7 5

7_4

7_6

(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!

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
Length is 8, width is 3, Braid index is 3		[{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]=

${\textrm {BR}}(3,\{-1,-1,-1,-1,-2,1,-2,-2\})$

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]]

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."

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]

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 Failed to parse (syntax error): {\displaystyle \text{$\$$Failed}} 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 $2$ Topological 4 genus $2$ Concordance genus ${\textrm {ConcordanceGenus}}({\textrm {Knot}}(7,5))$ Rasmussen s-Invariant -4

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

Polynomial invariants

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

Further Quantum Invariants

Further quantum knot invariants for 7_5.

A1 Invariants.

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

A2 Invariants.

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

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{86}-q^{84}+q^{82}-q^{80}+2q^{78}-2q^{76}+q^{74}-q^{72}+2q^{70}-q^{66}-2q^{62}+q^{60}-4q^{58}-4q^{54}+2q^{52}-3q^{50}+2q^{48}-3q^{46}+q^{44}+3q^{34}+q^{32}+4q^{30}+q^{28}+4q^{26}+q^{24}+2q^{22}+q^{18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{148}-q^{146}+2q^{144}-2q^{142}+q^{138}-2q^{136}+5q^{134}-5q^{132}+4q^{130}-2q^{128}-3q^{126}+4q^{124}-6q^{122}+5q^{120}-3q^{118}-q^{116}+3q^{114}-3q^{112}+q^{110}+2q^{108}-5q^{106}+4q^{104}-3q^{102}-2q^{100}+5q^{98}-7q^{96}+8q^{94}-7q^{92}+2q^{90}+2q^{88}-6q^{86}+6q^{84}-7q^{82}+4q^{80}-2q^{76}+3q^{74}-3q^{72}+2q^{70}+3q^{68}-5q^{66}+3q^{64}-2q^{60}+7q^{58}-5q^{56}+5q^{54}-q^{52}+4q^{48}-4q^{46}+5q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+2q^{34}+q^{30}$

.

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]=

$2t^{2}-4t+5-4t^{-1}+2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{4}+4z^{2}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 17, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{-9}+2q^{-8}-3q^{-7}+3q^{-6}-3q^{-5}+3q^{-4}-q^{-3}+q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{8}\left(-z^{2}\right)-a^{8}+a^{6}z^{4}+2a^{6}z^{2}+a^{4}z^{4}+3a^{4}z^{2}+2a^{4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{11}z^{3}-a^{11}z+2a^{10}z^{4}-2a^{10}z^{2}+2a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+a^{8}z^{6}+a^{8}z^{2}-a^{8}+3a^{7}z^{5}-4a^{7}z^{3}+a^{7}z+a^{6}z^{6}-a^{6}z^{4}+a^{5}z^{5}-a^{5}z^{3}-a^{5}z+a^{4}z^{4}-3a^{4}z^{2}+2a^{4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_130,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

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]=

{ $2t^{2}-4t+5-4t^{-1}+2t^{-2}$, $-q^{-9}+2q^{-8}-3q^{-7}+3q^{-6}-3q^{-5}+3q^{-4}-q^{-3}+q^{-2}$ }

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 $16$ $-64$ $128$ ${\frac {968}{3}}$ ${\frac {136}{3}}$ $-1024$ $-{\frac {5440}{3}}$ $-{\frac {928}{3}}$ $-224$ ${\frac {2048}{3}}$ $2048$ ${\frac {15488}{3}}$ ${\frac {2176}{3}}$ ${\frac {156422}{15}}$ ${\frac {5912}{15}}$ ${\frac {170888}{45}}$ ${\frac {730}{9}}$ ${\frac {7142}{15}}$

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$-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 -1 0 χ -3               1 1 -5             1 1 0 -7           2     2 -9         1 1     0 -11       2 2       0 -13     1 1         0 -15   1 2           -1 -17   1             1 -19 1               -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

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

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.

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