9 34

Polynomial invariants

Alexander polynomial	$-t^{3}+6t^{2}-16t+23-16t^{-1}+6t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 69, 0 }
Jones polynomial	$q^{4}-4q^{3}+8q^{2}-10q+12-12q^{-1}+10q^{-2}-7q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+z^{4}a^{-2}-3z^{4}-a^{4}z^{2}+3a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+a^{2}+a^{-2}-1$
Kauffman polynomial (db, data sources)	$3a^{2}z^{8}+3z^{8}+6a^{3}z^{7}+14az^{7}+8z^{7}a^{-1}+4a^{4}z^{6}+5a^{2}z^{6}+8z^{6}a^{-2}+9z^{6}+a^{5}z^{5}-11a^{3}z^{5}-26az^{5}-10z^{5}a^{-1}+4z^{5}a^{-3}-7a^{4}z^{4}-19a^{2}z^{4}-10z^{4}a^{-2}+z^{4}a^{-4}-23z^{4}-a^{5}z^{3}+5a^{3}z^{3}+12az^{3}+4z^{3}a^{-1}-2z^{3}a^{-3}+3a^{4}z^{2}+10a^{2}z^{2}+4z^{2}a^{-2}+11z^{2}-az-za^{-1}-a^{2}-a^{-2}-1$
The A2 invariant	$-q^{16}+q^{14}+2q^{12}-2q^{10}+2q^{8}-q^{6}-q^{4}+2q^{2}-2+3q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+q^{-12}$
The G2 invariant	$q^{80}-3q^{78}+7q^{76}-13q^{74}+14q^{72}-12q^{70}-q^{68}+27q^{66}-55q^{64}+83q^{62}-84q^{60}+44q^{58}+24q^{56}-112q^{54}+181q^{52}-188q^{50}+127q^{48}-11q^{46}-114q^{44}+205q^{42}-213q^{40}+135q^{38}-7q^{36}-116q^{34}+167q^{32}-131q^{30}+24q^{28}+103q^{26}-178q^{24}+183q^{22}-102q^{20}-37q^{18}+174q^{16}-269q^{14}+272q^{12}-182q^{10}+32q^{8}+130q^{6}-244q^{4}+280q^{2}-217+85q^{-2}+58q^{-4}-170q^{-6}+191q^{-8}-117q^{-10}-6q^{-12}+123q^{-14}-165q^{-16}+125q^{-18}-16q^{-20}-113q^{-22}+195q^{-24}-206q^{-26}+141q^{-28}-26q^{-30}-88q^{-32}+159q^{-34}-165q^{-36}+126q^{-38}-55q^{-40}-10q^{-42}+48q^{-44}-68q^{-46}+60q^{-48}-38q^{-50}+18q^{-52}+q^{-54}-8q^{-56}+10q^{-58}-10q^{-60}+6q^{-62}-3q^{-64}+q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 9_34.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+3q^{9}-3q^{7}+3q^{5}-2q^{3}+2q^{-1}-2q^{-3}+4q^{-5}-3q^{-7}+q^{-9}$
2	$q^{32}-3q^{30}-q^{28}+12q^{26}-8q^{24}-16q^{22}+25q^{20}+q^{18}-32q^{16}+22q^{14}+16q^{12}-28q^{10}+6q^{8}+19q^{6}-10q^{4}-13q^{2}+11+14q^{-2}-25q^{-4}-2q^{-6}+33q^{-8}-21q^{-10}-15q^{-12}+30q^{-14}-7q^{-16}-15q^{-18}+10q^{-20}+q^{-22}-3q^{-24}+q^{-26}$
3	$-q^{63}+3q^{61}+q^{59}-8q^{57}-7q^{55}+16q^{53}+30q^{51}-24q^{49}-64q^{47}+6q^{45}+107q^{43}+40q^{41}-138q^{39}-106q^{37}+135q^{35}+178q^{33}-96q^{31}-232q^{29}+37q^{27}+250q^{25}+28q^{23}-234q^{21}-86q^{19}+198q^{17}+119q^{15}-144q^{13}-137q^{11}+85q^{9}+150q^{7}-25q^{5}-151q^{3}-42q+150q^{-1}+110q^{-3}-132q^{-5}-180q^{-7}+98q^{-9}+234q^{-11}-44q^{-13}-255q^{-15}-22q^{-17}+239q^{-19}+81q^{-21}-187q^{-23}-113q^{-25}+121q^{-27}+105q^{-29}-53q^{-31}-82q^{-33}+15q^{-35}+48q^{-37}-2q^{-39}-18q^{-41}-3q^{-43}+7q^{-45}+q^{-47}-3q^{-49}+q^{-51}$
