7 5

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7 4.gif

7_4

7 6.gif

7_6

7 5.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 7 5's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 7 5 at Knotilus!

[edit 7 5 Quick Notes]
[edit 7 5 Further Notes and Views]


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]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif

Length is 8, width is 3,

Braid index is 3

7 5 ML.gif 7 5 AP.gif
[{9, 2}, {1, 7}, {6, 8}, {7, 9}, {8, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 1}]

[edit Notes on presentations of 7 5]

Knot 7_5.
A graph, knot 7_5.
Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["7 5"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= 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
In[5]:= GaussCode[K]
Out[5]= -1, 7, -2, 1, -3, 5, -4, 6, -7, 2, -6, 3, -5, 4
In[6]:= DTCode[K]
Out[6]= 4 10 12 14 2 8 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [322]
In[9]:= br = BR[K]
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
Out[9]= [math]\displaystyle{ \textrm{BR}(3,\{-1,-1,-1,-1,-2,1,-2,-2\}) }[/math]
In[10]:= {First[br], Crossings[br], BraidIndex[K]}
KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
KnotTheory::loading: Loading precomputed data in IndianaData`.
Out[10]= { 3, 8, 3 }
In[11]:= Show[BraidPlot[br]]
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif
Out[11]= -Graphics-
In[12]:= Show[DrawMorseLink[K]]
KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
7 5 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{9, 2}, {1, 7}, {6, 8}, {7, 9}, {8, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 1}]
In[14]:= Draw[ap]
7 5 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index 4
Nakanishi index 1
Maximal Thurston-Bennequin number [math]\displaystyle{ \text{$\$$Failed} }[/math]
Hyperbolic Volume 6.44354
A-Polynomial See Data:7 5/A-polynomial

[edit Notes for 7 5'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{ \textrm{ConcordanceGenus}(\textrm{Knot}(7,5)) }[/math]
Rasmussen s-Invariant -4

[edit Notes for 7 5's four dimensional invariants]

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]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_130,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["7 5"];
In[4]:= {A = Alexander[K][t], J = Jones[K][q]}
KnotTheory::loading: Loading precomputed data in PD4Knots`.
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[4]= { [math]\displaystyle{ 2 t^2-4 t+5-4 t^{-1} +2 t^{-2} }[/math], [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[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
Out[5]= {10_130,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {}

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]

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 7 5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-10χ
-3       11
-5      110
-7     2  2
-9    11  0
-11   22   0
-13  11    0
-15 12     -1
-17 1      1
-191       -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

