7 5

7_4

7_6

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Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X5,12,6,13 X7,14,8,1 X13,6,14,7 X11,8,12,9 X9,2,10,3
Gauss code	-1, 7, -2, 1, -3, 5, -4, 6, -7, 2, -6, 3, -5, 4
Dowker-Thistlethwaite code	4 10 12 14 2 8 6
Conway Notation	[322]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	2
Bridge index (super bridge index)	2 (4)
Nakanishi index	1

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^2-4 t+5-4 t^{-1} +2 t^{-2} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 17, -4 }
Jones polynomial	[math]\displaystyle{ - q^{-9} +2 q^{-8} -3 q^{-7} +3 q^{-6} -3 q^{-5} +3 q^{-4} - q^{-3} + q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^8 \left(-z^2\right)-a^8+a^6 z^4+2 a^6 z^2+a^4 z^4+3 a^4 z^2+2 a^4 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^{11} z^3-a^{11} z+2 a^{10} z^4-2 a^{10} z^2+2 a^9 z^5-2 a^9 z^3+a^9 z+a^8 z^6+a^8 z^2-a^8+3 a^7 z^5-4 a^7 z^3+a^7 z+a^6 z^6-a^6 z^4+a^5 z^5-a^5 z^3-a^5 z+a^4 z^4-3 a^4 z^2+2 a^4 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{28}-q^{22}-q^{18}+q^{16}+q^{14}+q^{12}+2 q^{10}+q^6 }[/math]
The G2 invariant	[math]\displaystyle{ q^{148}-q^{146}+2 q^{144}-2 q^{142}+q^{138}-2 q^{136}+5 q^{134}-5 q^{132}+4 q^{130}-2 q^{128}-3 q^{126}+4 q^{124}-6 q^{122}+5 q^{120}-3 q^{118}-q^{116}+3 q^{114}-3 q^{112}+q^{110}+2 q^{108}-5 q^{106}+4 q^{104}-3 q^{102}-2 q^{100}+5 q^{98}-7 q^{96}+8 q^{94}-7 q^{92}+2 q^{90}+2 q^{88}-6 q^{86}+6 q^{84}-7 q^{82}+4 q^{80}-2 q^{76}+3 q^{74}-3 q^{72}+2 q^{70}+3 q^{68}-5 q^{66}+3 q^{64}-2 q^{60}+7 q^{58}-5 q^{56}+5 q^{54}-q^{52}+4 q^{48}-4 q^{46}+5 q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+2 q^{34}+q^{30} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 7_5.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{19}+q^{17}-q^{15}+2 q^7+q^3 }[/math]
2	[math]\displaystyle{ q^{52}-q^{50}-q^{48}+3 q^{46}-q^{44}-3 q^{42}+3 q^{40}-2 q^{36}+q^{34}-q^{30}-2 q^{28}+q^{26}+q^{24}-3 q^{22}+q^{20}+3 q^{18}-2 q^{16}+q^{14}+3 q^{12}+q^6 }[/math]
3	[math]\displaystyle{ -q^{99}+q^{97}+q^{95}-q^{93}-2 q^{91}+q^{89}+5 q^{87}-q^{85}-7 q^{83}-q^{81}+7 q^{79}+3 q^{77}-7 q^{75}-3 q^{73}+7 q^{71}+4 q^{69}-4 q^{67}-3 q^{65}+2 q^{63}+2 q^{61}-3 q^{57}-2 q^{55}+q^{53}+4 q^{51}-2 q^{49}-6 q^{47}+7 q^{43}-q^{41}-7 q^{39}-2 q^{37}+6 q^{35}+3 q^{33}-6 q^{31}-3 q^{29}+4 q^{27}+5 q^{25}-q^{23}-2 q^{21}+q^{19}+3 q^{17}+q^{15}+q^9 }[/math]