4	$q^{104}-3q^{102}-q^{100}+8q^{98}+3q^{96}-q^{94}-30q^{92}-20q^{90}+45q^{88}+67q^{86}+55q^{84}-123q^{82}-211q^{80}-29q^{78}+231q^{76}+437q^{74}+26q^{72}-539q^{70}-606q^{68}-39q^{66}+965q^{64}+866q^{62}-237q^{60}-1287q^{58}-1138q^{56}+669q^{54}+1720q^{52}+979q^{50}-1012q^{48}-2141q^{46}-546q^{44}+1536q^{42}+2020q^{40}+111q^{38}-2061q^{36}-1539q^{34}+565q^{32}+2042q^{30}+996q^{28}-1229q^{26}-1668q^{24}-264q^{22}+1418q^{20}+1232q^{18}-399q^{16}-1348q^{14}-721q^{12}+758q^{10}+1234q^{8}+313q^{6}-1013q^{4}-1169q^{2}+46+1278q^{-2}+1203q^{-4}-513q^{-6}-1679q^{-8}-981q^{-10}+1003q^{-12}+2097q^{-14}+482q^{-16}-1631q^{-18}-2013q^{-20}+11q^{-22}+2191q^{-24}+1535q^{-26}-628q^{-28}-2103q^{-30}-1083q^{-32}+1160q^{-34}+1627q^{-36}+495q^{-38}-1100q^{-40}-1210q^{-42}+49q^{-44}+779q^{-46}+690q^{-48}-151q^{-50}-563q^{-52}-227q^{-54}+109q^{-56}+291q^{-58}+71q^{-60}-111q^{-62}-74q^{-64}-24q^{-66}+53q^{-68}+22q^{-70}-13q^{-72}-6q^{-74}-6q^{-76}+7q^{-78}+q^{-80}-3q^{-82}+q^{-84}$
5	$-q^{155}+3q^{153}+q^{151}-8q^{149}-3q^{147}+5q^{145}+15q^{143}+20q^{141}-q^{139}-59q^{137}-86q^{135}-15q^{133}+123q^{131}+240q^{129}+177q^{127}-129q^{125}-545q^{123}-631q^{121}-100q^{119}+802q^{117}+1419q^{115}+982q^{113}-623q^{111}-2384q^{109}-2611q^{107}-542q^{105}+2761q^{103}+4754q^{101}+3135q^{99}-1778q^{97}-6538q^{95}-6794q^{93}-1213q^{91}+6706q^{89}+10559q^{87}+6079q^{85}-4446q^{83}-12971q^{81}-11708q^{79}-320q^{77}+12798q^{75}+16578q^{73}+6682q^{71}-9776q^{69}-19250q^{67}-13013q^{65}+4536q^{63}+19015q^{61}+17848q^{59}+1556q^{57}-16212q^{55}-20243q^{53}-7017q^{51}+11839q^{49}+20042q^{47}+10881q^{45}-7083q^{43}-17954q^{41}-12766q^{39}+2989q^{37}+14851q^{35}+12931q^{33}+56q^{31}-11650q^{29}-12114q^{27}-1995q^{25}+8933q^{23}+11006q^{21}+3244q^{19}-6807q^{17}-10249q^{15}-4442q^{13}+5184q^{11}+10107q^{9}+6046q^{7}-3522q^{5}-10451q^{3}-8507q+1299q^{-1}+10908q^{-3}+11682q^{-5}+1980q^{-7}-10674q^{-9}-15162q^{-11}-6449q^{-13}+9048q^{-15}+18039q^{-17}+11723q^{-19}-5574q^{-21}-19230q^{-23}-16847q^{-25}+368q^{-27}+17919q^{-29}+20551q^{-31}+5661q^{-33}-13978q^{-35}-21629q^{-37}-11107q^{-39}+8117q^{-41}+19635q^{-43}+14545q^{-45}-1807q^{-47}-15140q^{-49}-15083q^{-51}-3294q^{-53}+9385q^{-55}+12979q^{-57}+6171q^{-59}-4122q^{-61}-9312q^{-63}-6523q^{-65}+394q^{-67