 [math]\displaystyle{ n }[/math]  [math]\displaystyle{ J_n }[/math]
2 [math]\displaystyle{ q^{-4} - q^{-5} +4 q^{-7} -3 q^{-8} -3 q^{-9} +9 q^{-10} -5 q^{-11} -7 q^{-12} +13 q^{-13} -5 q^{-14} -10 q^{-15} +14 q^{-16} -4 q^{-17} -9 q^{-18} +11 q^{-19} -2 q^{-20} -6 q^{-21} +5 q^{-22} -2 q^{-24} + q^{-25} }[/math]
3 [math]\displaystyle{ q^{-6} - q^{-7} + q^{-9} +3 q^{-10} -3 q^{-11} -3 q^{-12} +2 q^{-13} +9 q^{-14} -4 q^{-15} -10 q^{-16} - q^{-17} +18 q^{-18} - q^{-19} -18 q^{-20} -6 q^{-21} +24 q^{-22} +7 q^{-23} -25 q^{-24} -12 q^{-25} +28 q^{-26} +13 q^{-27} -28 q^{-28} -15 q^{-29} +27 q^{-30} +16 q^{-31} -26 q^{-32} -15 q^{-33} +22 q^{-34} +15 q^{-35} -18 q^{-36} -12 q^{-37} +12 q^{-38} +11 q^{-39} -8 q^{-40} -8 q^{-41} +4 q^{-42} +5 q^{-43} -2 q^{-44} -2 q^{-45} +2 q^{-47} - q^{-48} }[/math]
4 [math]\displaystyle{ q^{-8} - q^{-9} + q^{-11} +3 q^{-13} -4 q^{-14} -2 q^{-15} +3 q^{-16} + q^{-17} +10 q^{-18} -9 q^{-19} -9 q^{-20} +2 q^{-22} +26 q^{-23} -9 q^{-24} -17 q^{-25} -12 q^{-26} -5 q^{-27} +48 q^{-28} - q^{-29} -19 q^{-30} -28 q^{-31} -22 q^{-32} +66 q^{-33} +13 q^{-34} -13 q^{-35} -43 q^{-36} -43 q^{-37} +79 q^{-38} +26 q^{-39} -6 q^{-40} -54 q^{-41} -57 q^{-42} +84 q^{-43} +34 q^{-44} +2 q^{-45} -59 q^{-46} -64 q^{-47} +82 q^{-48} +37 q^{-49} +7 q^{-50} -55 q^{-51} -64 q^{-52} +69 q^{-53} +33 q^{-54} +13 q^{-55} -42 q^{-56} -56 q^{-57} +47 q^{-58} +22 q^{-59} +17 q^{-60} -22 q^{-61} -40 q^{-62} +23 q^{-63} +9 q^{-64} +15 q^{-65} -7 q^{-66} -21 q^{-67} +9 q^{-68} +7 q^{-70} -7 q^{-72} +3 q^{-73} - q^{-74} +2 q^{-75} -2 q^{-77} + q^{-78} }[/math]
5 [math]\displaystyle{ q^{-10} - q^{-11} + q^{-13} +2 q^{-16} -3 q^{-17} -2 q^{-18} +3 q^{-19} +3 q^{-20} + q^{-21} +4 q^{-22} -8 q^{-23} -9 q^{-24} +8 q^{-26} +10 q^{-27} +14 q^{-28} -10 q^{-29} -23 q^{-30} -15 q^{-31} + q^{-32} +22 q^{-33} +39 q^{-34} +5 q^{-35} -31 q^{-36} -40 q^{-37} -27 q^{-38} +18 q^{-39} +69 q^{-40} +40 q^{-41} -19 q^{-42} -61 q^{-43} -69 q^{-44} -7 q^{-45} +82 q^{-46} +87 q^{-47} +13 q^{-48} -69 q^{-49} -110 q^{-50} -44 q^{-51} +82 q^{-52} +125 q^{-53} +53 q^{-54} -70 q^{-55} -142 q^{-56} -74 q^{-57} +74 q^{-58} +152 q^{-59} +83 q^{-60} -66 q^{-61} -162 q^{-62} -95 q^{-63} +67 q^{-64} +166 q^{-65} +102 q^{-66} -62 q^{-67} -168 q^{-68} -107 q^{-69} +56 q^{-70} +167 q^{-71} +111 q^{-72} -51 q^{-73} -159 q^{-74} -111 q^{-75} +38 q^{-76} +148 q^{-77} +112 q^{-78} -30 q^{-79} -130 q^{-80} -103 q^{-81} +11 q^{-82} +110 q^{-83} +97 q^{-84} -3 q^{-85} -83 q^{-86} -81 q^{-87} -11 q^{-88} +58 q^{-89} +67 q^{-90} +16 q^{-91} -37 q^{-92} -47 q^{-93} -19 q^{-94} +20 q^{-95} +31 q^{-96} +15 q^{-97} -7 q^{-98} -18 q^{-99} -12 q^{-100} +3 q^{-101} +10 q^{-102} +3 q^{-103} +2 q^{-104} -2 q^{-105} -6 q^{-106} + q^{-107} +3 q^{-108} - q^{-109} + q^{-111} -2 q^{-112} +2 q^{-114} - q^{-115} }[/math]
6 [math]\displaystyle{ q^{-12} - q^{-13} + q^{-15} - q^{-18} +3 q^{-19} -3 q^{-20} -2 q^{-21} +4 q^{-22} +2 q^{-23} +2 q^{-24} -4 q^{-25} +5 q^{-26} -9 q^{-27} -9 q^{-28} +6 q^{-29} +8 q^{-30} +11 q^{-31} -2 q^{-32} +14 q^{-33} -21 q^{-34} -29 q^{-35} -5 q^{-36} +7 q^{-37} +26 q^{-38} +14 q^{-39} +48 q^{-40} -20 q^{-41} -52 q^{-42} -40 q^{-43} -23 q^{-44} +17 q^{-45} +30 q^{-46} +116 q^{-47} +17 q^{-48} -44 q^{-49} -76 q^{-50} -84 q^{-51} -41 q^{-52} +7 q^{-53} +188 q^{-54} +89 q^{-55} +17 q^{-56} -75 q^{-57} -143 q^{-58} -140 q^{-59} -70 q^{-60} +224 q^{-61} +165 q^{-62} +118 q^{-63} -27 q^{-64} -168 q^{-65} -246 q^{-66} -181 q^{-67} +220 q^{-68} +217 q^{-69} +223 q^{-70} +44 q^{-71} -162 q^{-72} -331 q^{-73} -287 q^{-74} +196 q^{-75} +244 q^{-76} +307 q^{-77} +107 q^{-78} -144 q^{-79} -390 