10 41

Polynomial invariants

Alexander polynomial	$t^{3}-7t^{2}+17t-21+17t^{-1}-7t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 71, -2 }
Jones polynomial	$q^{3}-3q^{2}+6q-8+11q^{-1}-12q^{-2}+11q^{-3}-9q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}+a^{6}-2z^{4}a^{4}-4z^{2}a^{4}-2a^{4}+z^{6}a^{2}+3z^{4}a^{2}+4z^{2}a^{2}+2a^{2}-2z^{4}-4z^{2}-1+z^{2}a^{-2}+a^{-2}$
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^3 z^9+a z^9+3 a^4 z^8+6 a^2 z^8+3 z^8+5 a^5 z^7+8 a^3 z^7+6 a z^7+3 z^7 a^{-1} +5 a^6 z^6+4 a^4 z^6-7 a^2 z^6+z^6 a^{-2} -5 z^6+3 a^7 z^5-4 a^5 z^5-18 a^3 z^5-20 a z^5-9 z^5 a^{-1} +a^8 z^4-6 a^6 z^4-14 a^4 z^4-8 a^2 z^4-3 z^4 a^{-2} -4 z^4-3 a^7 z^3+a^5 z^3+10 a^3 z^3+13 a z^3+7 z^3 a^{-1} -a^8 z^2+4 a^6 z^2+10 a^4 z^2+9 a^2 z^2+3 z^2 a^{-2} +7 z^2+a^7 z-2 a^3 z-2 a z-z a^{-1} -a^6-2 a^4-2 a^2- a^{-2} -1}
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{22}-q^{18}+2 q^{16}-2 q^{14}+q^{10}-2 q^8+2 q^6-2 q^4+2 q^2+1- q^{-2} +2 q^{-4} - q^{-6} + q^{-10} }
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+5 q^{106}-4 q^{104}-2 q^{102}+12 q^{100}-21 q^{98}+30 q^{96}-32 q^{94}+22 q^{92}-3 q^{90}-23 q^{88}+55 q^{86}-74 q^{84}+80 q^{82}-61 q^{80}+20 q^{78}+31 q^{76}-80 q^{74}+112 q^{72}-114 q^{70}+80 q^{68}-22 q^{66}-44 q^{64}+90 q^{62}-97 q^{60}+69 q^{58}-13 q^{56}-47 q^{54}+74 q^{52}-64 q^{50}+9 q^{48}+67 q^{46}-125 q^{44}+139 q^{42}-91 q^{40}+101 q^{36}-178 q^{34}+196 q^{32}-153 q^{30}+59 q^{28}+49 q^{26}-132 q^{24}+169 q^{22}-140 q^{20}+70 q^{18}+11 q^{16}-77 q^{14}+96 q^{12}-70 q^{10}+13 q^8+57 q^6-97 q^4+94 q^2-42-35 q^{-2} +106 q^{-4} -142 q^{-6} +126 q^{-8} -69 q^{-10} -11 q^{-12} +83 q^{-14} -120 q^{-16} +119 q^{-18} -77 q^{-20} +22 q^{-22} +24 q^{-24} -54 q^{-26} +57 q^{-28} -43 q^{-30} +24 q^{-32} -2 q^{-34} -9 q^{-36} +12 q^{-38} -10 q^{-40} +6 q^{-42} -2 q^{-44} + q^{-46} }

Further Quantum Invariants

Further quantum knot invariants for 10_41.

A1 Invariants.