}+5411q^{-69}+5270q^{-71}+1385q^{-73}-2442q^{-75}-3377q^{-77}-1635q^{-79}+677q^{-81}+1733q^{-83}+1211q^{-85}+37q^{-87}-739q^{-89}-650q^{-91}-151q^{-93}+224q^{-95}+283q^{-97}+113q^{-99}-60q^{-101}-110q^{-103}-41q^{-105}+20q^{-107}+27q^{-109}+11q^{-111}-q^{-113}-9q^{-115}-6q^{-117}+7q^{-119}+q^{-121}-3q^{-123}+q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{14}+2q^{12}-2q^{10}+2q^{8}-q^{6}-q^{4}+2q^{2}-2+3q^{-2}-2q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+q^{-12}$
1,1	$q^{44}-6q^{42}+20q^{40}-50q^{38}+107q^{36}-204q^{34}+344q^{32}-514q^{30}+701q^{28}-864q^{26}+950q^{24}-938q^{22}+791q^{20}-502q^{18}+98q^{16}+374q^{14}-840q^{12}+1276q^{10}-1606q^{8}+1788q^{6}-1806q^{4}+1650q^{2}-1352+934q^{-2}-456q^{-4}-16q^{-6}+432q^{-8}-726q^{-10}+888q^{-12}-918q^{-14}+850q^{-16}-712q^{-18}+538q^{-20}-376q^{-22}+248q^{-24}-148q^{-26}+77q^{-28}-38q^{-30}+18q^{-32}-6q^{-34}+q^{-36}$
2,0	$q^{42}-q^{40}-3q^{38}+7q^{34}+5q^{32}-10q^{30}-7q^{28}+9q^{26}+7q^{24}-13q^{22}-7q^{20}+13q^{18}+8q^{16}-11q^{14}-q^{12}+13q^{10}-5q^{8}-3q^{6}+4q^{4}-q^{2}-7+5q^{-2}+7q^{-4}-13q^{-6}-4q^{-8}+16q^{-10}+6q^{-12}-17q^{-14}+q^{-16}+14q^{-18}-9q^{-22}-3q^{-24}+6q^{-26}-2q^{-30}+q^{-32}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-3q^{32}+q^{30}+6q^{28}-12q^{26}+6q^{24}+13q^{22}-22q^{20}+10q^{18}+15q^{16}-24q^{14}+7q^{12}+14q^{10}-13q^{8}-2q^{6}+7q^{4}+3q^{2}-6-6q^{-2}+18q^{-4}-6q^{-6}-17q^{-8}+25q^{-10}-5q^{-12}-18q^{-14}+20q^{-16}-3q^{-18}-10q^{-20}+9q^{-22}-3q^{-26}+q^{-28}$
1,0,0	$-q^{21}+q^{19}+2q^{15}-2q^{13}+3q^{11}-2q^{9}+q^{7}-q^{5}+q^{3}-q^{-1}+2q^{-3}-2q^{-5}+2q^{-7}-q^{-9}+3q^{-11}-2q^{-13}+q^{-15}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{44}-q^{42}-3q^{40}+2q^{38}+4q^{36}-4q^{34}-5q^{32}+9q^{30}+7q^{28}-14q^{26}-4q^{24}+20q^{22}-q^{20}-20q^{18}+9q^{16}+17q^{14}-14q^{12}-12q^{10}+16q^{8}+2q^{6}-19q^{4}+10q^{2}+18-16q^{-2}-5q^{-4}+24q^{-6}-5q^{-8}-21q^{-10}+8q^{-12}+15q^{-14}-12q^{-16}-10q^{-18}+14q^{-20}+7q^{-22}-10q^{-24}-q^{-26}+7q^{-28}-2q^{-30}-2q^{-32}+q^{-34}$
1,0,0,0	$-q^{26}+q^{24}+2q^{18}-2q^{16}+3q^{14}-q^{12}+q^{8}-q^{6}+q^{4}-q^{2}+1-2q^{-2}+2q^{-4}-2q^{-6}+2q^{-8}+3q^{-14}-2q^{-16}+q^{-18}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+3q^{32}-7q^{30}+12q^{28}-18q^{26}+24q^{24}-25q^{22}+26q^{20}-22q^{18}+15q^{16}-4q^{14}-9q^{12}+22q^{10}-35q^{8}+44q^{6}-49q^{4}+49q^{2}-44+34q^{-2}-20q^{-4}+8q^{-6}+5q^{-8}-15q^{-10}+23q^{-12}-26q^{-14}+26q^{-16}-23q^{-18}+18q^{-20}-11q^{-22}+6q^{-24}-3q^{-26}+q^{-28}$
