10 103

Polynomial invariants

Alexander polynomial	$2t^{3}-8t^{2}+17t-21+17t^{-1}-8t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+4z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{5,t+1\}$
Determinant and Signature	{ 75, -2 }
Jones polynomial	$-q^{2}+3q-6+10q^{-1}-11q^{-2}+13q^{-3}-12q^{-4}+9q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{6}-2z^{2}a^{6}-a^{6}+z^{6}a^{4}+3z^{4}a^{4}+3z^{2}a^{4}+z^{6}a^{2}+3z^{4}a^{2}+4z^{2}a^{2}+3a^{2}-z^{4}-2z^{2}-1$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-2z^{3}a^{9}+3z^{6}a^{8}-6z^{4}a^{8}+z^{2}a^{8}+5z^{7}a^{7}-12z^{5}a^{7}+10z^{3}a^{7}-4za^{7}+5z^{8}a^{6}-12z^{6}a^{6}+13z^{4}a^{6}-6z^{2}a^{6}+a^{6}+2z^{9}a^{5}+3z^{7}a^{5}-16z^{5}a^{5}+21z^{3}a^{5}-6za^{5}+9z^{8}a^{4}-23z^{6}a^{4}+25z^{4}a^{4}-8z^{2}a^{4}+2z^{9}a^{3}+2z^{7}a^{3}-9z^{5}a^{3}+9z^{3}a^{3}-2za^{3}+4z^{8}a^{2}-5z^{6}a^{2}+2z^{2}a^{2}-3a^{2}+4z^{7}a-5z^{5}a-2z^{3}a+za+3z^{6}-6z^{4}+3z^{2}-1+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$
The A2 invariant	$-q^{24}+q^{22}-q^{20}-q^{18}+2q^{16}-3q^{14}+q^{12}+q^{8}+4q^{6}-q^{4}+3q^{2}-1-q^{-2}+q^{-4}-q^{-6}$
The G2 invariant	$q^{128}-2q^{126}+4q^{124}-7q^{122}+7q^{120}-6q^{118}+12q^{114}-22q^{112}+35q^{110}-42q^{108}+35q^{106}-15q^{104}-25q^{102}+72q^{100}-108q^{98}+118q^{96}-88q^{94}+18q^{92}+74q^{90}-153q^{88}+184q^{86}-149q^{84}+53q^{82}+55q^{80}-142q^{78}+160q^{76}-106q^{74}+11q^{72}+92q^{70}-143q^{68}+115q^{66}-28q^{64}-92q^{62}+180q^{60}-204q^{58}+150q^{56}-38q^{54}-94q^{52}+208q^{50}-252q^{48}+218q^{46}-117q^{44}-25q^{42}+148q^{40}-206q^{38}+189q^{36}-98q^{34}-12q^{32}+112q^{30}-142q^{28}+98q^{26}-2q^{24}-100q^{22}+159q^{20}-140q^{18}+58q^{16}+52q^{14}-136q^{12}+176q^{10}-149q^{8}+80q^{6}+3q^{4}-78q^{2}+111-107q^{-2}+78q^{-4}-34q^{-6}-3q^{-8}+28q^{-10}-42q^{-12}+39q^{-14}-29q^{-16}+15q^{-18}-3q^{-20}-6q^{-22}+8q^{-24}-8q^{-26}+5q^{-28}-2q^{-30}+q^{-32}$

