{14}+q^{12} }[/math]
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| 3,0
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[math]\displaystyle{ -q^{96}-q^{94}-2 q^{92}-2 q^{90}-q^{88}+q^{86}+3 q^{84}+4 q^{82}+5 q^{80}+4 q^{78}+4 q^{76}+2 q^{74}+q^{72}-q^{70}-2 q^{68}-3 q^{66}-4 q^{64}-5 q^{62}-5 q^{60}-5 q^{58}-4 q^{56}-3 q^{54}-2 q^{52}-q^{50}+q^{42}+q^{40}+2 q^{38}+2 q^{36}+3 q^{34}+3 q^{32}+4 q^{30}+3 q^{28}+3 q^{26}+2 q^{24}+2 q^{22}+q^{20}+q^{18} }[/math]
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A3 Invariants.
| Weight
|
Invariant
|
| 0,1,0
|
[math]\displaystyle{ q^{50}-q^{36}-2 q^{34}-3 q^{32}-3 q^{30}-2 q^{28}-q^{26}+2 q^{24}+3 q^{22}+4 q^{20}+3 q^{18}+3 q^{16}+q^{14}+q^{12} }[/math]
|
| 1,0,0
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[math]\displaystyle{ -q^{29}-q^{27}-2 q^{25}-q^{23}+q^{19}+2 q^{17}+2 q^{15}+2 q^{13}+q^{11}+q^9 }[/math]
|
| 1,0,1
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[math]\displaystyle{ q^{80}+q^{58}+q^{56}+3 q^{54}+3 q^{52}+2 q^{50}-q^{48}-5 q^{46}-9 q^{44}-12 q^{42}-12 q^{40}-9 q^{38}-3 q^{36}+2 q^{34}+7 q^{32}+10 q^{30}+11 q^{28}+10 q^{26}+7 q^{24}+5 q^{22}+2 q^{20}+q^{18} }[/math]
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A4 Invariants.
| Weight
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Invariant
|
| 0,1,0,0
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[math]\displaystyle{ q^{64}+q^{62}+q^{60}+q^{58}+q^{56}-q^{50}-2 q^{48}-4 q^{46}-5 q^{44}-6 q^{42}-6 q^{40}-4 q^{38}-q^{36}+2 q^{34}+4 q^{32}+7 q^{30}+6 q^{28}+6 q^{26}+4 q^{24}+3 q^{22}+q^{20}+q^{18} }[/math]
|
| 1,0,0,0
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[math]\displaystyle{ -q^{36}-q^{34}-2 q^{32}-2 q^{30}-q^{28}+q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+2 q^{16}+q^{14}+q^{12} }[/math]
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B2 Invariants.
| Weight
|
Invariant
|
| 0,1
|
[math]\displaystyle{ -q^{50}-q^{36}-q^{32}-q^{30}+q^{26}+q^{22}+2 q^{20}+q^{18}+q^{16}+q^{14}+q^{12} }[/math]
|
| 1,0
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[math]\displaystyle{ q^{80}-q^{58}-q^{56}-q^{54}-q^{52}-2 q^{50}-q^{48}-q^{46}-q^{44}+q^{38}+q^{36}+2 q^{34}+q^{32}+2 q^{30}+q^{28}+2 q^{26}+q^{24}+q^{22}+q^{18} }[/math]
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B3 Invariants.
| Weight
|
Invariant
|
| 1,0,0
|
[math]\displaystyle{ q^{120}-q^{86}-q^{82}-q^{80}-2 q^{78}-q^{76}-2 q^{74}-q^{72}-2 q^{70}-q^{68}-q^{66}-q^{64}+q^{58}+q^{56}+2 q^{54}+q^{52}+3 q^{50}+q^{48}+3 q^{46}+q^{44}+2 q^{42}+q^{40}+2 q^{38}+q^{34}+q^{30} }[/math]
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B4 Invariants.
| Weight
|
Invariant
|
| 1,0,0,0
|
[math]\displaystyle{ q^{160}-q^{114}-q^{110}-2 q^{106}-q^{104}-2 q^{102}-q^{100}-2 q^{98}-q^{96}-2 q^{94}-q^{92}-2 q^{90}-q^{88}-q^{86}+q^{78}+2 q^{74}+q^{72}+3 q^{70}+q^{68}+3 q^{66}+q^{64}+3 q^{62}+q^{60}+3 q^{58}+q^{56}+2 q^{54}+2 q^{50}+q^{46}+q^{42} }[/math]
|
C3 Invariants.