4	[math]\displaystyle{ q^{160}-q^{158}-q^{156}+q^{154}+2 q^{150}-3 q^{148}-3 q^{146}+2 q^{144}+3 q^{142}+9 q^{140}-5 q^{138}-12 q^{136}-4 q^{134}+5 q^{132}+19 q^{130}-15 q^{126}-13 q^{124}+24 q^{120}+8 q^{118}-12 q^{116}-16 q^{114}-5 q^{112}+17 q^{110}+9 q^{108}-4 q^{106}-10 q^{104}-6 q^{102}+7 q^{100}+7 q^{98}+3 q^{96}-2 q^{94}-5 q^{92}-3 q^{90}+4 q^{88}+9 q^{86}+q^{84}-7 q^{82}-12 q^{80}+2 q^{78}+13 q^{76}+6 q^{74}-7 q^{72}-20 q^{70}+q^{68}+16 q^{66}+10 q^{64}-4 q^{62}-22 q^{60}-5 q^{58}+11 q^{56}+13 q^{54}+5 q^{52}-17 q^{50}-10 q^{48}+2 q^{46}+10 q^{44}+10 q^{42}-6 q^{40}-7 q^{38}-4 q^{36}+3 q^{34}+8 q^{32}+q^{30}-2 q^{26}+3 q^{22}+q^{20}+q^{18}+q^{12} }[/math]
5	[math]\displaystyle{ -q^{235}+q^{233}+q^{231}-q^{229}+q^{221}+2 q^{219}-2 q^{217}-5 q^{215}-3 q^{213}+q^{211}+8 q^{209}+10 q^{207}+4 q^{205}-12 q^{203}-21 q^{201}-9 q^{199}+12 q^{197}+29 q^{195}+22 q^{193}-7 q^{191}-37 q^{189}-36 q^{187}+38 q^{183}+46 q^{181}+12 q^{179}-34 q^{177}-53 q^{175}-23 q^{173}+29 q^{171}+51 q^{169}+29 q^{167}-18 q^{165}-45 q^{163}-30 q^{161}+8 q^{159}+35 q^{157}+27 q^{155}-2 q^{153}-23 q^{151}-24 q^{149}-5 q^{147}+13 q^{145}+17 q^{143}+8 q^{141}-3 q^{139}-12 q^{137}-13 q^{135}-2 q^{133}+10 q^{131}+16 q^{129}+12 q^{127}-7 q^{125}-21 q^{123}-14 q^{121}+7 q^{119}+27 q^{117}+23 q^{115}-7 q^{113}-34 q^{111}-26 q^{109}+4 q^{107}+36 q^{105}+37 q^{103}-3 q^{101}-41 q^{99}-41 q^{97}-4 q^{95}+37 q^{93}+45 q^{91}+13 q^{89}-34 q^{87}-47 q^{85}-22 q^{83}+20 q^{81}+41 q^{79}+29 q^{77}-6 q^{75}-36 q^{73}-32 q^{71}-4 q^{69}+21 q^{67}+29 q^{65}+14 q^{63}-11 q^{61}-23 q^{59}-16 q^{57}-q^{55}+13 q^{53}+15 q^{51}+5 q^{49}-4 q^{47}-9 q^{45}-6 q^{43}+q^{41}+6 q^{39}+4 q^{37}+3 q^{35}-2 q^{31}+2 q^{27}+q^{25}+q^{23}+q^{21}+q^{15} }[/math]
6	[math]\displaystyle{ q^{324}-q^{322}-q^{320}+q^{318}-2 q^{312}+2 q^{310}-2 q^{306}+4 q^{304}+3 q^{302}+q^{300}-5 q^{298}-2 q^{296}-8 q^{294}-8 q^{292}+8 q^{290}+16 q^{288}+17 q^{286}+q^{284}-4 q^{282}-31 q^{280}-39 q^{278}-6 q^{276}+28 q^{274}+55 q^{272}+41 q^{270}+19 q^{268}-49 q^{266}-90 q^{264}-61 q^{262}+4 q^{260}+78 q^{258}+100 q^{256}+84 q^{254}-26 q^{252}-117 q^{250}-125 q^{248}-58 q^{246}+54 q^{244}+123 q^{242}+142 q^{240}+32 q^{238}-89 q^{236}-140 q^{234}-104 q^{232}+2 q^{230}+92 q^{228}+142 q^{226}+68 q^{224}-34 q^{222}-100 q^{220}-98 q^{218}-33 q^{216}+37 q^{214}+93 q^{212}+63 q^{210}+6 q^{208}-44 q^{206}-60 q^{204}-38 q^{202}-2 q^{200}+39 q^{198}+40 q^{196}+24 q^{194}-4 q^{192}-24 q^{190}-32 q^{188}-23 q^{186}+q^{184}+22 q^{182}+35 q^{180}+21 q^{178}-5 q^{176}-37 q^{174}-39 q^{172}-21 q^{170}+23 q^{168}+58 q^{166}+45 q^{164}+4 q^{162}-54 q^{160}-66 q^{158}-43 q^{156}+33 q^{154}+92 q^{152}+74 q^{150}+11 q^{148}-73 q^{146}-100 q^{144}-76 q^{142}+30 q^{140}+115 q^{138}+109 q^{136}+38 q^{134}-67 q^{132}-119 q^{130}-115 q^{128}-4 