10 7

10_6

10_8

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 7's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 7 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X5,14,6,15 X3,13,4,12 X13,3,14,2 X11,20,12,1 X19,6,20,7 X7,18,8,19 X9,16,10,17 X15,10,16,11 X17,8,18,9
Gauss code	-1, 4, -3, 1, -2, 6, -7, 10, -8, 9, -5, 3, -4, 2, -9, 8, -10, 7, -6, 5
Dowker-Thistlethwaite code	4 12 14 18 16 20 2 10 8 6
Conway Notation	[5212]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 12, width is 5, Braid index is 5		[{12, 7}, {1, 10}, {11, 8}, {7, 9}, {10, 12}, {6, 11}, {8, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {9, 2}]

[edit Notes on presentations of 10 7]

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["10 7"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X5,14,6,15 X3,13,4,12 X13,3,14,2 X11,20,12,1 X19,6,20,7 X7,18,8,19 X9,16,10,17 X15,10,16,11 X17,8,18,9

In[5]:=

GaussCode[K]

Out[5]=

-1, 4, -3, 1, -2, 6, -7, 10, -8, 9, -5, 3, -4, 2, -9, 8, -10, 7, -6, 5

In[6]:=

DTCode[K]

Out[6]=

4 12 14 18 16 20 2 10 8 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[5212]

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]=

${\textrm {BR}}(5,\{-1,-1,-2,1,-2,-3,2,-3,-3,4,-3,4\})$

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]=

{ 5, 12, 5 }

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[{12, 7}, {1, 10}, {11, 8}, {7, 9}, {10, 12}, {6, 11}, {8, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {9, 2}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-11][-1] Hyperbolic Volume 9.11591 A-Polynomial See Data:10 7/A-polynomial

[edit Notes for 10 7's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant -2

[edit Notes for 10 7's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{2}+11t-15+11t^{-1}-3t^{-2}$
Conway polynomial	$-3z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 43, -2 }
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-2+4 q^{-1} -5 q^{-2} +7 q^{-3} -7 q^{-4} +6 q^{-5} -5 q^{-6} +3 q^{-7} -2 q^{-8} + q^{-9} }
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 z^2 a^8+a^8-z^4 a^6-2 z^2 a^6-2 a^6-z^4 a^4+a^4-z^4 a^2-z^2 a^2+z^2+1}
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 z^6 a^{10}-4 z^4 a^{10}+3 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+8 z^3 a^9-3 z a^9+2 z^8 a^8-7 z^6 a^8+6 z^4 a^8-3 z^2 a^8+a^8+z^9 a^7-z^7 a^7-6 z^5 a^7+10 z^3 a^7-5 z a^7+4 z^8 a^6-15 z^6 a^6+20 z^4 a^6-10 z^2 a^6+2 a^6+z^9 a^5-z^7 a^5-2 z^5 a^5+6 z^3 a^5-2 z a^5+2 z^8 a^4-5 z^6 a^4+8 z^4 a^4-4 z^2 a^4+a^4+2 z^7 a^3-2 z^5 a^3+z^3 a^3+2 z^6 a^2-z^4 a^2-2 z^2 a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1}
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^{28}+q^{22}-2 q^{20}-q^{18}-q^{14}+q^{12}+q^8+q^6-q^4+2 q^2+ q^{-4} }
The G2 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^{142}-q^{140}+2 q^{138}-3 q^{136}+2 q^{134}-3 q^{132}-q^{130}+6 q^{128}-10 q^{126}+12 q^{124}-12 q^{122}+8 q^{120}+2 q^{118}-12 q^{116}+22 q^{114}-24 q^{112}+21 q^{110}-8 q^{108}-7 q^{106}+20 q^{104}-23 q^{102}+22 q^{100}-9 q^{98}-4 q^{96}+14 q^{94}-16 q^{92}+8 q^{90}+2 q^{88}-14 q^{86}+19 q^{84}-16 q^{82}+q^{80}+10 q^{78}-23 q^{76}+27 q^{74}-25 q^{72}+11 q^{70}+2 q^{68}-19 q^{66}+29 q^{64}-30 q^{62}+21 q^{60}-6 q^{58}-8 q^{56}+18 q^{54}-20 q^{52}+15 q^{50}-3 q^{48}-6 q^{46}+11 q^{44}-8 q^{42}+q^{40}+10 q^{38}-14 q^{36}+14 q^{34}-7 q^{32}-2 q^{30}+9 q^{28}-14 q^{26}+16 q^{24}-13 q^{22}+9 q^{20}-2 q^{18}-5 q^{16}+10 q^{14}-12 q^{12}+14 q^{10}-10 q^8+6 q^6-6 q^2+9-8 q^{-2} +7 q^{-4} -3 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} }

Further Quantum Invariants

Further quantum knot invariants for 10_7.

