7 1

6_3

7_2

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 7 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 7 1 at Knotilus!
	7_1 should perhaps be called "The Septafoil Knot", following the trefoil knot and the cinquefoil knot. See also T(7,2).

Interlaced form of 7/2 star polygon or "septagram"		Decorative interlaced form of 7/2 star polygon or "septagram"		3D depiction
Heptagram of intersecting circles.

Knot presentations

Planar diagram presentation	X1829 X3,10,4,11 X5,12,6,13 X7,14,8,1 X9,2,10,3 X11,4,12,5 X13,6,14,7
Gauss code	-1, 5, -2, 6, -3, 7, -4, 1, -5, 2, -6, 3, -7, 4
Dowker-Thistlethwaite code	8 10 12 14 2 4 6
Conway Notation	[7]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 7, width is 2, Braid index is 2		[{9, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 1}]

[edit Notes on presentations of 7 1]

Knot 7_1.

A graph, knot 7_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.org/wiki/KnotTheory.

In[3]:=

K = Knot["7 1"];

In[4]:=

PD[K]

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

Out[4]=

X1829 X3,10,4,11 X5,12,6,13 X7,14,8,1 X9,2,10,3 X11,4,12,5 X13,6,14,7

In[5]:=

GaussCode[K]

Out[5]=

-1, 5, -2, 6, -3, 7, -4, 1, -5, 2, -6, 3, -7, 4

In[6]:=

DTCode[K]

Out[6]=

8 10 12 14 2 4 6

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[7]

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]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(2,\{-1,-1,-1,-1,-1,-1,-1\})}

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]=

{ 2, 7, 2 }

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, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 2 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{$\$$Failed}} Hyperbolic Volume Not hyperbolic A-Polynomial See Data:7 1/A-polynomial

[edit Notes for 7 1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3} Topological 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3} Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{ConcordanceGenus}(\textrm{Knot}(7,1))} Rasmussen s-Invariant -6

[edit Notes for 7 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3+ t^{-3} -t^2- t^{-2} +t+ t^{-1} -1}
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6+5 z^4+6 z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 7, -6 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - q^{-10} + q^{-9} - q^{-8} + q^{-7} - q^{-6} + q^{-5} + q^{-3} }
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^8 \left(-z^4\right)-4 a^8 z^2-3 a^8+a^6 z^6+6 a^6 z^4+10 a^6 z^2+4 a^6}
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{13} z+a^{12} z^2+a^{11} z^3-a^{11} z+a^{10} z^4-2 a^{10} z^2+a^9 z^5-3 a^9 z^3+a^9 z+a^8 z^6-5 a^8 z^4+7 a^8 z^2-3 a^8+a^7 z^5-4 a^7 z^3+3 a^7 z+a^6 z^6-6 a^6 z^4+10 a^6 z^2-4 a^6}
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{30}-q^{28}-q^{26}+q^{18}+q^{16}+2 q^{14}+q^{12}+q^{10}}
The G2 invariant	$q^{168}-q^{136}-q^{134}-q^{128}-q^{126}-q^{124}-q^{118}-q^{116}-q^{102}-q^{96}-q^{94}-q^{92}+q^{88}-2q^{84}+2q^{80}+q^{78}+q^{72}+3q^{70}+2q^{68}+q^{64}+2q^{62}+2q^{60}+q^{58}+q^{54}+q^{52}+q^{50}$

Further Quantum Invariants

Further quantum knot invariants for 7_1.

A1 Invariants.

