10 86

Polynomial invariants

Alexander polynomial	$-2t^{3}+9t^{2}-19t+25-19t^{-1}+9t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-3z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 85, 0 }
Jones polynomial	$q^{6}-3q^{5}+6q^{4}-10q^{3}+13q^{2}-14q+14-11q^{-1}+8q^{-2}-4q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-3z^{4}a^{-2}+z^{4}a^{-4}-2z^{4}+a^{2}z^{2}-4z^{2}a^{-2}+2z^{2}a^{-4}-2a^{-2}+a^{-4}+2$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-1}+2z^{9}a^{-3}+10z^{8}a^{-2}+4z^{8}a^{-4}+6z^{8}+9az^{7}+9z^{7}a^{-1}+3z^{7}a^{-3}+3z^{7}a^{-5}+8a^{2}z^{6}-21z^{6}a^{-2}-10z^{6}a^{-4}+z^{6}a^{-6}-2z^{6}+4a^{3}z^{5}-11az^{5}-23z^{5}a^{-1}-17z^{5}a^{-3}-9z^{5}a^{-5}+a^{4}z^{4}-9a^{2}z^{4}+13z^{4}a^{-2}+7z^{4}a^{-4}-3z^{4}a^{-6}-7z^{4}-2a^{3}z^{3}+4az^{3}+13z^{3}a^{-1}+15z^{3}a^{-3}+8z^{3}a^{-5}+3a^{2}z^{2}-6z^{2}a^{-2}-2z^{2}a^{-4}+2z^{2}a^{-6}+z^{2}-az-3za^{-1}-4za^{-3}-2za^{-5}+2a^{-2}+a^{-4}+2$
The A2 invariant	$q^{12}-2q^{10}+q^{8}+q^{6}-2q^{4}+4q^{2}-1+2q^{-2}-2q^{-6}+2q^{-8}-3q^{-10}+q^{-12}+q^{-14}-q^{-16}+q^{-18}$
The G2 invariant	$q^{66}-3q^{64}+6q^{62}-10q^{60}+10q^{58}-8q^{56}+q^{54}+17q^{52}-36q^{50}+57q^{48}-65q^{46}+49q^{44}-16q^{42}-40q^{40}+104q^{38}-150q^{36}+163q^{34}-127q^{32}+39q^{30}+74q^{28}-178q^{26}+236q^{24}-219q^{22}+126q^{20}+8q^{18}-137q^{16}+206q^{14}-179q^{12}+77q^{10}+65q^{8}-170q^{6}+187q^{4}-97q^{2}-59+222q^{-2}-307q^{-4}+278q^{-6}-132q^{-8}-74q^{-10}+266q^{-12}-372q^{-14}+354q^{-16}-222q^{-18}+24q^{-20}+161q^{-22}-278q^{-24}+285q^{-26}-190q^{-28}+35q^{-30}+109q^{-32}-192q^{-34}+172q^{-36}-65q^{-38}-79q^{-40}+196q^{-42}-230q^{-44}+158q^{-46}-13q^{-48}-147q^{-50}+258q^{-52}-268q^{-54}+189q^{-56}-52q^{-58}-88q^{-60}+178q^{-62}-195q^{-64}+151q^{-66}-70q^{-68}-8q^{-70}+58q^{-72}-75q^{-74}+64q^{-76}-38q^{-78}+15q^{-80}+3q^{-82}-11q^{-84}+10q^{-86}-9q^{-88}+5q^{-90}-2q^{-92}+q^{-94}$

Further Quantum Invariants

Further quantum knot invariants for 10_86.

A1 Invariants.