A1 Invariants.

Weight	Invariant
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^{19}-q^{17}+q^{15}-2 q^{13}+q^{11}-q^9+2 q^5-q^3+2 q- q^{-1} + q^{-3} }
2	$q^{54}-q^{52}-q^{50}+3q^{48}-2q^{46}-4q^{44}+5q^{42}+q^{40}-6q^{38}+4q^{36}+4q^{34}-6q^{32}+q^{30}+5q^{28}-4q^{26}-q^{24}+3q^{22}-4q^{18}-q^{16}+6q^{14}-4q^{12}-3q^{10}+7q^{8}-2q^{6}-3q^{4}+5q^{2}-1-q^{-2}+3q^{-4}-q^{-6}-q^{-8}+q^{-10}$
3	$q^{105}-q^{103}-q^{101}+q^{99}+2q^{97}-2q^{95}-5q^{93}+2q^{91}+8q^{89}-10q^{85}-3q^{83}+12q^{81}+9q^{79}-11q^{77}-13q^{75}+4q^{73}+16q^{71}+q^{69}-15q^{67}-8q^{65}+12q^{63}+14q^{61}-7q^{59}-17q^{57}+3q^{55}+17q^{53}-16q^{49}-3q^{47}+13q^{45}+4q^{43}-9q^{41}-7q^{39}+9q^{37}+8q^{35}-13q^{31}-4q^{29}+13q^{27}+9q^{25}-12q^{23}-13q^{21}+7q^{19}+13q^{17}-2q^{15}-11q^{13}-q^{11}+6q^{9}+4q^{7}-q^{5}-q^{3}-2q+q^{-1}+3q^{-3}+2q^{-5}-2q^{-7}-3q^{-9}+q^{-11}+3q^{-13}-q^{-17}-q^{-19}+q^{-21}$
4	$q^{172}-q^{170}-q^{168}+q^{166}+2q^{162}-4q^{160}-3q^{158}+4q^{156}+3q^{154}+8q^{152}-8q^{150}-13q^{148}+2q^{146}+8q^{144}+23q^{142}-4q^{140}-26q^{138}-16q^{136}-2q^{134}+39q^{132}+20q^{130}-15q^{128}-29q^{126}-32q^{124}+24q^{122}+34q^{120}+21q^{118}-5q^{116}-47q^{114}-18q^{112}+3q^{110}+36q^{108}+44q^{106}-16q^{104}-42q^{102}-48q^{100}+13q^{98}+72q^{96}+31q^{94}-28q^{92}-73q^{90}-22q^{88}+62q^{86}+55q^{84}-4q^{82}-66q^{80}-36q^{78}+38q^{76}+47q^{74}+7q^{72}-43q^{70}-32q^{68}+17q^{66}+36q^{64}+12q^{62}-22q^{60}-30q^{58}-10q^{56}+29q^{54}+30q^{52}+12q^{50}-34q^{48}-53q^{46}+8q^{44}+46q^{42}+58q^{40}-11q^{38}-82q^{36}-38q^{34}+27q^{32}+85q^{30}+34q^{28}-63q^{26}-62q^{24}-20q^{22}+62q^{20}+60q^{18}-14q^{16}-42q^{14}-46q^{12}+18q^{10}+45q^{8}+15q^{6}-6q^{4}-38q^{2}-8+18q^{-2}+15q^{-4}+11q^{-6}-18q^{-8}-10q^{-10}+q^{-12}+5q^{-14}+12q^{-16}-4q^{-18}-4q^{-20}-3q^{-22}-q^{-24}+5q^{-26}-q^{-32}-q^{-34}+q^{-36}$
5	$q^{255}-q^{253}-q^{251}+q^{249}-2q^{241}-2q^{239}+4q^{237}+6q^{235}+q^{233}-4q^{231}-9q^{229}-8q^{227}+4q^{225}+19q^{223}+17q^{221}-3q^{219}-23q^{217}-33q^{215}-14q^{213}+25q^{211}+52q^{209}+35q^{207}-14q^{205}-59q^{203}-65q^{201}-15q^{199}+54q^{197}+88q^{195}+53q^{193}-24q^{191}-85q^{189}-85q^{187}-22q^{185}+52q^{183}+90q^{181}+65q^{179}-56q^{175}-80q^{173}-64q^{171}-14q^{169}+60q^{167}+104q^{165}+94q^{163}+16q^{161}-107q^{159}-172q^{157}-106q^{155}+66q^{153}+210q^{151}+200q^{149}+16q^{147}-212q^{145}-274q^{143}-104q^{141}+171q^{139}+310q^{137}+184q^{135}-104q^{133}-311q^{131}-244q^{129}+40q^{127}+281q^{125}+263q^{123}+17q^{121}-230q^{119}-259q^{117}-54q^{115}+179q^{113}+228q^{111}+69q^{109}-131q^{107}-184q^{105}-70q^{103}+92q^{101}+152q^{99}+64q^{97}-71q^{95}-116q^{93}-63q^{91}+38q^{89}+108q^{87}+84q^{85}-18q^{83}-98q^{81}-113q^{79}-46q^{77}+93q^{75}+169q^{73}+110q^{71}-60q^{69}-216q^{67}-205q^{65}+242q^{61}+300q^{59}+89q^{57}-230q^{55}-371q^{53}-187q^{51}+171q^{49}+397q^{47}+276q^{45}-85q^{43}-369q^{41}-326q^{39}-9q^{37}+294q^{35}+329q^{33}+89q^{31}-199q^{29}-294q^{27}-132q^{25}+115q^{23}+225q^{21}+143q^{19}-40q^{17}-163q^{15}-131q^{13}+4q^{11}+105q^{9}+103q^{7}+23q^{5}-63q^{3}-80q-30q^{-1}+37q^{-3}+58q^{-5}+30q^{-7}-16q^{-9}-40q^{-11}-27q^{-13}+3q^{-15}+27q^{-17}+24q^{-19}+q^{-21}-13q^{-23}-16q^{-25}-7q^{-27}+5q^{-29}+12q^{-31}+5q^{-33}-2q^{-35}-3q^{-37}-5q^{-39}-q^{-41}+3q^{-43}+2q^{-45}-q^{-51}-q^{-53}+q^{-55}$