Weight	Invariant
1	$-q^{21}+q^{9}+q^{7}+q^{5}$
2	$q^{56}-q^{44}-q^{42}-q^{40}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}$
3	$-q^{105}+q^{93}+q^{91}+q^{89}-q^{67}-q^{65}-q^{63}-q^{61}-q^{59}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15}$
4	$q^{168}-q^{156}-q^{154}-q^{152}+q^{130}+q^{128}+q^{126}+q^{124}+q^{122}-q^{90}-q^{88}-q^{86}-q^{84}-q^{82}-q^{80}-q^{78}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}$
5	$-q^{245}+q^{233}+q^{231}+q^{229}-q^{207}-q^{205}-q^{203}-q^{201}-q^{199}+q^{167}+q^{165}+q^{163}+q^{161}+q^{159}+q^{157}+q^{155}-q^{113}-q^{111}-q^{109}-q^{107}-q^{105}-q^{103}-q^{101}-q^{99}-q^{97}+q^{45}+q^{43}+q^{41}+q^{39}+q^{37}+q^{35}+q^{33}+q^{31}+q^{29}+q^{27}+q^{25}$
6	$q^{336}-q^{324}-q^{322}-q^{320}+q^{298}+q^{296}+q^{294}+q^{292}+q^{290}-q^{258}-q^{256}-q^{254}-q^{252}-q^{250}-q^{248}-q^{246}+q^{204}+q^{202}+q^{200}+q^{198}+q^{196}+q^{194}+q^{192}+q^{190}+q^{188}-q^{136}-q^{134}-q^{132}-q^{130}-q^{128}-q^{126}-q^{124}-q^{122}-q^{120}-q^{118}-q^{116}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}$
8	$q^{560}-q^{548}-q^{546}-q^{544}+q^{522}+q^{520}+q^{518}+q^{516}+q^{514}-q^{482}-q^{480}-q^{478}-q^{476}-q^{474}-q^{472}-q^{470}+q^{428}+q^{426}+q^{424}+q^{422}+q^{420}+q^{418}+q^{416}+q^{414}+q^{412}-q^{360}-q^{358}-q^{356}-q^{354}-q^{352}-q^{350}-q^{348}-q^{346}-q^{344}-q^{342}-q^{340}+q^{278}+q^{276}+q^{274}+q^{272}+q^{270}+q^{268}+q^{266}+q^{264}+q^{262}+q^{260}+q^{258}+q^{256}+q^{254}-q^{182}-q^{180}-q^{178}-q^{176}-q^{174}-q^{172}-q^{170}-q^{168}-q^{166}-q^{164}-q^{162}-q^{160}-q^{158}-q^{156}-q^{154}+q^{72}+q^{70}+q^{68}+q^{66}+q^{64}+q^{62}+q^{60}+q^{58}+q^{56}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+q^{40}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{30}-q^{28}-q^{26}+q^{18}+q^{16}+2q^{14}+q^{12}+q^{10}$
1,1	$q^{84}-2q^{48}-2q^{46}-4q^{44}-4q^{42}-4q^{40}-2q^{38}-q^{36}+2q^{34}+4q^{32}+4q^{30}+5q^{28}+4q^{26}+4q^{24}+2q^{22}+q^{20}$
2,0	$q^{74}+q^{72}+2q^{70}+q^{68}+q^{66}-q^{62}-2q^{60}-3q^{58}-3q^{56}-3q^{54}-2q^{52}-q^{50}+q^{36}+q^{34}+2q^{32}+2q^{30}+3q^{28}+2q^{26}+2q^{24}+q^{22}+q^{20}$
3,0	$-q^{132}-q^{130}-2q^{128}-2q^{126}-2q^{124}-q^{122}+2q^{118}+4q^{116}+4q^{114}+5q^{112}+4q^{110}+4q^{108}+2q^{106}+q^{104}-q^{94}-2q^{92}-3q^{90}-4q^{88}-5q^{86}-5q^{84}-5q^{82}-4q^{80}-3q^{78}-2q^{76}-q^{74}+q^{54}+q^{52}+2q^{50}+2q^{48}+3q^{46}+3q^{44}+4q^{42}+3q^{40}+3q^{38}+2q^{36}+2q^{34}+q^{32}+q^{30}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{70}-q^{48}-2q^{46}-3q^{44}-3q^{42}-3q^{40}-2q^{38}+q^{34}+3q^{32}+3q^{30}+4q^{28}+3q^{26}+3q^{24}+q^{22}+q^{20}$
1,0,0	$-q^{39}-q^{37}-2q^{35}-q^{33}-q^{31}+q^{27}+q^{25}+2q^{23}+2q^{21}+2q^{19}+q^{17}+q^{15}$
1,0,1	$q^{112}+q^{78}+q^{76}+3q^{74}+3q^{72}+4q^{70}+3q^{68}+q^{66}-3q^{64}-7q^{62}-10q^{60}-14q^{58}-14q^{56}-13q^{54}-8q^{52}-3q^{50}+3q^{48}+8q^{46}+11q^{44}+12q^{42}+11q^{40}+10q^{38}+7q^{36}+5q^{34}+2q^{32}+q^{30}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{88}+q^{86}+q^{84}+q^{82}+q^{80}-q^{66}-2q^{64}-4q^{62}-5q^{60}-7q^{58}-7q^{56}-6q^{54}-4q^{52}-q^{50}+2q^{48}+5q^{46}+6q^{44}+8q^{42}+6q^{40}+6q^{38}+4q^{36}+3q^{34}+q^{32}+q^{30}$
1,0,0,0	$-q^{48}-q^{46}-2q^{44}-2q^{42}-2q^{40}-q^{38}+q^{34}+2q^{32}+2q^{30}+3q^{28}+2q^{26}+2q^{24}+q^{22}+q^{20}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{70}-q^{48}-q^{44}-q^{42}-q^{40}+q^{34}+q^{32}+q^{30}+2q^{28}+q^{26}+q^{24}+q^{22}+q^{20}$
1,0	$q^{112}-q^{78}-q^{76}-q^{74}-q^{72}-2q^{70}-q^{68}-q^{66}-q^{64}-q^{62}+q^{54}+q^{50}+q^{48}+2q^{46}+q^{44}+2q^{42}+q^{40}+2q^{38}+q^{36}+q^{34}+q^{30}$