Weight	Invariant
1	$q^{9}-3q^{7}+4q^{5}-3q^{3}+3q-q^{-3}+3q^{-5}-4q^{-7}+3q^{-9}-2q^{-11}+q^{-13}$
2	$q^{26}-3q^{24}+q^{22}+9q^{20}-14q^{18}-2q^{16}+26q^{14}-24q^{12}-12q^{10}+39q^{8}-17q^{6}-23q^{4}+29q^{2}+3-21q^{-2}+2q^{-4}+21q^{-6}-7q^{-8}-24q^{-10}+26q^{-12}+11q^{-14}-38q^{-16}+16q^{-18}+24q^{-20}-30q^{-22}+q^{-24}+21q^{-26}-12q^{-28}-6q^{-30}+8q^{-32}-q^{-34}-2q^{-36}+q^{-38}$
3	$q^{51}-3q^{49}+q^{47}+6q^{45}-2q^{43}-13q^{41}-q^{39}+32q^{37}-q^{35}-54q^{33}-3q^{31}+87q^{29}+23q^{27}-135q^{25}-55q^{23}+172q^{21}+109q^{19}-190q^{17}-170q^{15}+171q^{13}+225q^{11}-122q^{9}-246q^{7}+48q^{5}+237q^{3}+31q-195q^{-1}-101q^{-3}+135q^{-5}+157q^{-7}-71q^{-9}-189q^{-11}+3q^{-13}+214q^{-15}+58q^{-17}-219q^{-19}-118q^{-21}+207q^{-23}+172q^{-25}-172q^{-27}-221q^{-29}+120q^{-31}+243q^{-33}-49q^{-35}-238q^{-37}-22q^{-39}+205q^{-41}+74q^{-43}-143q^{-45}-102q^{-47}+78q^{-49}+96q^{-51}-26q^{-53}-69q^{-55}-4q^{-57}+39q^{-59}+13q^{-61}-17q^{-63}-9q^{-65}+5q^{-67}+4q^{-69}-q^{-71}-2q^{-73}+q^{-75}$
4	$q^{84}-3q^{82}+q^{80}+6q^{78}-5q^{76}-q^{74}-12q^{72}+11q^{70}+30q^{68}-23q^{66}-16q^{64}-40q^{62}+50q^{60}+109q^{58}-74q^{56}-119q^{54}-128q^{52}+196q^{50}+387q^{48}-90q^{46}-451q^{44}-530q^{42}+328q^{40}+1057q^{38}+332q^{36}-802q^{34}-1451q^{32}-100q^{30}+1682q^{28}+1379q^{26}-426q^{24}-2225q^{22}-1219q^{20}+1349q^{18}+2197q^{16}+731q^{14}-1863q^{12}-2048q^{10}+93q^{8}+1859q^{6}+1649q^{4}-567q^{2}-1820-1039q^{-2}+727q^{-4}+1697q^{-6}+650q^{-8}-974q^{-10}-1531q^{-12}-298q^{-14}+1299q^{-16}+1376q^{-18}-222q^{-20}-1686q^{-22}-987q^{-24}+895q^{-26}+1871q^{-28}+425q^{-30}-1709q^{-32}-1618q^{-34}+319q^{-36}+2168q^{-38}+1243q^{-40}-1262q^{-42}-2098q^{-44}-697q^{-46}+1807q^{-48}+1980q^{-50}-132q^{-52}-1836q^{-54}-1671q^{-56}+615q^{-58}+1866q^{-60}+1011q^{-62}-689q^{-64}-1701q^{-66}-570q^{-68}+822q^{-70}+1180q^{-72}+392q^{-74}-805q^{-76}-790q^{-78}-128q^{-80}+522q^{-82}+569q^{-84}-38q^{-86}-324q^{-88}-294q^{-90}+10q^{-92}+230q^{-94}+108q^{-96}-9q^{-98}-104q^{-100}-59q^{-102}+31q^{-104}+28q^{-106}+22q^{-108}-11q^{-110}-15q^{-112}+2q^{-114}+q^{-116}+4q^{-118}-q^{-120}-2q^{-122}+q^{-124}$