q^{126}+102 q^{124}+127 q^{122}+79 q^{120}-22 q^{118}-98 q^{116}-134 q^{114}-57 q^{112}+42 q^{110}+101 q^{108}+100 q^{106}+39 q^{104}-33 q^{102}-105 q^{100}-82 q^{98}-24 q^{96}+37 q^{94}+73 q^{92}+65 q^{90}+28 q^{88}-40 q^{86}-56 q^{84}-47 q^{82}-17 q^{80}+18 q^{78}+41 q^{76}+40 q^{74}+6 q^{72}-10 q^{70}-25 q^{68}-24 q^{66}-13 q^{64}+7 q^{62}+19 q^{60}+10 q^{58}+8 q^{56}-q^{54}-7 q^{52}-9 q^{50}-2 q^{48}+4 q^{46}+2 q^{44}+5 q^{42}+3 q^{40}+q^{38}-2 q^{36}+2 q^{32}+q^{28}+q^{26}+q^{24}+q^{18} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{28}-q^{22}-q^{18}+q^{16}+q^{14}+q^{12}+2 q^{10}+q^6 }[/math]
1,1	[math]\displaystyle{ q^{76}-2 q^{74}+4 q^{72}-6 q^{70}+9 q^{68}-12 q^{66}+14 q^{64}-14 q^{62}+13 q^{60}-10 q^{58}+2 q^{56}+4 q^{54}-12 q^{52}+20 q^{50}-22 q^{48}+28 q^{46}-26 q^{44}+22 q^{42}-22 q^{40}+8 q^{38}-10 q^{36}-4 q^{34}+6 q^{32}-12 q^{30}+16 q^{28}-10 q^{26}+16 q^{24}-6 q^{22}+10 q^{20}-2 q^{18}+4 q^{16}+q^{12} }[/math]
2,0	[math]\displaystyle{ q^{70}+q^{62}-2 q^{58}+q^{54}+q^{48}-3 q^{44}-3 q^{42}-2 q^{40}-3 q^{38}-2 q^{36}+q^{34}+q^{32}+3 q^{28}+3 q^{26}+2 q^{24}+q^{22}+3 q^{20}+2 q^{18}+q^{12} }[/math]
3,0	[math]\displaystyle{ -q^{126}-q^{116}+2 q^{114}+2 q^{112}+2 q^{110}-2 q^{108}-5 q^{106}-q^{104}+2 q^{102}+3 q^{100}-2 q^{98}-3 q^{96}+2 q^{94}+6 q^{92}+4 q^{90}-2 q^{88}-2 q^{86}+2 q^{84}+5 q^{82}+q^{80}-3 q^{78}-q^{76}+q^{74}-4 q^{70}-4 q^{68}-q^{66}-2 q^{64}-6 q^{62}-6 q^{60}-3 q^{58}+q^{56}-3 q^{54}-5 q^{52}-2 q^{50}+5 q^{48}+6 q^{46}-q^{42}+2 q^{40}+8 q^{38}+6 q^{36}+q^{34}-q^{32}+2 q^{30}+4 q^{28}+3 q^{26}+q^{18} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{62}-q^{60}+2 q^{56}-2 q^{54}+3 q^{50}-2 q^{48}-q^{46}-2 q^{42}-2 q^{40}-q^{38}-q^{34}-2 q^{32}+q^{30}+q^{28}-2 q^{26}+4 q^{24}+3 q^{22}+q^{20}+3 q^{18}+2 q^{16}+q^{12} }[/math]
1,0,0	[math]\displaystyle{ -q^{37}-q^{33}-q^{29}-q^{25}+q^{21}+q^{19}+2 q^{17}+q^{15}+2 q^{13}+q^9 }[/math]
1,0,1	[math]\displaystyle{ q^{100}-2 q^{98}+3 q^{96}-q^{94}-3 q^{92}+8 q^{90}-10 q^{88}+6 q^{86}+3 q^{84}-12 q^{82}+18 q^{80}-15 q^{78}+4 q^{76}+6 q^{74}-17 q^{72}+17 q^{70}-10 q^{68}+3 q^{66}+9 q^{64}-2 q^{62}+3 q^{60}+6 q^{58}-6 q^{56}-7 q^{54}+5 q^{52}-26 q^{50}+6 q^{48}-14 q^{46}-12 q^{44}+13 q^{42}-17 q^{40}+17 q^{38}+3 q^{36}+3 q^{34}+15 q^{32}-q^{30}+10 q^{28}+4 q^{26}+2 q^{24}+4 q^{22}+q^{18} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{80}-q^{76}+q^{74}+2 q^{72}+2 q^{66}+q^{64}-2 q^{62}-2 q^{60}-q^{58}-4 q^{56}-5 q^{54}-q^{52}-2 q^{50}-3 q^{48}+q^{44}-2 q^{42}-q^{40}+2 q^{38}+2 q^{36}+q^{34}+3 q^{32}+6 q^{30}+3 q^{28}+3 q^{26}+3 q^{24}+2 q^{22}+q^{18} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{46}-q^{42}-q^{40}-q^{36}-q^{32}+q^{26}+q^{24}+2 q^{22}+2 q^{20}+q^{18}+2 q^{16}+q^{12} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{62}+q^{60}-2 q^{58}+2 q^{56}-2 q^{54}+2 q^{52}-q^{50}+q^{46}-2 q^{44}+2 q^{42}-4 q^{40}+3 q^{38}-4 q^{36}+3 q^{34}-2 q^{32}+q^{30}+q^{28}+2 q^{24}-q^{22}+3 q^{20}-q^{18}+2 q^{16}+q^{12} }[/math]