D4 Invariants.

Weight	Invariant
0,1,0,0	$q^{168}-q^{136}-q^{134}-3q^{132}-3q^{130}-4q^{128}-4q^{126}-q^{124}+3q^{120}+8q^{118}+11q^{116}+12q^{114}+15q^{112}+12q^{110}+12q^{108}+9q^{106}+6q^{104}+2q^{102}-6q^{98}-10q^{96}-16q^{94}-22q^{92}-27q^{90}-32q^{88}-33q^{86}-32q^{84}-27q^{82}-19q^{80}-8q^{78}+12q^{74}+19q^{72}+24q^{70}+26q^{68}+27q^{66}+22q^{64}+20q^{62}+14q^{60}+10q^{58}+6q^{56}+4q^{54}+q^{52}+q^{50}$
1,0,0,0	$q^{98}-q^{66}-q^{64}-3q^{62}-3q^{60}-4q^{58}-4q^{56}-3q^{54}-2q^{52}-q^{50}+2q^{48}+3q^{46}+4q^{44}+5q^{42}+4q^{40}+4q^{38}+3q^{36}+2q^{34}+q^{32}+q^{30}$

G2 Invariants.

Weight	Invariant
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{168}-q^{136}-q^{134}-q^{128}-q^{126}-q^{124}-q^{118}-q^{116}-q^{102}-q^{96}-q^{94}-q^{92}+q^{88}-2 q^{84}+2 q^{80}+q^{78}+q^{72}+3 q^{70}+2 q^{68}+q^{64}+2 q^{62}+2 q^{60}+q^{58}+q^{54}+q^{52}+q^{50}}

.

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 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3+ t^{-3} -t^2- t^{-2} +t+ t^{-1} -1}

In[5]:=

Conway[K][z]

Out[5]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6+5 z^4+6 z^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]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 7, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - q^{-10} + q^{-9} - q^{-8} + q^{-7} - q^{-6} + q^{-5} + q^{-3} }

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^8 \left(-z^4\right)-4 a^8 z^2-3 a^8+a^6 z^6+6 a^6 z^4+10 a^6 z^2+4 a^6}

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{13} z+a^{12} z^2+a^{11} z^3-a^{11} z+a^{10} z^4-2 a^{10} z^2+a^9 z^5-3 a^9 z^3+a^9 z+a^8 z^6-5 a^8 z^4+7 a^8 z^2-3 a^8+a^7 z^5-4 a^7 z^3+3 a^7 z+a^6 z^6-6 a^6 z^4+10 a^6 z^2-4 a^6}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 1"];

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]=

{ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3+ t^{-3} -t^2- t^{-2} +t+ t^{-1} -1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - q^{-10} + q^{-9} - q^{-8} + q^{-7} - q^{-6} + q^{-5} + q^{-3} } }

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]=

{}

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:

(6, -14)

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 24} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -112} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 288} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 684} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 100} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2688} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{13888}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{2464}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -560} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2304} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6272} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 16416} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2400} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{160231}{5}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{21548}{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{58148}{5}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 163} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{7351}{5}}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} -6 is the signature of 7 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 χ -5               1 1 -7               1 1 -9           1     1 -11                 0 -13       1 1       0 -15                 0 -17   1 1           0 -19                 0 -21 1               -1