5	$q^{125}-3q^{123}+q^{121}+6q^{119}-5q^{117}-4q^{115}+9q^{109}+14q^{107}-10q^{105}-33q^{103}-6q^{101}+34q^{99}+47q^{97}-6q^{95}-81q^{93}-111q^{91}+15q^{89}+268q^{87}+288q^{85}-72q^{83}-580q^{81}-712q^{79}-52q^{77}+1112q^{75}+1650q^{73}+526q^{71}-1751q^{69}-3171q^{67}-1795q^{65}+2062q^{63}+5301q^{61}+4267q^{59}-1540q^{57}-7597q^{55}-7883q^{53}-568q^{51}+9155q^{49}+12358q^{47}+4502q^{45}-9089q^{43}-16513q^{41}-9915q^{39}+6591q^{37}+19104q^{35}+15772q^{33}-1810q^{31}-19011q^{29}-20551q^{27}-4382q^{25}+15920q^{23}+22884q^{21}+10544q^{19}-10418q^{17}-22187q^{15}-15204q^{13}+3871q^{11}+18666q^{9}+17460q^{7}+2381q^{5}-13423q^{3}-17290q-7168q^{-1}+7751q^{-3}+15294q^{-5}+10139q^{-7}-2690q^{-9}-12540q^{-11}-11599q^{-13}-1091q^{-15}+9934q^{-17}+12136q^{-19}+3701q^{-21}-8012q^{-23}-12584q^{-25}-5544q^{-27}+6935q^{-29}+13321q^{-31}+7258q^{-33}-6222q^{-35}-14583q^{-37}-9386q^{-39}+5327q^{-41}+16022q^{-43}+12178q^{-45}-3551q^{-47}-17039q^{-49}-15465q^{-51}+462q^{-53}+16824q^{-55}+18717q^{-57}+3913q^{-59}-14777q^{-61}-20928q^{-63}-9066q^{-65}+10574q^{-67}+21240q^{-69}+13995q^{-71}-4718q^{-73}-19008q^{-75}-17365q^{-77}-1874q^{-79}+14294q^{-81}+18273q^{-83}+7721q^{-85}-8016q^{-87}-16293q^{-89}-11451q^{-91}+1500q^{-93}+11978q^{-95}+12437q^{-97}+3674q^{-99}-6636q^{-101}-10727q^{-103}-6539q^{-105}+1662q^{-107}+7387q^{-109}+6961q^{-111}+1732q^{-113}-3723q^{-115}-5546q^{-117}-3197q^{-119}+819q^{-121}+3395q^{-123}+3099q^{-125}+784q^{-127}-1467q^{-129}-2150q^{-131}-1229q^{-133}+225q^{-135}+1113q^{-137}+1012q^{-139}+274q^{-141}-394q^{-143}-576q^{-145}-322q^{-147}+35q^{-149}+243q^{-151}+216q^{-153}+51q^{-155}-74q^{-157}-91q^{-159}-44q^{-161}+4q^{-163}+35q^{-165}+25q^{-167}-3q^{-169}-9q^{-171}-4q^{-173}-2q^{-175}+q^{-177}+4q^{-179}-q^{-181}-2q^{-183}+q^{-185}$