q^{-80} -363 q^{-81} +173 q^{-82} +256 q^{-83} +365 q^{-84} +148 q^{-85} -131 q^{-86} -425 q^{-87} -407 q^{-88} +155 q^{-89} +258 q^{-90} +397 q^{-91} +174 q^{-92} -117 q^{-93} -438 q^{-94} -428 q^{-95} +131 q^{-96} +249 q^{-97} +405 q^{-98} +194 q^{-99} -89 q^{-100} -422 q^{-101} -429 q^{-102} +90 q^{-103} +213 q^{-104} +383 q^{-105} +210 q^{-106} -39 q^{-107} -365 q^{-108} -399 q^{-109} +34 q^{-110} +143 q^{-111} +318 q^{-112} +208 q^{-113} +27 q^{-114} -263 q^{-115} -325 q^{-116} -16 q^{-117} +53 q^{-118} +212 q^{-119} +172 q^{-120} +78 q^{-121} -142 q^{-122} -215 q^{-123} -35 q^{-124} -16 q^{-125} +100 q^{-126} +105 q^{-127} +86 q^{-128} -51 q^{-129} -105 q^{-130} -19 q^{-131} -38 q^{-132} +26 q^{-133} +40 q^{-134} +57 q^{-135} -10 q^{-136} -37 q^{-137} +3 q^{-138} -24 q^{-139} - q^{-140} +6 q^{-141} +24 q^{-142} -2 q^{-143} -10 q^{-144} +8 q^{-145} -8 q^{-146} -2 q^{-147} -2 q^{-148} +8 q^{-149} -2 q^{-150} -4 q^{-151} +5 q^{-152} -2 q^{-153} - q^{-155} +2 q^{-156} -2 q^{-158} + q^{-159} }[/math]
7 [math]\displaystyle{ q^{-14} - q^{-15} + q^{-17} - q^{-20} +3 q^{-22} -3 q^{-23} - q^{-24} +3 q^{-25} +2 q^{-26} +2 q^{-27} -3 q^{-28} -4 q^{-29} +5 q^{-30} -8 q^{-31} -4 q^{-32} +5 q^{-33} +8 q^{-34} +13 q^{-35} - q^{-36} -8 q^{-37} +3 q^{-38} -22 q^{-39} -19 q^{-40} -2 q^{-41} +10 q^{-42} +38 q^{-43} +22 q^{-44} +7 q^{-45} +15 q^{-46} -41 q^{-47} -53 q^{-48} -41 q^{-49} -23 q^{-50} +45 q^{-51} +57 q^{-52} +61 q^{-53} +82 q^{-54} -17 q^{-55} -75 q^{-56} -102 q^{-57} -120 q^{-58} -22 q^{-59} +38 q^{-60} +113 q^{-61} +205 q^{-62} +97 q^{-63} -13 q^{-64} -115 q^{-65} -239 q^{-66} -175 q^{-67} -94 q^{-68} +71 q^{-69} +311 q^{-70} +270 q^{-71} +168 q^{-72} -10 q^{-73} -292 q^{-74} -341 q^{-75} -320 q^{-76} -105 q^{-77} +307 q^{-78} +422 q^{-79} +409 q^{-80} +214 q^{-81} -232 q^{-82} -441 q^{-83} -558 q^{-84} -370 q^{-85} +193 q^{-86} +483 q^{-87} +633 q^{-88} +482 q^{-89} -84 q^{-90} -463 q^{-91} -744 q^{-92} -628 q^{-93} +23 q^{-94} +471 q^{-95} +791 q^{-96} +727 q^{-97} +72 q^{-98} -439 q^{-99} -864 q^{-100} -830 q^{-101} -127 q^{-102} +433 q^{-103} +893 q^{-104} +899 q^{-105} +193 q^{-106} -415 q^{-107} -934 q^{-108} -962 q^{-109} -228 q^{-110} +408 q^{-111} +950 q^{-112} +1007 q^{-113} +264 q^{-114} -398 q^{-115} -972 q^{-116} -1039 q^{-117} -289 q^{-118} +390 q^{-119} +979 q^{-120} +1063 q^{-121} +313 q^{-122} -376 q^{-123} -979 q^{-124} -1079 q^{-125} -341 q^{-126} +355 q^{-127} +970 q^{-128} +1087 q^{-129} +365 q^{-130} -321 q^{-131} -940 q^{-132} -1081 q^{-133} -402 q^{-134} +270 q^{-135} +900 q^{-136} +1064 q^{-137} +428 q^{-138} -211 q^{-139} -818 q^{-140} -1019 q^{-141} -466 q^{-142} +128 q^{-143} +731 q^{-144} +959 q^{-145} +471 q^{-146} -49 q^{-147} -597 q^{-148} -859 q^{-149} -480 q^{-150} -44 q^{-151} +472 q^{-152} +741 q^{-153} +450 q^{-154} +109 q^{-155} -323 q^{-156} -597 q^{-157} -406 q^{-158} -162 q^{-159} +193 q^{-160} +457 q^{-161} +336 q^{-162} +181 q^{-163} -89 q^{-164} -317 q^{-165} -252 q^{-166} -177 q^{-167} +13 q^{-168} +200 q^{-169} +173 q^{-170} +149 q^{-171} +30 q^{-172} -115 q^{-173} -105 q^{-174} -108 q^{-175} -41 q^{-176} +57 q^{-177} +44 q^{-178} +74 q^{-179} +45 q^{-180} -25 q^{-181} -22 q^{-182} -43 q^{-183} -22 q^{-184} +11 q^{-185} -7 q^{-186} +18 q^{-187} +26 q^{-188} -3 q^{-189} -12 q^{-191} -4 q^{-192} +7 q^{-193} -12 q^{-194} + q^{-195} +8 q^{-196} +2 q^{-198} -4 q^{-199} +5 q^{-201} -4 q^{-202} -2 q^{-203} +2 q^{-204} + q^{-206} -2 q^{-207} +2 q^{-209} - q^{-210} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

Back to the top.

7 4.gif

7_4

7 6.gif

7_6

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