Further Quantum Invariants

Further quantum knot invariants for 10_103.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+2q^{15}-3q^{13}+3q^{11}-3q^{9}+q^{7}+2q^{5}-q^{3}+4q-3q^{-1}+2q^{-3}-q^{-5}$
2	$q^{48}-2q^{46}+6q^{42}-8q^{40}-4q^{38}+19q^{36}-10q^{34}-19q^{32}+26q^{30}-27q^{26}+20q^{24}+11q^{22}-20q^{20}-q^{18}+13q^{16}-q^{14}-19q^{12}+13q^{10}+19q^{8}-26q^{6}+2q^{4}+27q^{2}-19-9q^{-2}+19q^{-4}-5q^{-6}-7q^{-8}+6q^{-10}-q^{-12}-2q^{-14}+q^{-16}$
3	$-q^{93}+2q^{91}-3q^{87}+7q^{83}+q^{81}-18q^{79}-5q^{77}+34q^{75}+22q^{73}-47q^{71}-58q^{69}+48q^{67}+103q^{65}-24q^{63}-143q^{61}-25q^{59}+165q^{57}+82q^{55}-158q^{53}-132q^{51}+125q^{49}+164q^{47}-83q^{45}-170q^{43}+36q^{41}+153q^{39}+16q^{37}-131q^{35}-50q^{33}+94q^{31}+91q^{29}-64q^{27}-128q^{25}+20q^{23}+159q^{21}+25q^{19}-175q^{17}-73q^{15}+168q^{13}+124q^{11}-134q^{9}-151q^{7}+82q^{5}+160q^{3}-29q-131q^{-1}-19q^{-3}+95q^{-5}+37q^{-7}-53q^{-9}-34q^{-11}+22q^{-13}+22q^{-15}-8q^{-17}-12q^{-19}+5q^{-21}+4q^{-23}-3q^{-25}-2q^{-27}+q^{-29}+2q^{-31}-q^{-33}$
4	$q^{152}-2q^{150}+3q^{146}-3q^{144}+q^{142}-5q^{140}+7q^{138}+16q^{136}-17q^{134}-17q^{132}-28q^{130}+31q^{128}+93q^{126}+6q^{124}-76q^{122}-177q^{120}-34q^{118}+246q^{116}+248q^{114}+47q^{112}-424q^{110}-459q^{108}+95q^{106}+597q^{104}+677q^{102}-224q^{100}-974q^{98}-687q^{96}+363q^{94}+1355q^{92}+666q^{90}-793q^{88}-1447q^{86}-557q^{84}+1248q^{82}+1443q^{80}+59q^{78}-1390q^{76}-1282q^{74}+494q^{72}+1410q^{70}+729q^{68}-738q^{66}-1277q^{64}-178q^{62}+874q^{60}+884q^{58}-132q^{56}-910q^{54}-543q^{52}+378q^{50}+870q^{48}+316q^{46}-600q^{44}-909q^{42}-79q^{40}+938q^{38}+866q^{36}-236q^{34}-1315q^{32}-732q^{30}+778q^{28}+1434q^{26}+479q^{24}-1294q^{22}-1398q^{20}+57q^{18}+1430q^{16}+1246q^{14}-530q^{12}-1410q^{10}-784q^{8}+623q^{6}+1304q^{4}+351q^{2}-631-903q^{-2}-218q^{-4}+619q^{-6}+523q^{-8}+85q^{-10}-394q^{-12}-359q^{-14}+53q^{-16}+187q^{-18}+189q^{-20}-26q^{-22}-128q^{-24}-41q^{-26}-7q^{-28}+58q^{-30}+17q^{-32}-16q^{-34}+2q^{-36}-13q^{-38}+6q^{-40}-q^{-42}-4q^{-44}+6q^{-46}-q^{-48}+2q^{-50}-q^{-52}-2q^{-54}+q^{-56}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{24}+q^{22}-q^{20}-q^{18}+2q^{16}-3q^{14}+q^{12}+q^{8}+4q^{6}-q^{4}+3q^{2}-1-q^{-2}+q^{-4}-q^{-6}$
1,1	$q^{68}-4q^{66}+10q^{64}-22q^{62}+46q^{60}-78q^{58}+124q^{56}-202q^{54}+301q^{52}-406q^{50}+528q^{48}-644q^{46}+711q^{44}-702q^{42}+606q^{40}-428q^{38}+145q^{36}+198q^{34}-556q^{32}+896q^{30}-1167q^{28}+1354q^{26}-1420q^{24}+1360q^{22}-1198q^{20}+912q^{18}-578q^{16}+228q^{14}+115q^{12}-388q^{10}+596q^{8}-672q^{6}+682q^{4}-632q^{2}+534-418q^{-2}+311q^{-4}-224q^{-6}+146q^{-8}-92q^{-10}+54q^{-12}-28q^{-14}+12q^{-16}-4q^{-18}+q^{-20}$
2,0	$q^{62}-q^{60}+2q^{56}-q^{54}-2q^{52}-q^{50}+7q^{48}+q^{46}-11q^{44}+9q^{40}-q^{38}-13q^{36}+2q^{34}+12q^{32}-2q^{30}-10q^{28}+5q^{26}+3q^{24}-11q^{22}+3q^{20}+5q^{18}-3q^{16}+12q^{12}+3q^{10}-10q^{8}+4q^{6}+14q^{4}-6q^{2}-11+7q^{-2}+7q^{-4}-5q^{-6}-6q^{-8}+2q^{-10}+3q^{-12}-2q^{-14}-q^{-16}+q^{-18}$

A3 Invariants.