A2 Invariants.

Weight	Invariant
1,0	$q^{28}+q^{22}-2q^{20}-q^{18}-q^{14}+q^{12}+q^{8}+q^{6}-q^{4}+2q^{2}+q^{-4}$
1,1	$q^{76}-2q^{74}+4q^{72}-8q^{70}+17q^{68}-26q^{66}+36q^{64}-54q^{62}+67q^{60}-78q^{58}+82q^{56}-80q^{54}+68q^{52}-38q^{50}+10q^{48}+26q^{46}-65q^{44}+96q^{42}-122q^{40}+132q^{38}-135q^{36}+134q^{34}-112q^{32}+92q^{30}-66q^{28}+36q^{26}-14q^{24}-10q^{22}+22q^{20}-36q^{18}+40q^{16}-38q^{14}+39q^{12}-42q^{10}+42q^{8}-36q^{6}+38q^{4}-32q^{2}+30-20q^{-2}+15q^{-4}-10q^{-6}+6q^{-8}-2q^{-10}+q^{-12}$
2,0	$q^{72}+q^{64}-q^{62}-4q^{60}-q^{58}+2q^{56}+q^{54}-3q^{52}+5q^{48}+4q^{46}-3q^{44}-2q^{42}+3q^{40}+q^{38}-3q^{36}-2q^{34}+3q^{32}+q^{30}-2q^{28}+q^{26}-q^{24}-3q^{22}+2q^{20}+q^{18}-4q^{16}-2q^{14}+5q^{12}+2q^{10}-5q^{8}-q^{6}+7q^{4}+2q^{2}-2+q^{-2}+2q^{-4}-q^{-8}+q^{-12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{60}-q^{58}+q^{54}-3q^{52}+q^{48}-3q^{46}+4q^{44}+4q^{42}-3q^{40}+5q^{38}+3q^{36}-6q^{34}-q^{32}+q^{30}-5q^{28}-2q^{26}+q^{24}+q^{22}-2q^{20}-q^{18}+6q^{16}-3q^{14}-2q^{12}+7q^{10}-q^{8}-4q^{6}+5q^{4}+q^{2}-2+3q^{-2}+q^{-4}-q^{-6}+q^{-8}$
1,0,0	$q^{37}+q^{33}+q^{29}-2q^{27}-q^{25}-2q^{23}-q^{19}+q^{17}+q^{15}+q^{11}+q^{7}-q^{5}+2q^{3}+q^{-1}+q^{-5}$

B2 Invariants.

