10 126

10_125

10_127

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 126's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 126 at Knotilus!
	10_126 is also known as the pretzel knot P(-5,3,2).

Knot presentations

Planar diagram presentation	X4251 X8493 X5,14,6,15 X15,20,16,1 X9,16,10,17 X11,18,12,19 X17,10,18,11 X19,12,20,13 X13,6,14,7 X2837
Gauss code	1, -10, 2, -1, -3, 9, 10, -2, -5, 7, -6, 8, -9, 3, -4, 5, -7, 6, -8, 4
Dowker-Thistlethwaite code	4 8 -14 2 -16 -18 -6 -20 -10 -12
Conway Notation	[41,3,2-]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 10, width is 3, Braid index is 3		[{3, 10}, {2, 4}, {1, 3}, {13, 11}, {10, 12}, {11, 5}, {4, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 13}, {12, 2}, {9, 1}]

[edit Notes on presentations of 10 126]

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["10 126"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X8493 X5,14,6,15 X15,20,16,1 X9,16,10,17 X11,18,12,19 X17,10,18,11 X19,12,20,13 X13,6,14,7 X2837

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 -14 2 -16 -18 -6 -20 -10 -12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[41,3,2-]

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]=

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

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, 10, 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[{3, 10}, {2, 4}, {1, 3}, {13, 11}, {10, 12}, {11, 5}, {4, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 13}, {12, 2}, {9, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-10][0] Hyperbolic Volume 6.90426 A-Polynomial See Data:10 126/A-polynomial

[edit Notes for 10 126's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for 10 126's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-2t^{2}+4t-5+4t^{-1}-2t^{-2}+t^{-3}$
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+4 z^4+5 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	{ 19, -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 -1+2 q^{-1} -2 q^{-2} +4 q^{-3} -3 q^{-4} +3 q^{-5} -2 q^{-6} + q^{-7} - q^{-8} }
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^4 a^6-4 z^2 a^6-4 a^6+z^6 a^4+6 z^4 a^4+12 z^2 a^4+7 a^4-z^4 a^2-3 z^2 a^2-2 a^2}
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^5 a^9-4 z^3 a^9+3 z a^9+z^6 a^8-3 z^4 a^8+z^2 a^8+z^7 a^7-3 z^5 a^7+2 z^3 a^7-z a^7+z^8 a^6-5 z^6 a^6+11 z^4 a^6-11 z^2 a^6+4 a^6+2 z^7 a^5-9 z^5 a^5+16 z^3 a^5-8 z a^5+z^8 a^4-6 z^6 a^4+16 z^4 a^4-16 z^2 a^4+7 a^4+z^7 a^3-5 z^5 a^3+11 z^3 a^3-6 z a^3+2 z^4 a^2-4 z^2 a^2+2 a^2+z^3 a-2 z a}
The A2 invariant	$-q^{24}-q^{22}-2q^{20}-q^{18}+q^{16}+q^{14}+3q^{12}+2q^{10}+2q^{8}+q^{6}-q^{4}-1$
The G2 invariant	$q^{128}+q^{124}-q^{122}+q^{120}-q^{116}+2q^{114}-2q^{112}+2q^{110}-2q^{108}-3q^{102}+4q^{100}-5q^{98}+q^{96}-q^{94}-4q^{92}+3q^{90}-5q^{88}-q^{86}+q^{84}-5q^{82}+2q^{80}-2q^{78}-4q^{76}+6q^{74}-5q^{72}+3q^{70}-3q^{66}+6q^{64}-4q^{62}+5q^{60}+2q^{56}+4q^{54}-2q^{52}+7q^{50}-q^{48}+3q^{46}+3q^{44}-2q^{42}+5q^{40}+2q^{38}-2q^{36}+6q^{34}-3q^{32}+q^{30}+2q^{28}-5q^{26}+5q^{24}-4q^{22}+q^{20}-3q^{16}+2q^{14}-2q^{12}-q^{8}-q^{4}-q^{2}+q^{-4}$

Further Quantum Invariants

Further quantum knot invariants for 10_126.

A1 Invariants.