A2 Invariants.

Weight	Invariant
1,0	$q^{12}-2q^{10}+q^{8}+q^{6}-2q^{4}+4q^{2}-1+2q^{-2}-2q^{-6}+2q^{-8}-3q^{-10}+q^{-12}+q^{-14}-q^{-16}+q^{-18}$
1,1	$q^{36}-6q^{34}+18q^{32}-38q^{30}+75q^{28}-142q^{26}+236q^{24}-352q^{22}+504q^{20}-680q^{18}+848q^{16}-992q^{14}+1085q^{12}-1088q^{10}+968q^{8}-708q^{6}+319q^{4}+188q^{2}-754+1320q^{-2}-1804q^{-4}+2160q^{-6}-2332q^{-8}+2294q^{-10}-2065q^{-12}+1666q^{-14}-1152q^{-16}+570q^{-18}+3q^{-20}-498q^{-22}+892q^{-24}-1134q^{-26}+1220q^{-28}-1168q^{-30}+1020q^{-32}-818q^{-34}+595q^{-36}-400q^{-38}+246q^{-40}-136q^{-42}+68q^{-44}-30q^{-46}+12q^{-48}-4q^{-50}+q^{-52}$
2,0	$q^{32}-2q^{30}-q^{28}+6q^{26}-q^{24}-8q^{22}+q^{20}+12q^{18}-2q^{16}-17q^{14}+6q^{12}+18q^{10}-9q^{8}-13q^{6}+13q^{4}+7q^{2}-8-3q^{-2}+9q^{-4}-4q^{-6}-7q^{-8}+11q^{-10}-13q^{-14}+6q^{-16}+14q^{-18}-10q^{-20}-10q^{-22}+10q^{-24}+11q^{-26}-9q^{-28}-10q^{-30}+9q^{-32}+5q^{-34}-6q^{-36}-4q^{-38}+2q^{-40}+3q^{-42}-q^{-44}-q^{-46}+q^{-48}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{28}-3q^{26}+9q^{22}-10q^{20}-4q^{18}+21q^{16}-17q^{14}-11q^{12}+29q^{10}-16q^{8}-14q^{6}+28q^{4}-5q^{2}-9+11q^{-2}+6q^{-4}-5q^{-6}-14q^{-8}+11q^{-10}+6q^{-12}-27q^{-14}+13q^{-16}+17q^{-18}-26q^{-20}+12q^{-22}+15q^{-24}-19q^{-26}+7q^{-28}+6q^{-30}-9q^{-32}+3q^{-34}+q^{-36}-2q^{-38}+q^{-40}$
1,0,0	$q^{15}-2q^{13}+2q^{11}-2q^{9}+2q^{7}-2q^{5}+4q^{3}+2q^{-1}+q^{-3}-q^{-5}-3q^{-9}+2q^{-11}-3q^{-13}+2q^{-15}-q^{-17}+2q^{-19}-q^{-21}+q^{-23}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{34}-2q^{32}-2q^{30}+6q^{28}-8q^{24}+3q^{22}+12q^{20}-5q^{18}-16q^{16}+8q^{14}+15q^{12}-16q^{10}-13q^{8}+25q^{6}+7q^{4}-17q^{2}+11+21q^{-2}-10q^{-4}-11q^{-6}+15q^{-8}-2q^{-10}-28q^{-12}+3q^{-14}+18q^{-16}-20q^{-18}-12q^{-20}+26q^{-22}+8q^{-24}-17q^{-26}+3q^{-28}+17q^{-30}-3q^{-32}-12q^{-34}+5q^{-36}+6q^{-38}-7q^{-40}-2q^{-42}+4q^{-44}-q^{-46}-q^{-48}+q^{-50}$
1,0,0,0	$q^{18}-2q^{16}+2q^{14}-q^{12}-q^{10}+2q^{8}-2q^{6}+4q^{4}+3+q^{-2}+q^{-4}-q^{-6}-q^{-8}-q^{-10}-3q^{-12}+2q^{-14}-3q^{-16}+2q^{-18}+2q^{-24}-q^{-26}+q^{-28}$

B2 Invariants.

Weight	Invariant
0,1	$q^{28}-3q^{26}+6q^{24}-11q^{22}+18q^{20}-24q^{18}+29q^{16}-33q^{14}+33q^{12}-29q^{10}+20q^{8}-6q^{6}-8q^{4}+27q^{2}-41+55q^{-2}-62q^{-4}+65q^{-6}-60q^{-8}+49q^{-10}-36q^{-12}+19q^{-14}-3q^{-16}-13q^{-18}+24q^{-20}-30q^{-22}+33q^{-24}-31q^{-26}+27q^{-28}-22q^{-30}+15q^{-32}-9q^{-34}+5q^{-36}-2q^{-38}+q^{-40}$
1,0	$q^{46}-3q^{42}-3q^{40}+3q^{38}+10q^{36}+3q^{34}-14q^{32}-14q^{30}+9q^{28}+25q^{26}+5q^{24}-28q^{22}-22q^{20}+18q^{18}+33q^{16}-2q^{14}-34q^{12}-12q^{10}+27q^{8}+23q^{6}-17q^{4}-23q^{2}+10+26q^{-2}-q^{-4}-23q^{-6}-4q^{-8}+21q^{-10}+7q^{-12}-21q^{-14}-13q^{-16}+19q^{-18}+19q^{-20}-17q^{-22}-30q^{-24}+6q^{-26}+36q^{-28}+10q^{-30}-31q^{-32}-25q^{-34}+19q^{-36}+33q^{-38}-q^{-40}-27q^{-42}-12q^{-44}+16q^{-46}+16q^{-48}-5q^{-50}-12q^{-52}-2q^{-54}+6q^{-56}+3q^{-58}-2q^{-60}-2q^{-62}+q^{-66}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{38}-3q^{36}+3q^{34}-4q^{32}+10q^{30}-14q^{28}+14q^{26}-17q^{24}+25q^{22}-26q^{20}+22q^{18}-26q^{16}+26q^{14}-18q^{12}+10q^{10}-7q^{8}+19q^{4}-20q^{2}+32-37q^{-2}+49q^{-4}-46q^{-6}+47q^{-8}-53q^{-10}+42q^{-12}-39q^{-14}+28q^{-16}-27q^{-18}+11q^{-20}-q^{-22}-4q^{-24}+13q^{-26}-18q^{-28}+27q^{-30}-23q^{-32}+26q^{-34}-25q^{-36}+23q^{-38}-19q^{-40}+15q^{-42}-13q^{-44}+8q^{-46}-5q^{-48}+3q^{-50}-2q^{-52}+q^{-54}$