Weight	Invariant
0,1	$q^{60}-q^{58}+2q^{56}-3q^{54}+5q^{52}-6q^{50}+7q^{48}-7q^{46}+6q^{44}-6q^{42}+3q^{40}-q^{38}-3q^{36}+6q^{34}-9q^{32}+11q^{30}-13q^{28}+14q^{26}-13q^{24}+11q^{22}-8q^{20}+5q^{18}-2q^{16}-q^{14}+4q^{12}-5q^{10}+7q^{8}-6q^{6}+7q^{4}-5q^{2}+4-3q^{-2}+3q^{-4}-q^{-6}+q^{-8}$
1,0	$q^{98}-q^{94}-q^{92}+q^{90}+2q^{88}-q^{86}-4q^{84}-q^{82}+4q^{80}+3q^{78}-4q^{76}-6q^{74}+2q^{72}+8q^{70}+4q^{68}-6q^{66}-5q^{64}+4q^{62}+8q^{60}+q^{58}-6q^{56}-3q^{54}+3q^{52}+2q^{50}-3q^{48}-4q^{46}+q^{44}+3q^{42}-2q^{40}-5q^{38}+5q^{34}+q^{32}-5q^{30}-3q^{28}+5q^{26}+5q^{24}-3q^{22}-6q^{20}+q^{18}+7q^{16}+4q^{14}-4q^{12}-5q^{10}+6q^{6}+3q^{4}-2q^{2}-3+3q^{-4}+2q^{-6}-q^{-8}-q^{-10}+q^{-14}$

G2 Invariants.

Weight

Invariant

1,0

$q^{142}-q^{140}+2q^{138}-3q^{136}+2q^{134}-3q^{132}-q^{130}+6q^{128}-10q^{126}+12q^{124}-12q^{122}+8q^{120}+2q^{118}-12q^{116}+22q^{114}-24q^{112}+21q^{110}-8q^{108}-7q^{106}+20q^{104}-23q^{102}+22q^{100}-9q^{98}-4q^{96}+14q^{94}-16q^{92}+8q^{90}+2q^{88}-14q^{86}+19q^{84}-16q^{82}+q^{80}+10q^{78}-23q^{76}+27q^{74}-25q^{72}+11q^{70}+2q^{68}-19q^{66}+29q^{64}-30q^{62}+21q^{60}-6q^{58}-8q^{56}+18q^{54}-20q^{52}+15q^{50}-3q^{48}-6q^{46}+11q^{44}-8q^{42}+q^{40}+10q^{38}-14q^{36}+14q^{34}-7q^{32}-2q^{30}+9q^{28}-14q^{26}+16q^{24}-13q^{22}+9q^{20}-2q^{18}-5q^{16}+10q^{14}-12q^{12}+14q^{10}-10q^{8}+6q^{6}-6q^{2}+9-8q^{-2}+7q^{-4}-3q^{-6}+q^{-8}+2q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

.

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["10 7"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{2}+11t-15+11t^{-1}-3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-3z^{4}-z^{2}+1$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 43, -2 }

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

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 z^2 a^8+a^8-z^4 a^6-2 z^2 a^6-2 a^6-z^4 a^4+a^4-z^4 a^2-z^2 a^2+z^2+1}

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 z^6 a^{10}-4 z^4 a^{10}+3 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+8 z^3 a^9-3 z a^9+2 z^8 a^8-7 z^6 a^8+6 z^4 a^8-3 z^2 a^8+a^8+z^9 a^7-z^7 a^7-6 z^5 a^7+10 z^3 a^7-5 z a^7+4 z^8 a^6-15 z^6 a^6+20 z^4 a^6-10 z^2 a^6+2 a^6+z^9 a^5-z^7 a^5-2 z^5 a^5+6 z^3 a^5-2 z a^5+2 z^8 a^4-5 z^6 a^4+8 z^4 a^4-4 z^2 a^4+a^4+2 z^7 a^3-2 z^5 a^3+z^3 a^3+2 z^6 a^2-z^4 a^2-2 z^2 a^2+2 z^5 a-3 z^3 a+z^4-2 z^2+1}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a59, K11n3,}

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["10 7"];

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]=

{ $-3t^{2}+11t-15+11t^{-1}-3t^{-2}$, $q-2+4q^{-1}-5q^{-2}+7q^{-3}-7q^{-4}+6q^{-5}-5q^{-6}+3q^{-7}-2q^{-8}+q^{-9}$ }

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]=

{K11a59, K11n3,}

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:

(-1, 3)

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 $-4$ $24$ $8$ $-{\frac {110}{3}}$ ${\frac {86}{3}}$ $-96$ $-176$ $-64$ $-168$ $-{\frac {32}{3}}$ $288$ ${\frac {440}{3}}$ $-{\frac {344}{3}}$ ${\frac {36929}{30}}$ $-{\frac {5498}{15}}$ ${\frac {47578}{45}}$ ${\frac {3967}{18}}$ ${\frac {2849}{30}}$

