9 25

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 -3 t^2+12 t-17+12 t^{-1} -3 t^{-2} }
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 1-3 z^4}
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	{ 47, -2 }
Jones polynomial	$q-2+5q^{-1}-7q^{-2}+8q^{-3}-8q^{-4}+7q^{-5}-5q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{8}+3z^{2}a^{6}+3a^{6}-2z^{4}a^{4}-4z^{2}a^{4}-3a^{4}-z^{4}a^{2}+a^{2}+z^{2}+1$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-7z^{4}a^{8}+4z^{2}a^{8}-a^{8}+3z^{7}a^{7}-4z^{5}a^{7}-2z^{3}a^{7}+za^{7}+z^{8}a^{6}+6z^{6}a^{6}-18z^{4}a^{6}+13z^{2}a^{6}-3a^{6}+6z^{7}a^{5}-10z^{5}a^{5}+5z^{3}a^{5}-za^{5}+z^{8}a^{4}+6z^{6}a^{4}-15z^{4}a^{4}+13z^{2}a^{4}-3a^{4}+3z^{7}a^{3}-3z^{5}a^{3}+3z^{3}a^{3}-za^{3}+3z^{6}a^{2}-3z^{4}a^{2}+2z^{2}a^{2}-a^{2}+2z^{5}a-2z^{3}a+z^{4}-2z^{2}+1$
The A2 invariant	$-q^{26}-q^{24}+2q^{22}+q^{18}+2q^{16}-2q^{14}-2q^{10}+q^{6}-q^{4}+3q^{2}+q^{-4}$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+7q^{120}-4q^{118}-6q^{116}+19q^{114}-29q^{112}+34q^{110}-28q^{108}+4q^{106}+23q^{104}-50q^{102}+63q^{100}-55q^{98}+26q^{96}+12q^{94}-46q^{92}+60q^{90}-48q^{88}+22q^{86}+15q^{84}-38q^{82}+41q^{80}-18q^{78}-14q^{76}+48q^{74}-60q^{72}+50q^{70}-13q^{68}-30q^{66}+68q^{64}-87q^{62}+79q^{60}-45q^{58}-6q^{56}+48q^{54}-78q^{52}+76q^{50}-50q^{48}+8q^{46}+26q^{44}-46q^{42}+38q^{40}-14q^{38}-20q^{36}+43q^{34}-43q^{32}+21q^{30}+13q^{28}-44q^{26}+63q^{24}-54q^{22}+31q^{20}-q^{18}-27q^{16}+43q^{14}-43q^{12}+34q^{10}-14q^{8}+11q^{4}-16q^{2}+15-11q^{-2}+8q^{-4}-2q^{-6}-q^{-8}+3q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 9_25.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+2q^{15}-2q^{13}+2q^{11}-q^{9}+q^{5}-2q^{3}+3q-q^{-1}+q^{-3}$
2	$q^{48}-2q^{46}-2q^{44}+7q^{42}-2q^{40}-9q^{38}+11q^{36}+3q^{34}-15q^{32}+7q^{30}+8q^{28}-12q^{26}+q^{24}+8q^{22}-3q^{20}-6q^{18}+3q^{16}+9q^{14}-10q^{12}-3q^{10}+16q^{8}-8q^{6}-8q^{4}+12q^{2}-2-5q^{-2}+4q^{-4}-q^{-8}+q^{-10}$
3	$-q^{93}+2q^{91}+2q^{89}-3q^{87}-7q^{85}+2q^{83}+16q^{81}+q^{79}-23q^{77}-12q^{75}+29q^{73}+29q^{71}-29q^{69}-46q^{67}+19q^{65}+59q^{63}-2q^{61}-68q^{59}-15q^{57}+66q^{55}+30q^{53}-55q^{51}-40q^{49}+44q^{47}+44q^{45}-29q^{43}-44q^{41}+11q^{39}+40q^{37}+7q^{35}-36q^{33}-28q^{31}+29q^{29}+46q^{27}-17q^{25}-61q^{23}+3q^{21}+70q^{19}+14q^{17}-69q^{15}-25q^{13}+55q^{11}+36q^{9}-39q^{7}-35q^{5}+24q^{3}+27q-8q^{-1}-18q^{-3}+3q^{-5}+9q^{-7}-q^{-9}-4q^{-11}+q^{-13}+q^{-15}-q^{-19}+q^{-21}$
