10 27

Polynomial invariants

Alexander polynomial	$2t^{3}-8t^{2}+16t-19+16t^{-1}-8t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+4z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 71, -2 }
Jones polynomial	$-q^{2}+3q-5+9q^{-1}-11q^{-2}+12q^{-3}-11q^{-4}+9q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{6}-2z^{2}a^{6}-a^{6}+z^{6}a^{4}+3z^{4}a^{4}+3z^{2}a^{4}+a^{4}+z^{6}a^{2}+3z^{4}a^{2}+3z^{2}a^{2}+a^{2}-z^{4}-2z^{2}$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-6z^{4}a^{8}+3z^{2}a^{8}+4z^{7}a^{7}-6z^{5}a^{7}+z^{3}a^{7}+3z^{8}a^{6}-z^{6}a^{6}-3z^{4}a^{6}-z^{2}a^{6}+a^{6}+z^{9}a^{5}+6z^{7}a^{5}-12z^{5}a^{5}+7z^{3}a^{5}-2za^{5}+6z^{8}a^{4}-9z^{6}a^{4}+7z^{4}a^{4}-4z^{2}a^{4}+a^{4}+z^{9}a^{3}+6z^{7}a^{3}-14z^{5}a^{3}+11z^{3}a^{3}-2za^{3}+3z^{8}a^{2}-2z^{6}a^{2}-3z^{4}a^{2}+4z^{2}a^{2}-a^{2}+4z^{7}a-8z^{5}a+5z^{3}a-za+3z^{6}-7z^{4}+4z^{2}+z^{5}a^{-1}-2z^{3}a^{-1}$
The A2 invariant	$-q^{24}+q^{22}-q^{20}-q^{18}+2q^{16}-2q^{14}+2q^{12}+2q^{6}-2q^{4}+3q^{2}+q^{-4}-q^{-6}$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+8q^{120}-6q^{118}-2q^{116}+17q^{114}-31q^{112}+44q^{110}-45q^{108}+26q^{106}+5q^{104}-49q^{102}+90q^{100}-111q^{98}+98q^{96}-52q^{94}-22q^{92}+93q^{90}-138q^{88}+142q^{86}-96q^{84}+21q^{82}+56q^{80}-107q^{78}+104q^{76}-55q^{74}-16q^{72}+76q^{70}-96q^{68}+62q^{66}+14q^{64}-99q^{62}+161q^{60}-167q^{58}+109q^{56}-6q^{54}-110q^{52}+193q^{50}-214q^{48}+172q^{46}-73q^{44}-38q^{42}+126q^{40}-160q^{38}+133q^{36}-60q^{34}-24q^{32}+80q^{30}-87q^{28}+47q^{26}+25q^{24}-86q^{22}+116q^{20}-96q^{18}+31q^{16}+47q^{14}-115q^{12}+145q^{10}-121q^{8}+70q^{6}-q^{4}-59q^{2}+94-96q^{-2}+73q^{-4}-36q^{-6}-2q^{-8}+27q^{-10}-38q^{-12}+36q^{-14}-25q^{-16}+14q^{-18}-q^{-20}-6q^{-22}+6q^{-24}-7q^{-26}+4q^{-28}-2q^{-30}+q^{-32}$