G2 Invariants.

Weight

Invariant

1,0

$q^{66}-3q^{64}+6q^{62}-10q^{60}+10q^{58}-8q^{56}+q^{54}+17q^{52}-36q^{50}+57q^{48}-65q^{46}+49q^{44}-16q^{42}-40q^{40}+104q^{38}-150q^{36}+163q^{34}-127q^{32}+39q^{30}+74q^{28}-178q^{26}+236q^{24}-219q^{22}+126q^{20}+8q^{18}-137q^{16}+206q^{14}-179q^{12}+77q^{10}+65q^{8}-170q^{6}+187q^{4}-97q^{2}-59+222q^{-2}-307q^{-4}+278q^{-6}-132q^{-8}-74q^{-10}+266q^{-12}-372q^{-14}+354q^{-16}-222q^{-18}+24q^{-20}+161q^{-22}-278q^{-24}+285q^{-26}-190q^{-28}+35q^{-30}+109q^{-32}-192q^{-34}+172q^{-36}-65q^{-38}-79q^{-40}+196q^{-42}-230q^{-44}+158q^{-46}-13q^{-48}-147q^{-50}+258q^{-52}-268q^{-54}+189q^{-56}-52q^{-58}-88q^{-60}+178q^{-62}-195q^{-64}+151q^{-66}-70q^{-68}-8q^{-70}+58q^{-72}-75q^{-74}+64q^{-76}-38q^{-78}+15q^{-80}+3q^{-82}-11q^{-84}+10q^{-86}-9q^{-88}+5q^{-90}-2q^{-92}+q^{-94}$

.

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 86"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+9t^{2}-19t+25-19t^{-1}+9t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 85, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-3q^{5}+6q^{4}-10q^{3}+13q^{2}-14q+14-11q^{-1}+8q^{-2}-4q^{-3}+q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-1}+2z^{9}a^{-3}+10z^{8}a^{-2}+4z^{8}a^{-4}+6z^{8}+9az^{7}+9z^{7}a^{-1}+3z^{7}a^{-3}+3z^{7}a^{-5}+8a^{2}z^{6}-21z^{6}a^{-2}-10z^{6}a^{-4}+z^{6}a^{-6}-2z^{6}+4a^{3}z^{5}-11az^{5}-23z^{5}a^{-1}-17z^{5}a^{-3}-9z^{5}a^{-5}+a^{4}z^{4}-9a^{2}z^{4}+13z^{4}a^{-2}+7z^{4}a^{-4}-3z^{4}a^{-6}-7z^{4}-2a^{3}z^{3}+4az^{3}+13z^{3}a^{-1}+15z^{3}a^{-3}+8z^{3}a^{-5}+3a^{2}z^{2}-6z^{2}a^{-2}-2z^{2}a^{-4}+2z^{2}a^{-6}+z^{2}-az-3za^{-1}-4za^{-3}-2za^{-5}+2a^{-2}+a^{-4}+2$