Weight	Invariant
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{15}-2 q^{13}+3 q^{11}-3 q^9+2 q^7-q^5-q^3+3 q-2 q^{-1} +3 q^{-3} -2 q^{-5} + q^{-7} }
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{42}-2 q^{40}+6 q^{36}-8 q^{34}-2 q^{32}+15 q^{30}-15 q^{28}-5 q^{26}+24 q^{24}-15 q^{22}-11 q^{20}+21 q^{18}-3 q^{16}-13 q^{14}+5 q^{12}+11 q^{10}-7 q^8-13 q^6+17 q^4+4 q^2-22+15 q^{-2} +12 q^{-4} -22 q^{-6} +5 q^{-8} +14 q^{-10} -12 q^{-12} -2 q^{-14} +8 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-22} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{81}-2 q^{79}+3 q^{75}+q^{73}-7 q^{71}-3 q^{69}+14 q^{67}+3 q^{65}-22 q^{63}-4 q^{61}+35 q^{59}+7 q^{57}-54 q^{55}-12 q^{53}+75 q^{51}+24 q^{49}-92 q^{47}-44 q^{45}+103 q^{43}+64 q^{41}-96 q^{39}-86 q^{37}+73 q^{35}+98 q^{33}-39 q^{31}-96 q^{29}-2 q^{27}+84 q^{25}+41 q^{23}-62 q^{21}-73 q^{19}+39 q^{17}+92 q^{15}-9 q^{13}-108 q^{11}-14 q^9+111 q^7+41 q^5-108 q^3-65 q+96 q^{-1} +87 q^{-3} -71 q^{-5} -103 q^{-7} +44 q^{-9} +103 q^{-11} -11 q^{-13} -92 q^{-15} -16 q^{-17} +70 q^{-19} +33 q^{-21} -45 q^{-23} -34 q^{-25} +20 q^{-27} +29 q^{-29} -4 q^{-31} -19 q^{-33} -2 q^{-35} +8 q^{-37} +3 q^{-39} -2 q^{-41} -2 q^{-43} + q^{-45} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{132}-2 q^{130}+3 q^{126}-2 q^{124}+2 q^{122}-8 q^{120}+2 q^{118}+13 q^{116}-8 q^{114}+3 q^{112}-21 q^{110}+11 q^{108}+38 q^{106}-28 q^{104}-17 q^{102}-43 q^{100}+57 q^{98}+107 q^{96}-67 q^{94}-106 q^{92}-117 q^{90}+148 q^{88}+287 q^{86}-56 q^{84}-279 q^{82}-324 q^{80}+196 q^{78}+566 q^{76}+118 q^{74}-396 q^{72}-638 q^{70}+41 q^{68}+734 q^{66}+432 q^{64}-251 q^{62}-806 q^{60}-287 q^{58}+555 q^{56}+629 q^{54}+121 q^{52}-619 q^{50}-529 q^{48}+110 q^{46}+521 q^{44}+435 q^{42}-188 q^{40}-522 q^{38}-316 q^{36}+238 q^{34}+554 q^{32}+214 q^{30}-381 q^{28}-570 q^{26}-15 q^{24}+538 q^{22}+492 q^{20}-219 q^{18}-708 q^{16}-230 q^{14}+452 q^{12}+695 q^{10}-4 q^8-727 q^6-459 q^4+226 q^2+784+302 q^{-2} -531 q^{-4} -614 q^{-6} -142 q^{-8} +627 q^{-10} +539 q^{-12} -138 q^{-14} -519 q^{-16} -444 q^{-18} +253 q^{-20} +500 q^{-22} +200 q^{-24} -198 q^{-26} -448 q^{-28} -78 q^{-30} +226 q^{-32} +261 q^{-34} +75 q^{-36} -226 q^{-38} -153 q^{-40} -4 q^{-42} +119 q^{-44} +122 q^{-46} -36 q^{-48} -66 q^{-50} -54 q^{-52} +9 q^{-54} +53 q^{-56} +11 q^{-58} -4 q^{-60} -19 q^{-62} -9 q^{-64} +8 q^{-66} +3 q^{-68} +3 q^{-70} -2 q^{-72} -2 q^{-74} + q^{-76} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{195}-2 q^{193}+3 q^{189}-2 q^{187}-q^{185}+q^{183}-3 q^{181}+q^{179}+8 q^{177}-2 q^{175}-8 q^{173}+3 q^{169}+8 q^{167}+3 q^{165}-15 q^{163}-25 q^{161}+4 q^{159}+57 q^{157}+54 q^{155}-24 q^{153}-122 q^{151}-130 q^{149}+25 q^{147}+255 q^{145}+291 q^{143}-14 q^{141}-446 q^{139}-559 q^{137}-87 q^{135}+679 q^{133}+1001 q^{131}+338 q^{129}-928 q^{127}-1605 q^{125}-797 q^{123}+1064 q^{121}+2328 q^{119}+1542 q^{117}-983 q^{115}-3066 q^{113}-2521 q^{111}+580 q^{109}+3604 q^{107}+3607 q^{105}+227 q^{103}-3758 q^{101}-4635 q^{99}-1307 q^{97}+3405 q^{95}+5278 q^{93}+2520 q^{91}-2506 q^{89}-5400 q^{87}-3601 q^{85}+1259 q^{83}+4895 q^{81}+4283 q^{79}+144 q^{77}-3856 q^{75}-4468 q^{73}-1437 q^{71}+2519 q^{69}+4152 q^{67}+2398 q^{65}-1093 q^{63}-3476 q^{61}-3006 q^{59}-185 q^{57}+2666 q^{55}+3284 q^{53}+1215 q^{51}-1880 q^{49}-3381 q^{47}-1971 q^{45}+1209 q^{43}+3423 q^{41}+2571 q^{39}-722 q^{37}-3474 q^{35}-3065 q^{33}+267 q^{31}+3539 q^{29}+3608 q^{27}+194 q^{25}-3571 q^{23}-4120 q^{21}-816 q^{19}+3413 q^{17}+4641 q^{15}+1596 q^{13}-3003 q^{11}-4973 q^9-2494 q^7+2226 q^5+5006 q^3+3405 q-1142 q^{-1} -4632 q^{-3} -4092 q^{-5} -130 q^{-7} +3773 q^{-9} +4413 q^{-11} +1406 q^{-13} -2576 q^{-15} -4216 q^{-17} -2395 q^{-19} +1172 q^{-21} +3521 q^{-23} +2931 q^{-25} +146 q^{-27} -2466 q^{-29} -2922 q^{-31} -1137 q^{-33} +1293 q^{-35} +2434 q^{-37} +1635 q^{-39} -230 q^{-41} -1667 q^{-43} -1674 q^{-45} -475 q^{-47} +861 q^{-49} +1333 q^{-51} +795 q^{-53} -192 q^{-55} -866 q^{-57} -787 q^{-59} -182 q^{-61} +409 q^{-63} +575 q^{-65} +321 q^{-67} -86 q^{-69} -338 q^{-71} -288 q^{-73} -60 q^{-75} +141 q^{-77} +182 q^{-79} +98 q^{-81} -23 q^{-83} -97 q^{-85} -74 q^{-87} -11 q^{-89} +33 q^{-91} +35 q^{-93} +20 q^{-95} -4 q^{-97} -19 q^{-99} -9 q^{-101} + q^{-103} +3 q^{-105} +3 q^{-107} +3 q^{-109} -2 q^{-111} -2 q^{-113} + q^{-115} }