Weight	Invariant
1	$-q^{17}-q^{13}+q^{11}+q^{7}+2q^{5}+q-q^{-1}$
2	$q^{48}+q^{42}-q^{40}-2q^{38}+q^{36}-3q^{32}+q^{28}-2q^{26}+2q^{22}+q^{16}+3q^{14}-q^{12}+3q^{8}-q^{6}-q^{4}+2q^{2}-q^{-2}$
3	$-q^{93}+q^{83}+q^{81}-q^{77}+q^{75}+3q^{73}+q^{71}-3q^{69}-3q^{67}+2q^{65}+4q^{63}-6q^{59}-3q^{57}+3q^{55}+4q^{53}-4q^{51}-5q^{49}+q^{47}+4q^{45}-2q^{43}-3q^{41}+3q^{37}-q^{31}+q^{29}+4q^{27}-q^{25}-4q^{23}+7q^{19}+2q^{17}-4q^{15}-3q^{13}+5q^{11}+4q^{9}-q^{7}-3q^{5}+2q+2q^{-1}-q^{-3}-2q^{-5}-q^{-7}+q^{-11}$
5	$-q^{225}+q^{217}+q^{215}-q^{207}+2q^{203}+q^{201}-2q^{195}-3q^{193}-q^{191}+2q^{189}+3q^{187}+3q^{185}-5q^{181}-8q^{179}-5q^{177}+q^{175}+9q^{173}+11q^{171}+6q^{169}-5q^{167}-15q^{165}-16q^{163}-4q^{161}+12q^{159}+23q^{157}+20q^{155}+3q^{153}-20q^{151}-32q^{149}-21q^{147}+7q^{145}+33q^{143}+40q^{141}+15q^{139}-25q^{137}-47q^{135}-34q^{133}+6q^{131}+45q^{129}+50q^{127}+11q^{125}-35q^{123}-53q^{121}-27q^{119}+22q^{117}+49q^{115}+32q^{113}-8q^{111}-38q^{109}-31q^{107}+2q^{105}+26q^{103}+22q^{101}+4q^{99}-16q^{97}-16q^{95}-2q^{93}+8q^{91}+6q^{89}+q^{87}-4q^{85}-6q^{83}-q^{81}+3q^{79}+q^{75}-3q^{73}-8q^{71}-8q^{69}+q^{67}+14q^{65}+14q^{63}+3q^{61}-16q^{59}-28q^{57}-12q^{55}+22q^{53}+40q^{51}+26q^{49}-14q^{47}-48q^{45}-41q^{43}+7q^{41}+49q^{39}+52q^{37}+12q^{35}-39q^{33}-55q^{31}-26q^{29}+20q^{27}+49q^{25}+40q^{23}+2q^{21}-31q^{19}-36q^{17}-19q^{15}+11q^{13}+29q^{11}+24q^{9}+6q^{7}-12q^{5}-20q^{3}-15q-q^{-1}+11q^{-3}+12q^{-5}+7q^{-7}-q^{-9}-7q^{-11}-8q^{-13}-3q^{-15}+2q^{-17}+3q^{-19}+3q^{-21}+q^{-23}-q^{-25}-q^{-27}$
6	$q^{312}-q^{304}-q^{302}-q^{300}+q^{298}+q^{292}-q^{288}-2q^{286}+q^{284}+q^{282}+2q^{278}+q^{276}-3q^{272}-q^{270}+4q^{264}+5q^{262}+4q^{260}-3q^{258}-4q^{256}-6q^{254}-8q^{252}-2q^{250}+6q^{248}+14q^{246}+10q^{244}+7q^{242}-4q^{240}-18q^{238}-24q^{236}-18q^{234}+14q^{230}+34q^{228}+34q^{226}+15q^{224}-14q^{222}-39q^{220}-48q^{218}-42q^{216}+q^{214}+43q^{212}+70q^{210}+64q^{208}+25q^{206}-36q^{204}-93q^{202}-93q^{200}-49q^{198}+30q^{196}+100q^{194}+127q^{192}+79q^{190}-25q^{188}-113q^{186}-146q^{184}-96q^{182}+11q^{180}+127q^{178}+166q^{176}+100q^{174}-21q^{172}-133q^{170}-165q^{168}-98q^{166}+39q^{164}+142q^{162}+148q^{160}+65q^{158}-53q^{156}-130q^{154}-120q^{152}-26q^{150}+69q^{148}+105q^{146}+72q^{144}-60q^{140}-73q^{138}-29q^{136}+20q^{134}+43q^{132}+34q^{130}+8q^{128}-15q^{126}-25q^{124}-11q^{122}+3q^{120}+8q^{118}+5q^{116}-q^{114}-5q^{112}-6q^{110}-2q^