Further Quantum Invariants

Further quantum knot invariants for 10_27.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+2q^{15}-3q^{13}+3q^{11}-2q^{9}+q^{7}+q^{5}-2q^{3}+4q-2q^{-1}+2q^{-3}-q^{-5}$
2	$q^{48}-2q^{46}-q^{44}+7q^{42}-6q^{40}-8q^{38}+18q^{36}-5q^{34}-19q^{32}+23q^{30}+2q^{28}-24q^{26}+15q^{24}+9q^{22}-16q^{20}-q^{18}+11q^{16}+2q^{14}-17q^{12}+7q^{10}+19q^{8}-23q^{6}-q^{4}+25q^{2}-15-6q^{-2}+16q^{-4}-6q^{-6}-6q^{-8}+6q^{-10}-q^{-12}-2q^{-14}+q^{-16}$
3	$-q^{93}+2q^{91}+q^{89}-3q^{87}-4q^{85}+6q^{83}+11q^{81}-11q^{79}-22q^{77}+12q^{75}+39q^{73}-7q^{71}-62q^{69}-4q^{67}+84q^{65}+26q^{63}-98q^{61}-58q^{59}+104q^{57}+87q^{55}-95q^{53}-112q^{51}+74q^{49}+126q^{47}-45q^{45}-126q^{43}+14q^{41}+112q^{39}+18q^{37}-88q^{35}-48q^{33}+56q^{31}+77q^{29}-22q^{27}-98q^{25}-17q^{23}+111q^{21}+52q^{19}-113q^{17}-87q^{15}+106q^{13}+107q^{11}-80q^{9}-117q^{7}+51q^{5}+117q^{3}-23q-95q^{-1}-q^{-3}+76q^{-5}+13q^{-7}-49q^{-9}-19q^{-11}+30q^{-13}+15q^{-15}-16q^{-17}-12q^{-19}+8q^{-21}+6q^{-23}-3q^{-25}-3q^{-27}+q^{-29}+2q^{-31}-q^{-33}$
4	$q^{152}-2q^{150}-q^{148}+3q^{146}+4q^{142}-9q^{140}-6q^{138}+12q^{136}+6q^{134}+17q^{132}-33q^{130}-36q^{128}+25q^{126}+41q^{124}+72q^{122}-71q^{120}-130q^{118}-13q^{116}+103q^{114}+240q^{112}-42q^{110}-288q^{108}-204q^{106}+82q^{104}+505q^{102}+186q^{100}-348q^{98}-529q^{96}-174q^{94}+662q^{92}+566q^{90}-145q^{88}-743q^{86}-572q^{84}+520q^{82}+819q^{80}+232q^{78}-654q^{76}-828q^{74}+169q^{72}+759q^{70}+510q^{68}-342q^{66}-780q^{64}-172q^{62}+465q^{60}+584q^{58}+13q^{56}-528q^{54}-418q^{52}+103q^{50}+531q^{48}+344q^{46}-202q^{44}-609q^{42}-272q^{40}+412q^{38}+637q^{36}+167q^{34}-698q^{32}-627q^{30}+170q^{28}+796q^{26}+559q^{24}-566q^{22}-826q^{20}-182q^{18}+667q^{16}+798q^{14}-209q^{12}-711q^{10}-459q^{8}+296q^{6}+723q^{4}+121q^{2}-360-459q^{-2}-35q^{-4}+416q^{-6}+208q^{-8}-54q^{-10}-266q^{-12}-138q^{-14}+146q^{-16}+118q^{-18}+53q^{-20}-93q^{-22}-90q^{-24}+32q^{-26}+33q^{-28}+40q^{-30}-21q^{-32}-33q^{-34}+8q^{-36}+3q^{-38}+14q^{-40}-3q^{-42}-9q^{-44}+3q^{-46}+3q^{-50}-q^{-52}-2q^{-54}+q^{-56}$
5	$-q^{225}+2q^{223}+q^{221}-3q^{219}-q^{213}+4q^{211}+5q^{209}-8q^{207}-9q^{205}+2q^{203}+8q^{201}+18q^{199}+11q^{197}-22q^{195}-52q^{193}-27q^{191}+46q^{189}+100q^{187}+81q^{185}-47q^{183}-197q^{181}-209q^{179}+14q^{177}+323q^{175}+420q^{173}+134q^{171}-413q^{169}-756q^{167}-468q^{165}+403q^{163}+1157q^{161}+1012q^{159}-140q^{157}-1494q^{155}-1785q^{153}-463q^{151}+1622q^{149}+2652q^{147}+1415q^{145}-1357q^{143}-3394q^{141}-2665q^{139}+621q^{137}+3830q^{135}+3964q^{133}+516q^{131}-3746q^{129}-5056q^{127}-1930q^{125}+3156q^{123}+5727q^{121}+3304q^{119}-2140q^{117}-5816q^{115}-4414q^{113}+895q^{111}+5379q^{109}+5066q^{107}+328q^{105}-4528q^{103}-5210q^{101}-1351q^{99}+3432q^{97}+4936q^{95}+2093q^{93}-2306q^{91}-4367q^{89}-2550q^{87}+1242q^{85}+3658q^{83}+