10 44

Polynomial invariants

Alexander polynomial	$t^{3}-7t^{2}+19t-25+19t^{-1}-7t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 79, -2 }
Jones polynomial	$q^{3}-3q^{2}+6q-9+12q^{-1}-13q^{-2}+13q^{-3}-10q^{-4}+7q^{-5}-4q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}-2z^{4}a^{4}-3z^{2}a^{4}-a^{4}+z^{6}a^{2}+3z^{4}a^{2}+5z^{2}a^{2}+3a^{2}-2z^{4}-4z^{2}-2+z^{2}a^{-2}+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{3}z^{9}+az^{9}+4a^{4}z^{8}+7a^{2}z^{8}+3z^{8}+7a^{5}z^{7}+12a^{3}z^{7}+8az^{7}+3z^{7}a^{-1}+7a^{6}z^{6}+5a^{4}z^{6}-7a^{2}z^{6}+z^{6}a^{-2}-4z^{6}+4a^{7}z^{5}-6a^{5}z^{5}-27a^{3}z^{5}-26az^{5}-9z^{5}a^{-1}+a^{8}z^{4}-8a^{6}z^{4}-18a^{4}z^{4}-12a^{2}z^{4}-3z^{4}a^{-2}-6z^{4}-3a^{7}z^{3}+15a^{3}z^{3}+20az^{3}+8z^{3}a^{-1}+3a^{6}z^{2}+10a^{4}z^{2}+13a^{2}z^{2}+3z^{2}a^{-2}+9z^{2}-2a^{3}z-4az-2za^{-1}-a^{4}-3a^{2}-a^{-2}-2$
The A2 invariant	$q^{22}-q^{20}-2q^{18}+2q^{16}-2q^{14}+q^{12}+2q^{10}-q^{8}+3q^{6}-2q^{4}+2q^{2}-2q^{-2}+2q^{-4}-q^{-6}+q^{-10}$
The G2 invariant	$q^{114}-3q^{112}+6q^{110}-10q^{108}+9q^{106}-6q^{104}-2q^{102}+19q^{100}-33q^{98}+50q^{96}-54q^{94}+36q^{92}-5q^{90}-41q^{88}+88q^{86}-122q^{84}+127q^{82}-93q^{80}+23q^{78}+63q^{76}-138q^{74}+176q^{72}-161q^{70}+92q^{68}-2q^{66}-90q^{64}+139q^{62}-122q^{60}+57q^{58}+35q^{56}-103q^{54}+114q^{52}-60q^{50}-44q^{48}+151q^{46}-212q^{44}+199q^{42}-102q^{40}-43q^{38}+188q^{36}-273q^{34}+272q^{32}-185q^{30}+40q^{28}+103q^{26}-197q^{24}+219q^{22}-156q^{20}+50q^{18}+60q^{16}-124q^{14}+119q^{12}-52q^{10}-43q^{8}+124q^{6}-153q^{4}+113q^{2}-20-91q^{-2}+177q^{-4}-199q^{-6}+154q^{-8}-64q^{-10}-43q^{-12}+121q^{-14}-154q^{-16}+137q^{-18}-81q^{-20}+15q^{-22}+36q^{-24}-63q^{-26}+63q^{-28}-44q^{-30}+23q^{-32}-q^{-34}-10q^{-36}+12q^{-38}-10q^{-40}+6q^{-42}-2q^{-44}+q^{-46}$

Further Quantum Invariants

Further quantum knot invariants for 10_44.

A1 Invariants.