4	$q^{152}-2q^{150}-2q^{148}+3q^{146}+3q^{144}+7q^{142}-9q^{140}-16q^{138}-q^{136}+11q^{134}+41q^{132}-q^{130}-47q^{128}-44q^{126}-10q^{124}+101q^{122}+72q^{120}-33q^{118}-123q^{116}-129q^{114}+98q^{112}+190q^{110}+105q^{108}-121q^{106}-295q^{104}-53q^{102}+211q^{100}+299q^{98}+33q^{96}-349q^{94}-252q^{92}+79q^{90}+374q^{88}+218q^{86}-243q^{84}-338q^{82}-85q^{80}+301q^{78}+291q^{76}-93q^{74}-289q^{72}-165q^{70}+170q^{68}+258q^{66}+32q^{64}-191q^{62}-191q^{60}+36q^{58}+196q^{56}+152q^{54}-78q^{52}-215q^{50}-129q^{48}+113q^{46}+288q^{44}+76q^{42}-208q^{40}-305q^{38}-32q^{36}+357q^{34}+254q^{32}-95q^{30}-388q^{28}-210q^{26}+269q^{24}+329q^{22}+82q^{20}-286q^{18}-287q^{16}+78q^{14}+232q^{12}+171q^{10}-100q^{8}-202q^{6}-36q^{4}+74q^{2}+119+7q^{-2}-73q^{-4}-33q^{-6}-2q^{-8}+40q^{-10}+13q^{-12}-14q^{-14}-4q^{-16}-7q^{-18}+7q^{-20}+2q^{-22}-3q^{-24}+2q^{-26}-2q^{-28}+q^{-30}-q^{-34}+q^{-36}$
5	$-q^{225}+2q^{223}+2q^{221}-3q^{219}-3q^{217}-3q^{215}+9q^{211}+16q^{209}+q^{207}-20q^{205}-29q^{203}-19q^{201}+19q^{199}+61q^{197}+65q^{195}-8q^{193}-97q^{191}-128q^{189}-59q^{187}+101q^{185}+232q^{183}+192q^{181}-50q^{179}-314q^{177}-380q^{175}-132q^{173}+317q^{171}+608q^{169}+433q^{167}-179q^{165}-764q^{163}-819q^{161}-168q^{159}+769q^{157}+1205q^{155}+673q^{153}-549q^{151}-1463q^{149}-1236q^{147}+92q^{145}+1502q^{143}+1753q^{141}+499q^{139}-1299q^{137}-2079q^{135}-1103q^{133}+880q^{131}+2172q^{129}+1600q^{127}-375q^{125}-2039q^{123}-1894q^{121}-105q^{119}+1723q^{117}+1983q^{115}+488q^{113}-1347q^{111}-1887q^{109}-723q^{107}+973q^{105}+1668q^{103}+846q^{101}-645q^{99}-1424q^{97}-882q^{95}+374q^{93}+1185q^{91}+903q^{89}-124q^{87}-979q^{85}-964q^{83}-132q^{81}+798q^{79}+1067q^{77}+447q^{75}-597q^{73}-1215q^{71}-837q^{69}+336q^{67}+1349q^{65}+1278q^{63}+33q^{61}-1400q^{59}-1722q^{57}-511q^{55}+1301q^{53}+2085q^{51}+1038q^{49}-1011q^{47}-2255q^{45}-1542q^{43}+537q^{41}+2191q^{39}+1913q^{37}-10q^{35}-1840q^{33}-2044q^{31}-521q^{29}+1324q^{27}+1924q^{25}+878q^{23}-749q^{21}-1559q^{19}-1024q^{17}+235q^{15}+1098q^{13}+964q^{11}+102q^{9}-649q^{7}-735q^{5}-264q^{3}+292q+489q^{-1}+271q^{-3}-86q^{-5}-266q^{-7}-199q^{-9}-12q^{-11}+122q^{-13}+116q^{-15}+34q^{-17}-43q^{-19}-59q^{-21}-23q^{-23}+13q^{-25}+20q^{-27}+12q^{-29}+3q^{-31}-10q^{-33}-6q^{-35}+3q^{-37}+3q^{-43}-q^{-45}-2q^{-47}+q^{-49}-q^{-53}+q^{-55}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{26}-q^{24}+2q^{22}+q^{18}+2q^{16}-2q^{14}-2q^{10}+q^{6}-q^{4}+3q^{2}+q^{-4}$
1,1	$q^{68}-4q^{66}+12q^{64}-28q^{62}+52q^{60}-86q^{58}+130q^{56}-176q^{54}+212q^{52}-234q^{50}+232q^{48}-194q^{46}+126q^{44}-30q^{42}-80q^{40}+196q^{38}-303q^{36}+380q^{34}-432q^{32}+438q^{30}-410q^{28}+342q^{26}-244q^{24}+138q^{22}-19q^{20}-76q^{18}+154q^{16}-198q^{14}+214q^{12}-206q^{10}+174q^{8}-142q^{6}+107q^{4}-74q^{2}+50-30q^{-2}+21q^{-4}-10q^{-6}+6q^{-8}-2q^{-10}+q^{-12}$
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^{66}+q^{64}-q^{62}-4 q^{60}-2 q^{58}+4 q^{56}+2 q^{54}-3 q^{52}+8 q^{48}+5 q^{46}-8 q^{44}-5 q^{42}+4 q^{40}-q^{38}-8 q^{36}-3 q^{34}+6 q^{32}+q^{30}-q^{28}+4 q^{26}+q^{24}-2 q^{22}+5 q^{20}+4 q^{18}-7 q^{16}-2 q^{14}+7 q^{12}+q^{10}-9 q^8-2 q^6+9 q^4-5+ q^{-2} +4 q^{-4} + q^{-6} - q^{-8} + q^{-12} }