2838q^{81}-284q^{79}-2964q^{77}-3032q^{75}-594q^{73}+2291q^{71}+3251q^{69}+1472q^{67}-1657q^{65}-3511q^{63}-2387q^{61}+967q^{59}+3766q^{57}+3378q^{55}-154q^{53}-3890q^{51}-4390q^{49}-844q^{47}+3782q^{45}+5257q^{43}+1987q^{41}-3270q^{39}-5835q^{37}-3212q^{35}+2406q^{33}+5936q^{31}+4234q^{29}-1168q^{27}-5475q^{25}-4935q^{23}-182q^{21}+4512q^{19}+5098q^{17}+1384q^{15}-3165q^{13}-4701q^{11}-2265q^{9}+1751q^{7}+3884q^{5}+2613q^{3}-514q-2766q^{-1}-2529q^{-3}-364q^{-5}+1708q^{-7}+2072q^{-9}+809q^{-11}-805q^{-13}-1487q^{-15}-902q^{-17}+225q^{-19}+909q^{-21}+769q^{-23}+92q^{-25}-486q^{-27}-543q^{-29}-187q^{-31}+195q^{-33}+335q^{-35}+184q^{-37}-66q^{-39}-181q^{-41}-121q^{-43}+3q^{-45}+81q^{-47}+78q^{-49}+11q^{-51}-40q^{-53}-36q^{-55}-5q^{-57}+13q^{-59}+14q^{-61}+8q^{-63}-7q^{-65}-10q^{-67}+2q^{-69}+4q^{-71}-3q^{-79}+q^{-81}+2q^{-83}-q^{-85}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{24}+q^{22}-q^{20}-q^{18}+2q^{16}-2q^{14}+2q^{12}+2q^{6}-2q^{4}+3q^{2}+q^{-4}-q^{-6}$
1,1	$q^{68}-4q^{66}+12q^{64}-28q^{62}+56q^{60}-102q^{58}+168q^{56}-252q^{54}+351q^{52}-462q^{50}+560q^{48}-622q^{46}+639q^{44}-592q^{42}+466q^{40}-252q^{38}-18q^{36}+318q^{34}-636q^{32}+920q^{30}-1141q^{28}+1266q^{26}-1286q^{24}+1198q^{22}-1011q^{20}+748q^{18}-446q^{16}+132q^{14}+156q^{12}-374q^{10}+534q^{8}-612q^{6}+626q^{4}-570q^{2}+488-394q^{-2}+296q^{-4}-204q^{-6}+132q^{-8}-84q^{-10}+46q^{-12}-22q^{-14}+10q^{-16}-4q^{-18}+q^{-20}$
2,0	$q^{62}-q^{60}-q^{58}+2q^{56}+q^{54}-3q^{52}-3q^{50}+5q^{48}+3q^{46}-8q^{44}-q^{42}+10q^{40}-10q^{36}+2q^{34}+10q^{32}-5q^{30}-9q^{28}+6q^{26}+2q^{24}-10q^{22}+2q^{20}+7q^{18}-6q^{16}-2q^{14}+11q^{12}+3q^{10}-9q^{8}+3q^{6}+13q^{4}-4q^{2}-7+6q^{-2}+5q^{-4}-6q^{-6}-4q^{-8}+3q^{-10}+2q^{-12}-2q^{-14}-q^{-16}+q^{-18}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{54}-2q^{52}+q^{50}+4q^{48}-8q^{46}+3q^{44}+10q^{42}-16q^{40}+4q^{38}+15q^{36}-21q^{34}+16q^{30}-14q^{28}-4q^{26}+11q^{24}-5q^{20}-q^{18}+13q^{16}-3q^{14}-13q^{12}+19q^{10}-19q^{6}+17q^{4}+3q^{2}-13+10q^{-2}+3q^{-4}-7q^{-6}+3q^{-8}-2q^{-12}+q^{-14}$
1,0,0	$-q^{31}+q^{29}-2q^{27}+q^{25}-2q^{23}+2q^{21}-2q^{19}+2q^{17}+q^{13}+q^{11}+2q^{7}-2q^{5}+3q^{3}-q+2q^{-1}-q^{-3}+q^{-5}-q^{-7}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{68}-q^{66}-q^{64}+4q^{62}-6q^{58}+4q^{56}+8q^{54}-7q^{52}-8q^{50}+9q^{48}+5q^{46}-16q^{44}-6q^{42}+13q^{40}-5q^{38}-15q^{36}+10q^{34}+9q^{32}-11q^{30}+3q^{28}+15q^{26}-3q^{24}-9q^{22}+12q^{20}+8q^{18}-14q^{16}-2q^{14}+15q^{12}-2q^{10}-12q^{8}+7q^{6}+8q^{4}-3q^{2}-3+5q^{-2}+2q^{-4}-3q^{-6}-q^{-8}+q^{-10}-q^{-12}-q^{-14}+q^{-16}$
1,0,0,0	$-q^{38}+q^{36}-2q^{34}-2q^{28}+2q^{26}-2q^{24}+2q^{22}+q^{18}+q^{16}+q^{14}+q^{12}+2q^{8}-2q^{6}+3q^{4}-q^{2}+1+q^{-2}-q^{-4}+q^{-6}-q^{-8}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{54}+2q^{52}-5q^{50}+8q^{48}-12q^{46}+17q^{44}-20q^{42}+22q^{40}-22q^{38}+19q^{36}-13q^{34}+4q^{32}+6q^{30}-18q^{28}+28q^{26}-37q^{24}+42q^{22}-43q^{20}+41q^{18}-33q^{16}+25q^{14}-13q^{12}+3q^{10}+8q^{8}-15q^{6}+21q^{4}-21q^{2}+21-18q^{-2}+15q^{-4}-11q^{-6}+7q^{-8}-4q^{-10}+2q^{-12}-q^{-14}$
