10 126

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

Further Quantum Invariants

Further quantum knot invariants for 10_126.

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^{17}-q^{13}+q^{11}+q^7+2 q^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	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^{68}+2 q^{64}-2 q^{62}+4 q^{60}-4 q^{58}+4 q^{56}-8 q^{54}+7 q^{52}-8 q^{50}+8 q^{48}-6 q^{46}+q^{44}+2 q^{42}-10 q^{40}+10 q^{38}-18 q^{36}+14 q^{34}-18 q^{32}+12 q^{30}-13 q^{28}+10 q^{26}-2 q^{24}+6 q^{22}+10 q^{20}-2 q^{18}+16 q^{16}-6 q^{14}+10 q^{12}-8 q^{10}+2 q^8-4 q^6-q^4-2 q^2+ q^{-4} }
2,0	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^{62}+q^{60}+2 q^{58}+2 q^{56}+2 q^{54}-q^{50}-3 q^{48}-4 q^{46}-7 q^{44}-6 q^{42}-3 q^{40}-2 q^{38}+q^{34}+4 q^{32}+3 q^{30}+4 q^{28}+4 q^{26}+5 q^{24}+2 q^{22}+3 q^{20}+q^{18}-q^{16}-q^{14}-q^8+q^4-1}

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

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

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

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