{108}+3q^{106}+7q^{104}+4q^{102}-q^{100}-9q^{98}-16q^{96}-20q^{94}-7q^{92}+24q^{90}+34q^{88}+31q^{86}+2q^{84}-39q^{82}-70q^{80}-52q^{78}+19q^{76}+79q^{74}+104q^{72}+57q^{70}-38q^{68}-128q^{66}-135q^{64}-42q^{62}+77q^{60}+161q^{58}+142q^{56}+32q^{54}-110q^{52}-179q^{50}-131q^{48}-12q^{46}+117q^{44}+171q^{42}+125q^{40}+2q^{38}-108q^{36}-143q^{34}-105q^{32}-10q^{30}+83q^{28}+123q^{26}+88q^{24}+18q^{22}-47q^{20}-87q^{18}-77q^{16}-27q^{14}+27q^{12}+54q^{10}+55q^{8}+35q^{6}-3q^{4}-33q^{2}-39-27q^{-2}-9q^{-4}+9q^{-6}+24q^{-8}+23q^{-10}+11q^{-12}-q^{-14}-10q^{-16}-13q^{-18}-12q^{-20}-3q^{-22}+3q^{-24}+5q^{-26}+5q^{-28}+4q^{-30}+2q^{-32}-2q^{-34}-q^{-36}-q^{-38}-q^{-40}-q^{-42}+q^{-46}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{24}-q^{22}-2q^{20}-q^{18}+q^{16}+q^{14}+3q^{12}+2q^{10}+2q^{8}+q^{6}-q^{4}-1$
1,1	$q^{68}+2q^{64}-2q^{62}+4q^{60}-4q^{58}+4q^{56}-8q^{54}+7q^{52}-8q^{50}+8q^{48}-6q^{46}+q^{44}+2q^{42}-10q^{40}+10q^{38}-18q^{36}+14q^{34}-18q^{32}+12q^{30}-13q^{28}+10q^{26}-2q^{24}+6q^{22}+10q^{20}-2q^{18}+16q^{16}-6q^{14}+10q^{12}-8q^{10}+2q^{8}-4q^{6}-q^{4}-2q^{2}+q^{-4}$
2,0	$q^{62}+q^{60}+2q^{58}+2q^{56}+2q^{54}-q^{50}-3q^{48}-4q^{46}-7q^{44}-6q^{42}-3q^{40}-2q^{38}+q^{34}+4q^{32}+3q^{30}+4q^{28}+4q^{26}+5q^{24}+2q^{22}+3q^{20}+q^{18}-q^{16}-q^{14}-q^{8}+q^{4}-1$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{54}+q^{50}+q^{48}-3q^{40}-q^{38}-2q^{36}-6q^{34}-4q^{32}-4q^{30}-3q^{28}+5q^{24}+7q^{22}+9q^{20}+7q^{18}+7q^{16}+q^{14}-2q^{12}-q^{10}-4q^{8}-4q^{6}+q^{-2}$
1,0,0	$-q^{31}-q^{29}-3q^{27}-2q^{25}-2q^{23}+q^{21}+2q^{19}+4q^{17}+4q^{15}+3q^{13}+2q^{11}-2q^{5}-q$
1,0,1	$q^{88}+2q^{84}+q^{80}+q^{78}-2q^{76}+q^{74}-2q^{72}-3q^{70}+q^{68}-4q^{66}+5q^{64}+q^{62}+3q^{60}+9q^{58}-4q^{56}+11q^{54}-9q^{52}-3q^{50}-12q^{48}-18q^{46}-14q^{44}-20q^{42}-7q^{40}-11q^{38}+12q^{36}+3q^{34}+24q^{32}+17q^{30}+19q^{28}+24q^{26}+3q^{24}+15q^{22}-4q^{20}-2q^{18}-5q^{16}-8q^{14}-5q^{12}-3q^{10}-3q^{8}-q^{6}-q^{2}+2+q^{-4}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{68}+q^{66}+2q^{64}+3q^{62}+3q^{60}+2q^{58}+3q^{56}-4q^{52}-6q^{50}-8q^{48}-12q^{46}-15q^{44}-12q^{42}-9q^{40}-6q^{38}+q^{36}+11q^{34}+14q^{32}+18q^{30}+21q^{28}+17q^{26}+10q^{24}+5q^{22}-q^{20}-7q^{18}-10q^{16}-7q^{14}-5q^{12}-4q^{10}-q^{8}+2q^{6}+q^{4}+q^{2}+1$
1,0,0,0	$-q^{38}-q^{36}-3q^{34}-3q^{32}-3q^{30}-2q^{28}+q^{26}+2q^{24}+5q^{22}+5q^{20}+5q^{18}+3q^{16}+2q^{14}-q^{10}-q^{8}-2q^{6}-q^{2}$