1,0	$q^{88}-2q^{84}-2q^{82}+3q^{80}+6q^{78}-q^{76}-10q^{74}-6q^{72}+11q^{70}+15q^{68}-5q^{66}-21q^{64}-7q^{62}+20q^{60}+18q^{58}-12q^{56}-24q^{54}-q^{52}+21q^{50}+9q^{48}-16q^{46}-13q^{44}+10q^{42}+14q^{40}-6q^{38}-14q^{36}+3q^{34}+15q^{32}+q^{30}-15q^{28}-3q^{26}+16q^{24}+9q^{22}-15q^{20}-14q^{18}+12q^{16}+22q^{14}-3q^{12}-24q^{10}-9q^{8}+20q^{6}+18q^{4}-7q^{2}-19-3q^{-2}+14q^{-4}+10q^{-6}-5q^{-8}-9q^{-10}-q^{-12}+5q^{-14}+2q^{-16}-2q^{-18}-2q^{-20}+q^{-24}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{74}-2q^{72}+3q^{70}-4q^{68}+7q^{66}-10q^{64}+11q^{62}-13q^{60}+17q^{58}-18q^{56}+16q^{54}-16q^{52}+16q^{50}-13q^{48}+3q^{46}-3q^{44}-3q^{42}+10q^{40}-20q^{38}+21q^{36}-25q^{34}+34q^{32}-32q^{30}+32q^{28}-31q^{26}+33q^{24}-22q^{22}+19q^{20}-15q^{18}+9q^{16}+2q^{14}-4q^{12}+7q^{10}-14q^{8}+18q^{6}-15q^{4}+16q^{2}-16+16q^{-2}-11q^{-4}+10q^{-6}-9q^{-8}+6q^{-10}-4q^{-12}+2q^{-14}-2q^{-16}+q^{-18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{128}-2q^{126}+5q^{124}-8q^{122}+8q^{120}-6q^{118}-2q^{116}+17q^{114}-31q^{112}+44q^{110}-45q^{108}+26q^{106}+5q^{104}-49q^{102}+90q^{100}-111q^{98}+98q^{96}-52q^{94}-22q^{92}+93q^{90}-138q^{88}+142q^{86}-96q^{84}+21q^{82}+56q^{80}-107q^{78}+104q^{76}-55q^{74}-16q^{72}+76q^{70}-96q^{68}+62q^{66}+14q^{64}-99q^{62}+161q^{60}-167q^{58}+109q^{56}-6q^{54}-110q^{52}+193q^{50}-214q^{48}+172q^{46}-73q^{44}-38q^{42}+126q^{40}-160q^{38}+133q^{36}-60q^{34}-24q^{32}+80q^{30}-87q^{28}+47q^{26}+25q^{24}-86q^{22}+116q^{20}-96q^{18}+31q^{16}+47q^{14}-115q^{12}+145q^{10}-121q^{8}+70q^{6}-q^{4}-59q^{2}+94-96q^{-2}+73q^{-4}-36q^{-6}-2q^{-8}+27q^{-10}-38q^{-12}+36q^{-14}-25q^{-16}+14q^{-18}-q^{-20}-6q^{-22}+6q^{-24}-7q^{-26}+4q^{-28}-2q^{-30}+q^{-32}$

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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 27"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-8t^{2}+16t-19+16t^{-1}-8t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{6}+4z^{4}+2z^{2}+1$

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]=

{ 71, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{2}+3q-5+9q^{-1}-11q^{-2}+12q^{-3}-11q^{-4}+9q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-6z^{4}a^{8}+3z^{2}a^{8}+4z^{7}a^{7}-6z^{5}a^{7}+z^{3}a^{7}+3z^{8}a^{6}-z^{6}a^{6}-3z^{4}a^{6}-z^{2}a^{6}+a^{6}+z^{9}a^{5}+6z^{7}a^{5}-12z^{5}a^{5}+7z^{3}a^{5}-2za^{5}+6z^{8}a^{4}-9z^{6}a^{4}+7z^{4}a^{4}-4z^{2}a^{4}+a^{4}+z^{9}a^{3}+6z^{7}a^{3}-14z^{5}a^{3}+11z^{3}a^{3}-2za^{3}+3z^{8}a^{2}-2z^{6}a^{2}-3z^{4}a^{2}+4z^{2}a^{2}-a^{2}+4z^{7}a-8z^{5}a+5z^{3}a-za+3z^{6}-7z^{4}+4z^{2}+z^{5}a^{-1}-2z^{3}a^{-1}$