Weight	Invariant
1	$q^{15}-3q^{13}+3q^{11}-3q^{9}+3q^{7}-q^{3}+3q-3q^{-1}+3q^{-3}-2q^{-5}+q^{-7}$
2	$q^{42}-3q^{40}+9q^{36}-11q^{34}-3q^{32}+22q^{30}-19q^{28}-10q^{26}+31q^{24}-16q^{22}-18q^{20}+24q^{18}-16q^{14}+3q^{12}+16q^{10}-5q^{8}-19q^{6}+21q^{4}+9q^{2}-30+15q^{-2}+18q^{-4}-26q^{-6}+3q^{-8}+17q^{-10}-12q^{-12}-3q^{-14}+8q^{-16}-2q^{-18}-2q^{-20}+q^{-22}$
3	$q^{81}-3q^{79}+6q^{75}+q^{73}-11q^{71}-6q^{69}+26q^{67}+6q^{65}-41q^{63}-15q^{61}+64q^{59}+29q^{57}-91q^{55}-50q^{53}+113q^{51}+82q^{49}-126q^{47}-111q^{45}+118q^{43}+142q^{41}-95q^{39}-155q^{37}+49q^{35}+154q^{33}-4q^{31}-132q^{29}-47q^{27}+97q^{25}+92q^{23}-57q^{21}-121q^{19}+12q^{17}+142q^{15}+33q^{13}-148q^{11}-74q^{9}+144q^{7}+111q^{5}-124q^{3}-141q+94q^{-1}+157q^{-3}-52q^{-5}-159q^{-7}+12q^{-9}+142q^{-11}+23q^{-13}-110q^{-15}-46q^{-17}+74q^{-19}+53q^{-21}-41q^{-23}-45q^{-25}+15q^{-27}+32q^{-29}-q^{-31}-19q^{-33}-3q^{-35}+8q^{-37}+3q^{-39}-2q^{-41}-2q^{-43}+q^{-45}$
4	$q^{132}-3q^{130}+6q^{126}-2q^{124}+q^{122}-14q^{120}+4q^{118}+26q^{116}-12q^{114}-6q^{112}-46q^{110}+24q^{108}+96q^{106}-24q^{104}-65q^{102}-143q^{100}+74q^{98}+278q^{96}+19q^{94}-203q^{92}-394q^{90}+76q^{88}+597q^{86}+250q^{84}-323q^{82}-816q^{80}-125q^{78}+878q^{76}+691q^{74}-188q^{72}-1169q^{70}-566q^{68}+799q^{66}+1059q^{64}+256q^{62}-1085q^{60}-953q^{58}+282q^{56}+1003q^{54}+719q^{52}-529q^{50}-968q^{48}-343q^{46}+539q^{44}+887q^{42}+163q^{40}-642q^{38}-779q^{36}-22q^{34}+789q^{32}+703q^{30}-230q^{28}-983q^{26}-474q^{24}+563q^{22}+1054q^{20}+174q^{18}-1009q^{16}-836q^{14}+222q^{12}+1213q^{10}+602q^{8}-783q^{6}-1065q^{4}-272q^{2}+1061+956q^{-2}-277q^{-4}-984q^{-6}-740q^{-8}+560q^{-10}+978q^{-12}+273q^{-14}-541q^{-16}-866q^{-18}-9q^{-20}+608q^{-22}+500q^{-24}-27q^{-26}-580q^{-28}-278q^{-30}+148q^{-32}+348q^{-34}+208q^{-36}-201q^{-38}-208q^{-40}-71q^{-42}+106q^{-44}+157q^{-46}-10q^{-48}-61q^{-50}-67q^{-52}-2q^{-54}+53q^{-56}+14q^{-58}-q^{-60}-19q^{-62}-10q^{-64}+8q^{-66}+3q^{-68}+3q^{-70}-2q^{-72}-2q^{-74}+q^{-76}$
5	$q^{195}-3q^{193}+6q^{189}-2q^{187}-2q^{185}-2q^{183}-4q^{181}+4q^{179}+14q^{177}-2q^{175}-24q^{173}-14q^{171}+19q^{169}+49q^{167}+26q^{165}-39q^{163}-118q^{161}-86q^{159}+117q^{157}+261q^{155}+152q^{153}-180q^{151}-495q^{149}-384q^{147}+275q^{145}+907q^{143}+740q^{141}-311q^{139}-1429q^{137}-1394q^{135}+188q^{133}+2101q^{131}+2367q^{129}+194q^{127}-2761q^{125}-3626q^{123}-1019q^{121}+3211q^{119}+5121q^{117}+2297q^{115}-3264q^{113}-6516q^{111}-3991q^{109}+2658q^{107}+7562q^{105}+5861q^{103}-1406q^{101}-7874q^{99}-7558q^{97}-403q^{95}+7331q^{93}+8665q^{91}+2434q^{89}-5866q^{87}-8984q^{85}-4316q^{83}+3826q^{81}+8346q^{79}+5688q^{77}-1450q^{75}-6959q^{73}-6439q^{71}-772q^{69}+5090q^{67}+6497q^{65}+2661q^{63}-3064q^{61}-6100q^{59}-4108q^{57}+1207q^{55}+5447q^{53}+5118q^{51}+409q^{49}-4750q^{47}-5883q^{45}-1752q^{43}+4151q^{41}+6476q^{39}+2915q^{37}-3562q^{35}-7031q^{33}-4042q^{31}+2928q^{29}+7489q^{27}+5207q^{25}-2074q^{23}-7724q^{21}-6406q^{19}+881q^{17}+7578q^{15}+7529q^{13}+652q^{11}-6891q^{9}-8327q^{7}-2426q^{5}+5551q^{3}+8619q+4197q^{-1}-3690q^{-3}-8172q^{-5}-5610q^{-7}+1470q^{-9}+6965q^{-11}+6426q^{-13}+696q^{-15}-5152q^{-17}-6402q^{-19}-2458q^{-21}+3025q^{-23}+5584q^{-25}+3538q^{-27}-1005q^{-29}-4206q^{-31}-3791q^{-33}-565q^{-35}+2596q^{-37}+3342q^{-39}+1505q^{-41}-1144q^{-43}-2481q^{-45}-1767q^{-47}+92q^{-49}+1499q^{-51}+1549q^{-53}+494q^{-55}-690q^{-57}-1105q^{-59}-642q^{-61}+155q^{-63}+630q^{-65}+549q^{-67}+113q^{-69}-286q^{-71}-367q^{-73}-169q^{-75}+83q^{-77}+190q^{-79}+138q^{-81}+12q^{-83}-86q^{-85}-85q^{-87}-24q^{-89}+27q^{-91}+35q^{-93}+23q^{-95}-q^{-97}-19q^{-99}-10q^{-101}+q^{-103}+3q^{-105}+3q^{-107}+3q^{-109}-2q^{-111}-2q^{-113}+q^{-115}$