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{74}+q^{70}+2q^{66}-q^{64}+q^{62}-2q^{60}+q^{58}-2q^{56}-2q^{52}-2q^{50}-3q^{48}-6q^{46}-4q^{44}-7q^{42}-3q^{40}-6q^{38}+3q^{36}+q^{34}+10q^{32}+7q^{30}+12q^{28}+8q^{26}+9q^{24}+5q^{22}+q^{20}-q^{18}-4q^{16}-2q^{14}-5q^{12}-2q^{10}-4q^{8}+q^{6}-q^{4}+q^{2}+q^{-2}$

G2 Invariants.

Weight

Invariant

1,0

$q^{128}+q^{124}-q^{122}+q^{120}-q^{116}+2q^{114}-2q^{112}+2q^{110}-2q^{108}-3q^{102}+4q^{100}-5q^{98}+q^{96}-q^{94}-4q^{92}+3q^{90}-5q^{88}-q^{86}+q^{84}-5q^{82}+2q^{80}-2q^{78}-4q^{76}+6q^{74}-5q^{72}+3q^{70}-3q^{66}+6q^{64}-4q^{62}+5q^{60}+2q^{56}+4q^{54}-2q^{52}+7q^{50}-q^{48}+3q^{46}+3q^{44}-2q^{42}+5q^{40}+2q^{38}-2q^{36}+6q^{34}-3q^{32}+q^{30}+2q^{28}-5q^{26}+5q^{24}-4q^{22}+q^{20}-3q^{16}+2q^{14}-2q^{12}-q^{8}-q^{4}-q^{2}+q^{-4}$

.

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["10 126"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-2t^{2}+4t-5+4t^{-1}-2t^{-2}+t^{-3}$

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+4 z^4+5 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]=

{ 19, -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 -1+2 q^{-1} -2 q^{-2} +4 q^{-3} -3 q^{-4} +3 q^{-5} -2 q^{-6} + q^{-7} - q^{-8} }

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^4 a^6-4 z^2 a^6-4 a^6+z^6 a^4+6 z^4 a^4+12 z^2 a^4+7 a^4-z^4 a^2-3 z^2 a^2-2 a^2}

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^5 a^9-4 z^3 a^9+3 z a^9+z^6 a^8-3 z^4 a^8+z^2 a^8+z^7 a^7-3 z^5 a^7+2 z^3 a^7-z a^7+z^8 a^6-5 z^6 a^6+11 z^4 a^6-11 z^2 a^6+4 a^6+2 z^7 a^5-9 z^5 a^5+16 z^3 a^5-8 z a^5+z^8 a^4-6 z^6 a^4+16 z^4 a^4-16 z^2 a^4+7 a^4+z^7 a^3-5 z^5 a^3+11 z^3 a^3-6 z a^3+2 z^4 a^2-4 z^2 a^2+2 a^2+z^3 a-2 z a}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["10 126"];

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]=

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

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:

(5, -9)

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 $20$ $-72$ $200$ ${\frac {1222}{3}}$ ${\frac {194}{3}}$ $-1440$ $-2416$ $-384$ $-392$ ${\frac {4000}{3}}$ $2592$ ${\frac {24440}{3}}$ ${\frac {3880}{3}}$ ${\frac {87967}{6}}$ $-{\frac {310}{3}}$ ${\frac {56342}{9}}$ ${\frac {3109}{18}}$ ${\frac {5119}{6}}$