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}-2 q^{52}+q^{50}+3 q^{48}-7 q^{46}+3 q^{44}+4 q^{42}-11 q^{40}+6 q^{38}+7 q^{36}-10 q^{34}+4 q^{32}+7 q^{30}-5 q^{28}-q^{26}+3 q^{24}+2 q^{22}-4 q^{20}-4 q^{18}+8 q^{16}-5 q^{14}-8 q^{12}+12 q^{10}-3 q^8-7 q^6+10 q^4-3+4 q^{-2} + q^{-4} - q^{-6} + q^{-8} }
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^{35}-q^{33}-q^{31}+2 q^{29}+3 q^{25}+q^{23}+2 q^{21}-2 q^{19}-q^{17}-2 q^{15}-2 q^{13}+2 q^7-q^5+3 q^3+ q^{-1} + q^{-5} }

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}+2 q^{52}-5 q^{50}+7 q^{48}-9 q^{46}+11 q^{44}-12 q^{42}+11 q^{40}-8 q^{38}+5 q^{36}+2 q^{34}-6 q^{32}+13 q^{30}-17 q^{28}+21 q^{26}-23 q^{24}+20 q^{22}-18 q^{20}+12 q^{18}-8 q^{16}+q^{14}+4 q^{12}-8 q^{10}+11 q^8-11 q^6+12 q^4-8 q^2+7-4 q^{-2} +3 q^{-4} - q^{-6} + q^{-8} }
1,0	$q^{88}-2q^{84}-2q^{82}+3q^{80}+5q^{78}-2q^{76}-8q^{74}-3q^{72}+9q^{70}+8q^{68}-7q^{66}-12q^{64}+q^{62}+13q^{60}+6q^{58}-10q^{56}-8q^{54}+5q^{52}+9q^{50}-2q^{48}-8q^{46}+8q^{42}+2q^{40}-8q^{38}-3q^{36}+7q^{34}+6q^{32}-6q^{30}-8q^{28}+4q^{26}+10q^{24}-2q^{22}-12q^{20}-3q^{18}+11q^{16}+8q^{14}-7q^{12}-11q^{10}+q^{8}+11q^{6}+4q^{4}-4q^{2}-5+q^{-2}+4q^{-4}+2q^{-6}-q^{-8}-q^{-10}+q^{-14}$

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}-2 q^{126}+5 q^{124}-8 q^{122}+7 q^{120}-4 q^{118}-6 q^{116}+19 q^{114}-29 q^{112}+34 q^{110}-28 q^{108}+4 q^{106}+23 q^{104}-50 q^{102}+63 q^{100}-55 q^{98}+26 q^{96}+12 q^{94}-46 q^{92}+60 q^{90}-48 q^{88}+22 q^{86}+15 q^{84}-38 q^{82}+41 q^{80}-18 q^{78}-14 q^{76}+48 q^{74}-60 q^{72}+50 q^{70}-13 q^{68}-30 q^{66}+68 q^{64}-87 q^{62}+79 q^{60}-45 q^{58}-6 q^{56}+48 q^{54}-78 q^{52}+76 q^{50}-50 q^{48}+8 q^{46}+26 q^{44}-46 q^{42}+38 q^{40}-14 q^{38}-20 q^{36}+43 q^{34}-43 q^{32}+21 q^{30}+13 q^{28}-44 q^{26}+63 q^{24}-54 q^{22}+31 q^{20}-q^{18}-27 q^{16}+43 q^{14}-43 q^{12}+34 q^{10}-14 q^8+11 q^4-16 q^2+15-11 q^{-2} +8 q^{-4} -2 q^{-6} - q^{-8} +3 q^{-10} -3 q^{-12} +3 q^{-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["9 25"];

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 -3 t^2+12 t-17+12 t^{-1} -3 t^{-2} }

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 1-3 z^4}

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]=

{ 47, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{8}+3z^{2}a^{6}+3a^{6}-2z^{4}a^{4}-4z^{2}a^{4}-3a^{4}-z^{4}a^{2}+a^{2}+z^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-7z^{4}a^{8}+4z^{2}a^{8}-a^{8}+3z^{7}a^{7}-4z^{5}a^{7}-2z^{3}a^{7}+za^{7}+z^{8}a^{6}+6z^{6}a^{6}-18z^{4}a^{6}+13z^{2}a^{6}-3a^{6}+6z^{7}a^{5}-10z^{5}a^{5}+5z^{3}a^{5}-za^{5}+z^{8}a^{4}+6z^{6}a^{4}-15z^{4}a^{4}+13z^{2}a^{4}-3a^{4}+3z^{7}a^{3}-3z^{5}a^{3}+3z^{3}a^{3}-za^{3}+3z^{6}a^{2}-3z^{4}a^{2}+2z^{2}a^{2}-a^{2}+2z^{5}a-2z^{3}a+z^{4}-2z^{2}+1$