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

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{68}-q^{66}-2q^{64}+4q^{62}+q^{60}-8q^{58}+4q^{56}+11q^{54}-6q^{52}-8q^{50}+12q^{48}+9q^{46}-18q^{44}-10q^{42}+14q^{40}-6q^{38}-22q^{36}+8q^{34}+10q^{32}-18q^{30}-2q^{28}+15q^{26}-3q^{24}-9q^{22}+17q^{20}+19q^{18}-9q^{16}+2q^{14}+22q^{12}+q^{10}-16q^{8}+5q^{6}+7q^{4}-8q^{2}-8+2q^{-2}+2q^{-4}-4q^{-6}-q^{-8}+3q^{-10}-q^{-14}+q^{-16}$
1,0,0,0	$-q^{38}+q^{36}-2q^{34}-2q^{28}+2q^{26}-3q^{24}+q^{22}-q^{20}+q^{16}+2q^{14}+3q^{12}+2q^{10}+4q^{8}-q^{6}+3q^{4}-2q^{2}-2q^{-4}+q^{-6}-q^{-8}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{54}+2q^{52}-4q^{50}+8q^{48}-12q^{46}+16q^{44}-23q^{42}+25q^{40}-26q^{38}+23q^{36}-16q^{34}+9q^{32}+3q^{30}-16q^{28}+30q^{26}-42q^{24}+47q^{22}-50q^{20}+47q^{18}-42q^{16}+32q^{14}-16q^{12}+6q^{10}+9q^{8}-14q^{6}+24q^{4}-26q^{2}+25-23q^{-2}+17q^{-4}-12q^{-6}+8q^{-8}-5q^{-10}+2q^{-12}-q^{-14}$
1,0	$q^{88}-2q^{84}-2q^{82}+2q^{80}+6q^{78}-9q^{74}-6q^{72}+9q^{70}+17q^{68}-3q^{66}-23q^{64}-10q^{62}+21q^{60}+22q^{58}-12q^{56}-28q^{54}-3q^{52}+25q^{50}+12q^{48}-19q^{46}-17q^{44}+11q^{42}+17q^{40}-7q^{38}-18q^{36}+q^{34}+17q^{32}+2q^{30}-17q^{28}-6q^{26}+18q^{24}+14q^{22}-14q^{20}-16q^{18}+13q^{16}+28q^{14}-q^{12}-26q^{10}-12q^{8}+23q^{6}+21q^{4}-10q^{2}-23-4q^{-2}+16q^{-4}+9q^{-6}-7q^{-8}-10q^{-10}-q^{-12}+6q^{-14}+3q^{-16}-2q^{-18}-2q^{-20}+q^{-24}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{74}-2q^{72}+2q^{70}-4q^{68}+7q^{66}-9q^{64}+11q^{62}-13q^{60}+19q^{58}-20q^{56}+20q^{54}-19q^{52}+19q^{50}-17q^{48}+6q^{46}-7q^{44}-q^{42}+8q^{40}-21q^{38}+22q^{36}-29q^{34}+37q^{32}-38q^{30}+36q^{28}-38q^{26}+38q^{24}-27q^{22}+26q^{20}-17q^{18}+15q^{16}+4q^{14}+10q^{10}-14q^{8}+20q^{6}-20q^{4}+18q^{2}-22+17q^{-2}-15q^{-4}+10q^{-6}-10q^{-8}+7q^{-10}-4q^{-12}+3q^{-14}-2q^{-16}+q^{-18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{128}-2q^{126}+4q^{124}-7q^{122}+7q^{120}-6q^{118}+12q^{114}-22q^{112}+35q^{110}-42q^{108}+35q^{106}-15q^{104}-25q^{102}+72q^{100}-108q^{98}+118q^{96}-88q^{94}+18q^{92}+74q^{90}-153q^{88}+184q^{86}-149q^{84}+53q^{82}+55q^{80}-142q^{78}+160q^{76}-106q^{74}+11q^{72}+92q^{70}-143q^{68}+115q^{66}-28q^{64}-92q^{62}+180q^{60}-204q^{58}+150q^{56}-38q^{54}-94q^{52}+208q^{50}-252q^{48}+218q^{46}-117q^{44}-25q^{42}+148q^{40}-206q^{38}+189q^{36}-98q^{34}-12q^{32}+112q^{30}-142q^{28}+98q^{26}-2q^{24}-100q^{22}+159q^{20}-140q^{18}+58q^{16}+52q^{14}-136q^{12}+176q^{10}-149q^{8}+80q^{6}+3q^{4}-78q^{2}+111-107q^{-2}+78q^{-4}-34q^{-6}-3q^{-8}+28q^{-10}-42q^{-12}+39q^{-14}-29q^{-16}+15q^{-18}-3q^{-20}-6q^{-22}+8q^{-24}-8q^{-26}+5q^{-28}-2q^{-30}+q^{-32}$

.

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["10 103"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-8t^{2}+17t-21+17t^{-1}-8t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{5,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 75, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{2}+3q-6+10q^{-1}-11q^{-2}+13q^{-3}-12q^{-4}+9q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{9}-2z^{3}a^{9}+3z^{6}a^{8}-6z^{4}a^{8}+z^{2}a^{8}+5z^{7}a^{7}-12z^{5}a^{7}+10z^{3}a^{7}-4za^{7}+5z^{8}a^{6}-12z^{6}a^{6}+13z^{4}a^{6}-6z^{2}a^{6}+a^{6}+2z^{9}a^{5}+3z^{7}a^{5}-16z^{5}a^{5}+21z^{3}a^{5}-6za^{5}+9z^{8}a^{4}-23z^{6}a^{4}+25z^{4}a^{4}-8z^{2}a^{4}+2z^{9}a^{3}+2z^{7}a^{3}-9z^{5}a^{3}+9z^{3}a^{3}-2za^{3}+4z^{8}a^{2}-5z^{6}a^{2}+2z^{2}a^{2}-3a^{2}+4z^{7}a-5z^{5}a-2z^{3}a+za+3z^{6}-6z^{4}+3z^{2}-1+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$