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 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=} -2 is the signature of 10 126. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 χ 1                 1 -1 -1               1   1 -3             2 2   0 -5           2       2 -7         1 2       1 -9       2 2         0 -11       1           1 -13   1 2             -1 -15                   0 -17 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=-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 i=-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=-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}^{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}\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=-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}^{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=-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}\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=-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^{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=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}} 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}} $r=1$ 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} 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 -1+ q^{-1} +2 q^{-2} -4 q^{-3} + q^{-4} +6 q^{-5} -7 q^{-6} +10 q^{-8} -9 q^{-9} - q^{-10} +10 q^{-11} -7 q^{-12} -3 q^{-13} +8 q^{-14} -4 q^{-15} -4 q^{-16} +5 q^{-17} - q^{-18} -3 q^{-19} +2 q^{-20} - q^{-22} + q^{-23} }
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^4-q^3-q^2-q+2+2 q^{-1} - q^{-2} -3 q^{-3} - q^{-4} +4 q^{-5} +4 q^{-6} -2 q^{-7} -9 q^{-8} +3 q^{-9} +10 q^{-10} +3 q^{-11} -16 q^{-12} - q^{-13} +13 q^{-14} +8 q^{-15} -19 q^{-16} -3 q^{-17} +14 q^{-18} +8 q^{-19} -16 q^{-20} -6 q^{-21} +11 q^{-22} +9 q^{-23} -10 q^{-24} -9 q^{-25} +5 q^{-26} +10 q^{-27} -2 q^{-28} -10 q^{-29} - q^{-30} +7 q^{-31} +4 q^{-32} -6 q^{-33} -3 q^{-34} +2 q^{-35} +4 q^{-36} -2 q^{-37} - q^{-38} +2 q^{-40} - q^{-41} + q^{-44} - q^{-45} }
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^8+q^7+2 q^6-q^4-5 q^3-q^2+5 q+4+4 q^{-1} -8 q^{-2} -10 q^{-3} +3 q^{-4} +4 q^{-5} +15 q^{-6} + q^{-7} -15 q^{-8} -7 q^{-9} -10 q^{-10} +22 q^{-11} +19 q^{-12} -7 q^{-13} -13 q^{-14} -33 q^{-15} +18 q^{-16} +34 q^{-17} +6 q^{-18} -10 q^{-19} -51 q^{-20} +11 q^{-21} +39 q^{-22} +13 q^{-23} -3 q^{-24} -60 q^{-25} +8 q^{-26} +40 q^{-27} +14 q^{-28} - q^{-29} -59 q^{-30} +6 q^{-31} +36 q^{-32} +14 q^{-33} +5 q^{-34} -54 q^{-35} +26 q^{-37} +14 q^{-38} +16 q^{-39} -41 q^{-40} -8 q^{-41} +8 q^{-42} +9 q^{-43} +27 q^{-44} -21 q^{-45} -9 q^{-46} -7 q^{-47} -3 q^{-48} +26 q^{-49} -4 q^{-50} -10 q^{-52} -11 q^{-53} +14 q^{-54} +7 q^{-56} -3 q^{-57} -9 q^{-58} +5 q^{-59} -3 q^{-60} +5 q^{-61} + q^{-62} -4 q^{-63} +3 q^{-64} -3 q^{-65} + q^{-66} + q^{-67} -2 q^{-68} +2 q^{-69} - q^{-70} - q^{-73} + q^{-74} }