| Weight
|
Invariant
|
| 1,0,0
|
[math]\displaystyle{ -q^{70}-q^{50}-q^{48}-q^{46}-q^{44}-q^{42}-q^{40}+q^{34}+q^{32}+2 q^{30}+2 q^{28}+2 q^{26}+q^{24}+2 q^{22}+q^{20}+q^{18} }[/math]
|
C4 Invariants.
| Weight
|
Invariant
|
| 1,0,0,0
|
[math]\displaystyle{ -q^{90}-q^{64}-q^{62}-2 q^{60}-q^{58}-q^{56}-q^{54}-q^{52}-q^{50}-q^{48}+q^{44}+2 q^{42}+2 q^{40}+2 q^{38}+2 q^{36}+2 q^{34}+2 q^{32}+2 q^{30}+2 q^{28}+q^{26}+q^{24} }[/math]
|
D4 Invariants.
| Weight
|
Invariant
|
| 0,1,0,0
|
[math]\displaystyle{ q^{120}-q^{100}-q^{98}-3 q^{96}-3 q^{94}-q^{92}-q^{90}+2 q^{88}+6 q^{86}+9 q^{84}+11 q^{82}+14 q^{80}+11 q^{78}+9 q^{76}+3 q^{74}-4 q^{72}-12 q^{70}-18 q^{68}-24 q^{66}-27 q^{64}-27 q^{62}-24 q^{60}-17 q^{58}-11 q^{56}+7 q^{52}+14 q^{50}+19 q^{48}+22 q^{46}+19 q^{44}+19 q^{42}+14 q^{40}+10 q^{38}+6 q^{36}+4 q^{34}+q^{32}+q^{30} }[/math]
|
| 1,0,0,0
|
[math]\displaystyle{ q^{70}-q^{50}-q^{48}-3 q^{46}-3 q^{44}-3 q^{42}-3 q^{40}-2 q^{38}+q^{34}+3 q^{32}+4 q^{30}+4 q^{28}+4 q^{26}+3 q^{24}+2 q^{22}+q^{20}+q^{18} }[/math]
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G2 Invariants.
| Weight
|
Invariant
|
| 0,1
|
[math]\displaystyle{ q^{240}-q^{198}-q^{192}+q^{178}+q^{176}+q^{172}+2 q^{170}+q^{168}+q^{166}+q^{164}+q^{162}+2 q^{160}+q^{158}+q^{154}+q^{152}-q^{148}-q^{146}-q^{144}-q^{142}-2 q^{140}-3 q^{138}-2 q^{136}-2 q^{134}-3 q^{132}-4 q^{130}-4 q^{128}-3 q^{126}-3 q^{124}-4 q^{122}-4 q^{120}-3 q^{118}-2 q^{116}-2 q^{114}-3 q^{112}-2 q^{110}+q^{102}+q^{100}+2 q^{98}+3 q^{96}+2 q^{94}+2 q^{92}+4 q^{90}+3 q^{88}+3 q^{86}+4 q^{84}+3 q^{82}+3 q^{80}+4 q^{78}+2 q^{76}+2 q^{74}+3 q^{72}+2 q^{70}+q^{68}+2 q^{66}+q^{64}+q^{62}+q^{60}+q^{54} }[/math]
|
| 1,0
|
[math]\displaystyle{ q^{120}-q^{100}-q^{98}-q^{92}-q^{90}-q^{88}-q^{82}-q^{80}-q^{78}-q^{72}+q^{58}+q^{56}+q^{52}+2 q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+2 q^{40}+q^{38}+q^{34}+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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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[math]\displaystyle{ t^2+ t^{-2} -t- t^{-1} +1 }[/math]
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Out[5]=
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[math]\displaystyle{ z^4+3 z^2+1 }[/math]
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In[6]:=
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Alexander[K, 2][t]
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|
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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.