A2 Invariants.

Weight	Invariant
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{22}-q^{18}+2 q^{16}-2 q^{14}+q^{10}-2 q^8+2 q^6-2 q^4+2 q^2+1- q^{-2} +2 q^{-4} - q^{-6} + q^{-10} }
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{60}-4 q^{58}+10 q^{56}-20 q^{54}+40 q^{52}-70 q^{50}+110 q^{48}-162 q^{46}+227 q^{44}-302 q^{42}+372 q^{40}-436 q^{38}+485 q^{36}-506 q^{34}+478 q^{32}-394 q^{30}+262 q^{28}-66 q^{26}-170 q^{24}+432 q^{22}-689 q^{20}+904 q^{18}-1060 q^{16}+1128 q^{14}-1105 q^{12}+994 q^{10}-798 q^8+548 q^6-275 q^4+6 q^2+240-432 q^{-2} +554 q^{-4} -602 q^{-6} +588 q^{-8} -524 q^{-10} +427 q^{-12} -316 q^{-14} +218 q^{-16} -134 q^{-18} +74 q^{-20} -36 q^{-22} +14 q^{-24} -4 q^{-26} + q^{-28} }
2,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{56}-q^{52}+2 q^{48}-5 q^{44}+q^{42}+6 q^{40}-5 q^{38}-8 q^{36}+6 q^{34}+11 q^{32}-7 q^{30}-9 q^{28}+11 q^{26}+8 q^{24}-10 q^{22}-3 q^{20}+8 q^{18}-4 q^{16}-4 q^{14}+4 q^{12}+q^{10}-7 q^8+3 q^6+10 q^4-7 q^2-6+10 q^{-2} +6 q^{-4} -11 q^{-6} -4 q^{-8} +8 q^{-10} +4 q^{-12} -6 q^{-14} -3 q^{-16} +6 q^{-18} +3 q^{-20} -2 q^{-22} -2 q^{-24} + q^{-28} }