1,0	[math]\displaystyle{ q^{100}-q^{96}-q^{94}+q^{92}+2 q^{90}-2 q^{86}-q^{84}+2 q^{82}+2 q^{80}-q^{78}-2 q^{76}+q^{72}-q^{70}-2 q^{68}-q^{66}+q^{64}-q^{60}-2 q^{58}+q^{54}-q^{52}-2 q^{50}+2 q^{46}-q^{42}-q^{40}+3 q^{38}+3 q^{36}+q^{34}-q^{32}+q^{30}+2 q^{28}+2 q^{26}+q^{18} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{86}-q^{84}+q^{82}-q^{80}+2 q^{78}-2 q^{76}+q^{74}-q^{72}+2 q^{70}-q^{66}-2 q^{62}+q^{60}-4 q^{58}-4 q^{54}+2 q^{52}-3 q^{50}+2 q^{48}-3 q^{46}+q^{44}+3 q^{34}+q^{32}+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^{148}-q^{146}+2 q^{144}-2 q^{142}+q^{138}-2 q^{136}+5 q^{134}-5 q^{132}+4 q^{130}-2 q^{128}-3 q^{126}+4 q^{124}-6 q^{122}+5 q^{120}-3 q^{118}-q^{116}+3 q^{114}-3 q^{112}+q^{110}+2 q^{108}-5 q^{106}+4 q^{104}-3 q^{102}-2 q^{100}+5 q^{98}-7 q^{96}+8 q^{94}-7 q^{92}+2 q^{90}+2 q^{88}-6 q^{86}+6 q^{84}-7 q^{82}+4 q^{80}-2 q^{76}+3 q^{74}-3 q^{72}+2 q^{70}+3 q^{68}-5 q^{66}+3 q^{64}-2 q^{60}+7 q^{58}-5 q^{56}+5 q^{54}-q^{52}+4 q^{48}-4 q^{46}+5 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["7 5"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^2-4 t+5-4 t^{-1} +2 t^{-2} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^4+4 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]=

{ 17, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ a^8 \left(-z^2\right)-a^8+a^6 z^4+2 a^6 z^2+a^4 z^4+3 a^4 z^2+2 a^4 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

Vassiliev invariants

V2 and V3:

(4, -8)

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{ 16 }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{968}{3} }[/math] [math]\displaystyle{ \frac{136}{3} }[/math] [math]\displaystyle{ -1024 }[/math] [math]\displaystyle{ -\frac{5440}{3} }[/math] [math]\displaystyle{ -\frac{928}{3} }[/math] [math]\displaystyle{ -224 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 2048 }[/math] [math]\displaystyle{ \frac{15488}{3} }[/math] [math]\displaystyle{ \frac{2176}{3} }[/math] [math]\displaystyle{ \frac{156422}{15} }[/math] [math]\displaystyle{ \frac{5912}{15} }[/math] [math]\displaystyle{ \frac{170888}{45} }[/math] [math]\displaystyle{ \frac{730}{9} }[/math] [math]\displaystyle{ \frac{7142}{15} }[/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.