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^{11}+2 q^9+2 q^8-q^6-6 q^5-5 q^4+3 q^3+8 q^2+8 q+4-7 q^{-1} -17 q^{-2} -11 q^{-3} +3 q^{-4} +16 q^{-5} +22 q^{-6} +11 q^{-7} -12 q^{-8} -29 q^{-9} -27 q^{-10} - q^{-11} +27 q^{-12} +44 q^{-13} +26 q^{-14} -20 q^{-15} -56 q^{-16} -47 q^{-17} -2 q^{-18} +60 q^{-19} +77 q^{-20} +20 q^{-21} -59 q^{-22} -89 q^{-23} -50 q^{-24} +53 q^{-25} +111 q^{-26} +60 q^{-27} -45 q^{-28} -107 q^{-29} -84 q^{-30} +37 q^{-31} +123 q^{-32} +79 q^{-33} -34 q^{-34} -107 q^{-35} -97 q^{-36} +28 q^{-37} +123 q^{-38} +84 q^{-39} -30 q^{-40} -108 q^{-41} -94 q^{-42} +24 q^{-43} +118 q^{-44} +86 q^{-45} -25 q^{-46} -103 q^{-47} -92 q^{-48} +14 q^{-49} +104 q^{-50} +86 q^{-51} -5 q^{-52} -85 q^{-53} -88 q^{-54} -10 q^{-55} +71 q^{-56} +79 q^{-57} +25 q^{-58} -45 q^{-59} -71 q^{-60} -37 q^{-61} +22 q^{-62} +53 q^{-63} +43 q^{-64} + q^{-65} -32 q^{-66} -42 q^{-67} -17 q^{-68} +13 q^{-69} +30 q^{-70} +23 q^{-71} +8 q^{-72} -17 q^{-73} -24 q^{-74} -13 q^{-75} +2 q^{-76} +12 q^{-77} +20 q^{-78} +6 q^{-79} -7 q^{-80} -10 q^{-81} -9 q^{-82} -4 q^{-83} +8 q^{-84} +7 q^{-85} +3 q^{-86} + q^{-87} -4 q^{-88} -6 q^{-89} + q^{-91} +4 q^{-93} + q^{-94} -3 q^{-95} -3 q^{-98} +2 q^{-99} +2 q^{-100} - q^{-101} + q^{-103} -2 q^{-104} + q^{-106} + q^{-109} - q^{-110} }
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^{20}-q^{19}-q^{18}-q^{14}+5 q^{13}+q^{12}-2 q^9-6 q^8-10 q^7+4 q^6+4 q^5+9 q^4+12 q^3+10 q^2-5 q-25-14 q^{-1} -14 q^{-2} -3 q^{-3} +18 q^{-4} +40 q^{-5} +33 q^{-6} -5 q^{-7} -15 q^{-8} -41 q^{-9} -57 q^{-10} -32 q^{-11} +30 q^{-12} +73 q^{-13} +60 q^{-14} +55 q^{-15} -6 q^{-16} -97 q^{-17} -125 q^{-18} -65 q^{-19} +35 q^{-20} +95 q^{-21} +165 q^{-22} +117 q^{-23} -51 q^{-24} -179 q^{-25} -194 q^{-26} -84 q^{-27} +47 q^{-28} +234 q^{-29} +259 q^{-30} +59 q^{-31} -160 q^{-32} -278 q^{-33} -203 q^{-34} -46 q^{-35} +241 q^{-36} +349 q^{-37} +154 q^{-38} -113 q^{-39} -303 q^{-40} -263 q^{-41} -117 q^{-42} +223 q^{-43} +380 q^{-44} +195 q^{-45} -84 q^{-46} -300 q^{-47} -273 q^{-48} -148 q^{-49} +210 q^{-50} +384 q^{-51} +202 q^{-52} -76 q^{-53} -295 q^{-54} -270 q^{-55} -152 q^{-56} +205 q^{-57} +380 q^{-58} +203 q^{-59} -72 q^{-60} -289 q^{-61} -267 q^{-62} -157 q^{-63} +191 q^{-64} +366 q^{-65} +213 q^{-66} -49 q^{-67} -263 q^{-68} -258 q^{-69} -180 q^{-70} +142 q^{-71} +322 q^{-72} +226 q^{-73} +11 q^{-74} -191 q^{-75} -225 q^{-76} -216 q^{-77} +47 q^{-78} +228 q^{-79} +216 q^{-80} +88 q^{-81} -73 q^{-82} -142 q^{-83} -222 q^{-84} -56 q^{-85} +91 q^{-86} +149 q^{-87} +120 q^{-88} +39 q^{-89} -21 q^{-90} -156 q^{-91} -95 q^{-92} -25 q^{-93} +42 q^{-94} +70 q^{-95} +72 q^{-96} +67 q^{-97} -52 q^{-98} -47 q^{-99} -52 q^{-100} -28 q^{-101} -9 q^{-102} +28 q^{-103} +67 q^{-104} +5 q^{-105} +14 q^{-106} -13 q^{-107} -22 q^{-108} -36 q^{-109} -14 q^{-110} +24 q^{-111} - q^{-112} +23 q^{-113} +12 q^{-114} +7 q^{-115} -17 q^{-116} -14 q^{-117} +4 q^{-118} -15 q^{-119} +5 q^{-120} +6 q^{-121} +13 q^{-122} -3 q^{-123} -3 q^{-124} +7 q^{-125} -11 q^{-126} -3 q^{-127} -2 q^{-128} +7 q^{-129} - q^{-130} - q^{-131} +8 q^{-132} -4 q^{-133} -2 q^{-134} -3 q^{-135} +3 q^{-136} - q^{-137} -2 q^{-138} +6 q^{-139} - q^{-140} - q^{-141} -2 q^{-142} + q^{-143} -2 q^{-145} +3 q^{-146} - q^{-149} - q^{-152} + q^{-153} }

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.

10_125 10_127