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-7} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-7} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-6} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-6} + q^{-9} - q^{-11} + q^{-12} - q^{-14} + q^{-15} - q^{-17} + q^{-18} - q^{-20} - q^{-23} + q^{-24} - q^{-26} + q^{-27} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-9} + q^{-13} - q^{-16} + q^{-17} - q^{-20} + q^{-21} - q^{-24} + q^{-25} - q^{-28} + q^{-29} - q^{-31} - q^{-32} + q^{-33} - q^{-35} + q^{-37} - q^{-39} + q^{-41} - q^{-43} + q^{-45} + q^{-46} - q^{-47} + q^{-50} - q^{-51} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-12} + q^{-17} - q^{-21} + q^{-22} - q^{-26} + q^{-27} - q^{-31} + q^{-32} - q^{-36} + q^{-37} -2 q^{-41} + q^{-42} -2 q^{-46} + q^{-47} + q^{-48} -2 q^{-51} + q^{-52} + q^{-53} -2 q^{-56} + q^{-57} + q^{-58} -2 q^{-61} + q^{-62} +2 q^{-63} -2 q^{-66} + q^{-67} + q^{-68} -2 q^{-71} + q^{-72} + q^{-73} -2 q^{-76} + q^{-77} - q^{-81} + q^{-82} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-15} + q^{-21} - q^{-26} + q^{-27} - q^{-32} + q^{-33} - q^{-38} + q^{-39} - q^{-44} + q^{-45} - q^{-50} - q^{-56} + q^{-60} - q^{-62} + q^{-66} - q^{-68} + q^{-72} - q^{-74} + q^{-78} + q^{-84} - q^{-87} + q^{-90} - q^{-93} + q^{-96} - q^{-99} - q^{-105} + q^{-107} - q^{-111} + q^{-113} + q^{-119} - q^{-120} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-18} + q^{-25} - q^{-31} + q^{-32} - q^{-38} + q^{-39} - q^{-45} + q^{-46} - q^{-52} + q^{-53} - q^{-59} + q^{-60} - q^{-61} - q^{-66} + q^{-67} - q^{-68} + q^{-72} - q^{-73} + q^{-74} - q^{-75} + q^{-79} - q^{-80} + q^{-81} - q^{-82} + q^{-86} - q^{-87} + q^{-88} - q^{-89} + q^{-93} - q^{-94} + q^{-95} - q^{-96} + q^{-97} + q^{-100} - q^{-101} + q^{-102} - q^{-103} + q^{-104} - q^{-106} + q^{-107} - q^{-108} + q^{-109} - q^{-110} + q^{-111} - q^{-113} + q^{-114} - q^{-115} + q^{-116} - q^{-117} + q^{-118} - q^{-120} + q^{-121} - q^{-122} + q^{-123} - q^{-124} + q^{-125} - q^{-126} - q^{-127} + q^{-128} - q^{-129} + q^{-130} - q^{-131} + q^{-132} - q^{-134} + q^{-135} - q^{-136} + q^{-137} - q^{-138} + q^{-139} - q^{-141} + q^{-142} - q^{-143} + q^{-144} - q^{-145} + q^{-146} + q^{-149} - q^{-150} + q^{-151} - q^{-152} + q^{-156} - q^{-157} + q^{-158} - q^{-159} - q^{-164} + q^{-165} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-21} + q^{-29} - q^{-36} + q^{-37} - q^{-44} + q^{-45} - q^{-52} + q^{-53} - q^{-60} + q^{-61} - q^{-68} + q^{-69} - q^{-71} - q^{-76} + q^{-77} - q^{-79} + q^{-85} - q^{-87} + q^{-93} - q^{-95} + q^{-101} - q^{-103} + q^{-109} - q^{-111} + q^{-114} + q^{-117} - q^{-119} + q^{-122} - q^{-127} + q^{-130} - q^{-135} + q^{-138} - q^{-143} + q^{-146} - q^{-150} - q^{-151} + q^{-154} - q^{-158} + q^{-162} - q^{-166} + q^{-170} - q^{-174} + q^{-178} + q^{-179} - q^{-182} + q^{-187} - q^{-190} + q^{-195} - q^{-198} - q^{-201} + q^{-203} - q^{-209} + q^{-211} + q^{-216} - q^{-217} }

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.

6_3 7_2