A2 Invariants.

Weight	Invariant
1,0	$q^{22}-q^{20}-2q^{18}+2q^{16}-2q^{14}+q^{12}+2q^{10}-q^{8}+3q^{6}-2q^{4}+2q^{2}-2q^{-2}+2q^{-4}-q^{-6}+q^{-10}$
1,1	$q^{60}-6q^{58}+18q^{56}-38q^{54}+71q^{52}-128q^{50}+208q^{48}-304q^{46}+419q^{44}-552q^{42}+682q^{40}-776q^{38}+829q^{36}-818q^{34}+712q^{32}-508q^{30}+203q^{28}+168q^{26}-586q^{24}+1002q^{22}-1362q^{20}+1632q^{18}-1768q^{16}+1758q^{14}-1595q^{12}+1316q^{10}-940q^{8}+510q^{6}-77q^{4}-304q^{2}+606-814q^{-2}+911q^{-4}-900q^{-6}+814q^{-8}-680q^{-10}+525q^{-12}-370q^{-14}+242q^{-16}-144q^{-18}+76q^{-20}-36q^{-22}+14q^{-24}-4q^{-26}+q^{-28}$
2,0	$q^{56}-q^{54}-3q^{52}+q^{50}+5q^{48}+q^{46}-8q^{44}+2q^{42}+11q^{40}-5q^{38}-13q^{36}+5q^{34}+14q^{32}-9q^{30}-14q^{28}+10q^{26}+9q^{24}-11q^{22}-q^{20}+10q^{18}-3q^{16}-3q^{14}+8q^{12}-11q^{8}+4q^{6}+14q^{4}-8q^{2}-11+12q^{-2}+9q^{-4}-11q^{-6}-7q^{-8}+8q^{-10}+6q^{-12}-6q^{-14}-4q^{-16}+5q^{-18}+3q^{-20}-2q^{-22}-2q^{-24}+q^{-28}$