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 $s=$-2 is the signature of 10 7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                     1 1 1                   1   -1 -1                 3 1   2 -3               3 2     -1 -5             4 2       2 -7           3 3         0 -9         3 4           -1 -11       2 3             1 -13     1 3               -2 -15   1 2                 1 -17   1                   -1 -19 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ 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=-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}\oplus{\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}^{3}\oplus{\mathbb Z}_2^{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 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}^{3}\oplus{\mathbb Z}_2^{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}^{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 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}^{4}\oplus{\mathbb Z}_2^{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}^{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 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}^{3}\oplus{\mathbb Z}_2^{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}^{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 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 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{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}^{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 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}^{2}\oplus{\mathbb Z}_2^{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}^{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 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 {\mathbb Z}\oplus{\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=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}}

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^4-2 q^3+5 q-6+11 q^{-2} -14 q^{-3} + q^{-4} +20 q^{-5} -24 q^{-6} +30 q^{-8} -31 q^{-9} -3 q^{-10} +34 q^{-11} -28 q^{-12} -7 q^{-13} +31 q^{-14} -19 q^{-15} -11 q^{-16} +24 q^{-17} -9 q^{-18} -11 q^{-19} +14 q^{-20} -2 q^{-21} -7 q^{-22} +5 q^{-23} -2 q^{-25} + q^{-26} }
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^9-2 q^8+q^6+4 q^5-4 q^4-4 q^3+2 q^2+8 q-3-6 q^{-1} - q^{-2} +9 q^{-3} -3 q^{-4} - q^{-5} + q^{-6} +2 q^{-7} -13 q^{-8} +8 q^{-9} +16 q^{-10} -4 q^{-11} -33 q^{-12} +9 q^{-13} +37 q^{-14} -50 q^{-16} +50 q^{-18} +8 q^{-19} -49 q^{-20} -16 q^{-21} +48 q^{-22} +21 q^{-23} -40 q^{-24} -32 q^{-25} +35 q^{-26} +37 q^{-27} -23 q^{-28} -46 q^{-29} +15 q^{-30} +47 q^{-31} -2 q^{-32} -48 q^{-33} -5 q^{-34} +40 q^{-35} +14 q^{-36} -33 q^{-37} -17 q^{-38} +23 q^{-39} +16 q^{-40} -13 q^{-41} -14 q^{-42} +8 q^{-43} +9 q^{-44} -3 q^{-45} -6 q^{-46} +2 q^{-47} +2 q^{-48} -2 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^{16}-2 q^{15}+q^{13}+6 q^{11}-8 q^{10}-2 q^9+22 q^6-15 q^5-6 q^4-11 q^3-8 q^2+51 q-11-3 q^{-1} -37 q^{-2} -38 q^{-3} +83 q^{-4} +10 q^{-5} +27 q^{-6} -64 q^{-7} -102 q^{-8} +87 q^{-9} +38 q^{-10} +101 q^{-11} -62 q^{-12} -184 q^{-13} +45 q^{-14} +37 q^{-15} +198 q^{-16} -11 q^{-17} -242 q^{-18} -20 q^{-19} -7 q^{-20} +269 q^{-21} +58 q^{-22} -254 q^{-23} -58 q^{-24} -68 q^{-25} +288 q^{-26} +104 q^{-27} -236 q^{-28} -59 q^{-29} -107 q^{-30} +268 q^{-31} +112 q^{-32} -202 q^{-33} -35 q^{-34} -126 q^{-35} +219 q^{-36} +101 q^{-37} -152 q^{-38} +5 q^{-39} -135 q^{-40} +145 q^{-41} +71 q^{-42} -90 q^{-43} +64 q^{-44} -128 q^{-45} +61 q^{-46} +20 q^{-47} -45 q^{-48} +123 q^{-49} -87 q^{-50} +2 q^{-51} -41 q^{-52} -39 q^{-53} +149 q^{-54} -27 q^{-55} -6 q^{-56} -74 q^{-57} -60 q^{-58} +120 q^{-59} +15 q^{-60} +20 q^{-61} -61 q^{-62} -70 q^{-63} +64 q^{-64} +18 q^{-65} +34 q^{-66} -26 q^{-67} -51 q^{-68} +23 q^{-69} +4 q^{-70} +24 q^{-71} -4 q^{-72} -24 q^{-73} +8 q^{-74} -2 q^{-75} +9 q^{-76} + q^{-77} -8 q^{-78} +3 q^{-79} - q^{-80} +2 q^{-81} -2 q^{-83} + q^{-84} }

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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