1,0	$q^{56}-3q^{52}-3q^{50}+4q^{48}+9q^{46}-q^{44}-15q^{42}-8q^{40}+18q^{38}+19q^{36}-10q^{34}-27q^{32}-3q^{30}+28q^{28}+15q^{26}-21q^{24}-22q^{22}+10q^{20}+23q^{18}-q^{16}-20q^{14}-3q^{12}+18q^{10}+7q^{8}-16q^{6}-11q^{4}+13q^{2}+16-10q^{-2}-20q^{-4}+7q^{-6}+23q^{-8}+q^{-10}-25q^{-12}-10q^{-14}+23q^{-16}+22q^{-18}-14q^{-20}-27q^{-22}+q^{-24}+23q^{-26}+10q^{-28}-13q^{-30}-14q^{-32}+3q^{-34}+10q^{-36}+3q^{-38}-3q^{-40}-3q^{-42}+q^{-46}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{46}-3q^{44}+4q^{42}-6q^{40}+10q^{38}-15q^{36}+17q^{34}-18q^{32}+22q^{30}-21q^{28}+18q^{26}-15q^{24}+14q^{22}-5q^{20}-5q^{18}+10q^{16}-14q^{14}+25q^{12}-33q^{10}+33q^{8}-35q^{6}+40q^{4}-37q^{2}+31-29q^{-2}+25q^{-4}-12q^{-6}+4q^{-8}-2q^{-10}-5q^{-12}+16q^{-14}-17q^{-16}+18q^{-18}-22q^{-20}+23q^{-22}-16q^{-24}+14q^{-26}-14q^{-28}+10q^{-30}-4q^{-32}+3q^{-34}-3q^{-36}+q^{-38}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-3q^{78}+7q^{76}-13q^{74}+14q^{72}-12q^{70}-q^{68}+27q^{66}-55q^{64}+83q^{62}-84q^{60}+44q^{58}+24q^{56}-112q^{54}+181q^{52}-188q^{50}+127q^{48}-11q^{46}-114q^{44}+205q^{42}-213q^{40}+135q^{38}-7q^{36}-116q^{34}+167q^{32}-131q^{30}+24q^{28}+103q^{26}-178q^{24}+183q^{22}-102q^{20}-37q^{18}+174q^{16}-269q^{14}+272q^{12}-182q^{10}+32q^{8}+130q^{6}-244q^{4}+280q^{2}-217+85q^{-2}+58q^{-4}-170q^{-6}+191q^{-8}-117q^{-10}-6q^{-12}+123q^{-14}-165q^{-16}+125q^{-18}-16q^{-20}-113q^{-22}+195q^{-24}-206q^{-26}+141q^{-28}-26q^{-30}-88q^{-32}+159q^{-34}-165q^{-36}+126q^{-38}-55q^{-40}-10q^{-42}+48q^{-44}-68q^{-46}+60q^{-48}-38q^{-50}+18q^{-52}+q^{-54}-8q^{-56}+10q^{-58}-10q^{-60}+6q^{-62}-3q^{-64}+q^{-66}$

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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["9 34"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+6t^{2}-16t+23-16t^{-1}+6t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 69, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{4}-4q^{3}+8q^{2}-10q+12-12q^{-1}+10q^{-2}-7q^{-3}+4q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{2}z^{8}+3z^{8}+6a^{3}z^{7}+14az^{7}+8z^{7}a^{-1}+4a^{4}z^{6}+5a^{2}z^{6}+8z^{6}a^{-2}+9z^{6}+a^{5}z^{5}-11a^{3}z^{5}-26az^{5}-10z^{5}a^{-1}+4z^{5}a^{-3}-7a^{4}z^{4}-19a^{2}z^{4}-10z^{4}a^{-2}+z^{4}a^{-4}-23z^{4}-a^{5}z^{3}+5a^{3}z^{3}+12az^{3}+4z^{3}a^{-1}-2z^{3}a^{-3}+3a^{4}z^{2}+10a^{2}z^{2}+4z^{2}a^{-2}+11z^{2}-az-za^{-1}-a^{2}-a^{-2}-1$