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Out[6]=
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[math]\displaystyle{ \{1\} }[/math]
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In[7]:=
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{KnotDet[K], KnotSignature[K]}
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|
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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[math]\displaystyle{ - q^{-7} + q^{-6} - q^{-5} + q^{-4} + q^{-2} }[/math]
|
In[9]:=
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HOMFLYPT[K][a, z]
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|
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
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[math]\displaystyle{ a^6 \left(-z^2\right)-2 a^6+a^4 z^4+4 a^4 z^2+3 a^4 }[/math]
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In[10]:=
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Kauffman[K][a, z]
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|
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
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[math]\displaystyle{ a^9 z+a^8 z^2+a^7 z^3-a^7 z+a^6 z^4-3 a^6 z^2+2 a^6+a^5 z^3-2 a^5 z+a^4 z^4-4 a^4 z^2+3 a^4 }[/math]
|
| V2,1 through V6,9:
|
| V2,1
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V3,1
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V4,1
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V4,2
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V4,3
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V5,1
|
V5,2
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V5,3
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V5,4
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V6,1
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V6,2
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V6,3
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V6,4
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V6,5
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V6,6
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V6,7
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V6,8
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V6,9
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| [math]\displaystyle{ 12 }[/math]
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[math]\displaystyle{ -40 }[/math]
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[math]\displaystyle{ 72 }[/math]
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[math]\displaystyle{ 174 }[/math]
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[math]\displaystyle{ 26 }[/math]
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[math]\displaystyle{ -480 }[/math]
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[math]\displaystyle{ -\frac{2512}{3} }[/math]
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[math]\displaystyle{ -\frac{448}{3} }[/math]
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[math]\displaystyle{ -104 }[/math]
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[math]\displaystyle{ 288 }[/math]
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[math]\displaystyle{ 800 }[/math]
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[math]\displaystyle{ 2088 }[/math]
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[math]\displaystyle{ 312 }[/math]
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[math]\displaystyle{ \frac{41151}{10} }[/math]
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[math]\displaystyle{ \frac{2494}{15} }[/math]
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[math]\displaystyle{ \frac{7634}{5} }[/math]
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[math]\displaystyle{ \frac{43}{2} }[/math]
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[math]\displaystyle{ \frac{1951}{10} }[/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.
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 5 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
|
-5 | -4 | -3 | -2 | -1 | 0 | χ |
| -3 | | | | | | 1 | 1 |
| -5 | | | | | | 1 | 1 |
| -7 | | | | 1 | | | 1 |
| -9 | | | | | | | 0 |
| -11 | | 1 | 1 | | | | 0 |
| -13 | | | | | | | 0 |
| -15 | 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[5, 1]] |
Out[2]= | 5 |
In[3]:= | PD[Knot[5, 1]] |
Out[3]= | PD[X[1, 6, 2, 7], X[3, 8, 4, 9], X[5, 10, 6, 1], X[7, 2, 8, 3],
X[9, 4, 10, 5]] |
In[4]:= | GaussCode[Knot[5, 1]] |
Out[4]= | GaussCode[-1, 4, -2, 5, -3, 1, -4, 2, -5, 3] |
In[5]:= | BR[Knot[5, 1]] |
Out[5]= | BR[2, {-1, -1, -1, -1, -1}] |
In[6]:= | alex = Alexander[Knot[5, 1]][t] |
Out[6]= | -2 1 2
1 + t - - - t + t
t |
In[7]:= | Conway[Knot[5, 1]][z] |
Out[7]= | 2 4
1 + 3 z + z |
In[8]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[8]= | {Knot[5, 1], Knot[10, 132]} |
In[9]:= | {KnotDet[Knot[5, 1]], KnotSignature[Knot[5, 1]]} |
Out[9]= | {5, -4} |
In[10]:= | J=Jones[Knot[5, 1]][q] |
Out[10]= | -7 -6 -5 -4 -2
-q + q - q + q + q |
In[11]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[11]= | {Knot[5, 1], Knot[10, 132]} |
In[12]:= | A2Invariant[Knot[5, 1]][q] |
Out[12]= | -22 -20 -18 -14 -12 2 -8 -6
-q - q - q + q + q + --- + q + q
10
q |
In[13]:= | Kauffman[Knot[5, 1]][a, z] |
Out[13]= | 4 6 5 7 9 4 2 6 2 8 2
3 a + 2 a - 2 a z - a z + a z - 4 a z - 3 a z + a z +
5 3 7 3 4 4 6 4
a z + a z + a z + a z |
In[14]:= | {Vassiliev[2][Knot[5, 1]], Vassiliev[3][Knot[5, 1]]} |
Out[14]= | {0, -5} |
In[15]:= | Kh[Knot[5, 1]][q, t] |
Out[15]= | -5 -3 1 1 1 1
q + q + ------ + ------ + ------ + -----
15 5 11 4 11 3 7 2
q t q t q t q t |
.png)