A3 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$q^{48}-3q^{46}+6q^{44}-10q^{42}+15q^{40}-20q^{38}+24q^{36}-27q^{34}+26q^{32}-24q^{30}+16q^{28}-6q^{26}-7q^{24}+21q^{22}-33q^{20}+45q^{18}-50q^{16}+54q^{14}-49q^{12}+43q^{10}-31q^{8}+18q^{6}-4q^{4}-8q^{2}+17-24q^{-2}+27q^{-4}-27q^{-6}+23q^{-8}-19q^{-10}+14q^{-12}-9q^{-14}+6q^{-16}-2q^{-18}+q^{-20}$
1,0	$q^{78}-3q^{74}-3q^{72}+3q^{70}+9q^{68}+2q^{66}-12q^{64}-11q^{62}+9q^{60}+21q^{58}+3q^{56}-24q^{54}-18q^{52}+14q^{50}+27q^{48}-2q^{46}-28q^{44}-11q^{42}+21q^{40}+19q^{38}-13q^{36}-20q^{34}+6q^{32}+21q^{30}-18q^{26}-4q^{24}+17q^{22}+7q^{20}-15q^{18}-10q^{16}+16q^{14}+16q^{12}-13q^{10}-24q^{8}+5q^{6}+29q^{4}+9q^{2}-25-22q^{-2}+14q^{-4}+28q^{-6}+q^{-8}-23q^{-10}-12q^{-12}+13q^{-14}+15q^{-16}-3q^{-18}-11q^{-20}-3q^{-22}+6q^{-24}+4q^{-26}-2q^{-28}-2q^{-30}+q^{-34}$

G2 Invariants.

Weight

Invariant

1,0

$q^{114}-3q^{112}+6q^{110}-10q^{108}+9q^{106}-6q^{104}-2q^{102}+19q^{100}-33q^{98}+50q^{96}-54q^{94}+36q^{92}-5q^{90}-41q^{88}+88q^{86}-122q^{84}+127q^{82}-93q^{80}+23q^{78}+63q^{76}-138q^{74}+176q^{72}-161q^{70}+92q^{68}-2q^{66}-90q^{64}+139q^{62}-122q^{60}+57q^{58}+35q^{56}-103q^{54}+114q^{52}-60q^{50}-44q^{48}+151q^{46}-212q^{44}+199q^{42}-102q^{40}-43q^{38}+188q^{36}-273q^{34}+272q^{32}-185q^{30}+40q^{28}+103q^{26}-197q^{24}+219q^{22}-156q^{20}+50q^{18}+60q^{16}-124q^{14}+119q^{12}-52q^{10}-43q^{8}+124q^{6}-153q^{4}+113q^{2}-20-91q^{-2}+177q^{-4}-199q^{-6}+154q^{-8}-64q^{-10}-43q^{-12}+121q^{-14}-154q^{-16}+137q^{-18}-81q^{-20}+15q^{-22}+36q^{-24}-63q^{-26}+63q^{-28}-44q^{-30}+23q^{-32}-q^{-34}-10q^{-36}+12q^{-38}-10q^{-40}+6q^{-42}-2q^{-44}+q^{-46}$

.

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 44"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-7t^{2}+19t-25+19t^{-1}-7t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 79, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{2}a^{6}-2z^{4}a^{4}-3z^{2}a^{4}-a^{4}+z^{6}a^{2}+3z^{4}a^{2}+5z^{2}a^{2}+3a^{2}-2z^{4}-4z^{2}-2+z^{2}a^{-2}+a^{-2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{3}z^{9}+az^{9}+4a^{4}z^{8}+7a^{2}z^{8}+3z^{8}+7a^{5}z^{7}+12a^{3}z^{7}+8az^{7}+3z^{7}a^{-1}+7a^{6}z^{6}+5a^{4}z^{6}-7a^{2}z^{6}+z^{6}a^{-2}-4z^{6}+4a^{7}z^{5}-6a^{5}z^{5}-27a^{3}z^{5}-26az^{5}-9z^{5}a^{-1}+a^{8}z^{4}-8a^{6}z^{4}-18a^{4}z^{4}-12a^{2}z^{4}-3z^{4}a^{-2}-6z^{4}-3a^{7}z^{3}+15a^{3}z^{3}+20az^{3}+8z^{3}a^{-1}+3a^{6}z^{2}+10a^{4}z^{2}+13a^{2}z^{2}+3z^{2}a^{-2}+9z^{2}-2a^{3}z-4az-2za^{-1}-a^{4}-3a^{2}-a^{-2}-2$