A3 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$q^{48}-2q^{46}+4q^{44}-7q^{42}+11q^{40}-14q^{38}+18q^{36}-20q^{34}+21q^{32}-20q^{30}+15q^{28}-8q^{26}-2q^{24}+12q^{22}-23q^{20}+32q^{18}-39q^{16}+42q^{14}-41q^{12}+37q^{10}-29q^{8}+19q^{6}-7q^{4}-2q^{2}+11-17q^{-2}+22q^{-4}-22q^{-6}+20q^{-8}-17q^{-10}+13q^{-12}-9q^{-14}+6q^{-16}-2q^{-18}+q^{-20}$
1,0	$q^{78}-2q^{74}-2q^{72}+2q^{70}+6q^{68}+q^{66}-9q^{64}-8q^{62}+6q^{60}+16q^{58}+3q^{56}-17q^{54}-14q^{52}+10q^{50}+22q^{48}+q^{46}-21q^{44}-11q^{42}+15q^{40}+17q^{38}-8q^{36}-18q^{34}+2q^{32}+17q^{30}+3q^{28}-15q^{26}-6q^{24}+12q^{22}+8q^{20}-11q^{18}-10q^{16}+10q^{14}+13q^{12}-8q^{10}-19q^{8}+2q^{6}+22q^{4}+10q^{2}-18-19q^{-2}+9q^{-4}+24q^{-6}+3q^{-8}-19q^{-10}-12q^{-12}+11q^{-14}+14q^{-16}-2q^{-18}-10q^{-20}-3q^{-22}+6q^{-24}+4q^{-26}-2q^{-28}-2q^{-30}+q^{-34}$

G2 Invariants.

Weight

Invariant

1,0

$q^{114}-2q^{112}+4q^{110}-6q^{108}+5q^{106}-4q^{104}-2q^{102}+12q^{100}-21q^{98}+30q^{96}-32q^{94}+22q^{92}-3q^{90}-23q^{88}+55q^{86}-74q^{84}+80q^{82}-61q^{80}+20q^{78}+31q^{76}-80q^{74}+112q^{72}-114q^{70}+80q^{68}-22q^{66}-44q^{64}+90q^{62}-97q^{60}+69q^{58}-13q^{56}-47q^{54}+74q^{52}-64q^{50}+9q^{48}+67q^{46}-125q^{44}+139q^{42}-91q^{40}+101q^{36}-178q^{34}+196q^{32}-153q^{30}+59q^{28}+49q^{26}-132q^{24}+169q^{22}-140q^{20}+70q^{18}+11q^{16}-77q^{14}+96q^{12}-70q^{10}+13q^{8}+57q^{6}-97q^{4}+94q^{2}-42-35q^{-2}+106q^{-4}-142q^{-6}+126q^{-8}-69q^{-10}-11q^{-12}+83q^{-14}-120q^{-16}+119q^{-18}-77q^{-20}+22q^{-22}+24q^{-24}-54q^{-26}+57q^{-28}-43q^{-30}+24q^{-32}-2q^{-34}-9q^{-36}+12q^{-38}-10q^{-40}+6q^{-42}-2q^{-44}+q^{-46}$

.

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 41"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-7t^{2}+17t-21+17t^{-1}-7t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^3 z^9+a z^9+3 a^4 z^8+6 a^2 z^8+3 z^8+5 a^5 z^7+8 a^3 z^7+6 a z^7+3 z^7 a^{-1} +5 a^6 z^6+4 a^4 z^6-7 a^2 z^6+z^6 a^{-2} -5 z^6+3 a^7 z^5-4 a^5 z^5-18 a^3 z^5-20 a z^5-9 z^5 a^{-1} +a^8 z^4-6 a^6 z^4-14 a^4 z^4-8 a^2 z^4-3 z^4 a^{-2} -4 z^4-3 a^7 z^3+a^5 z^3+10 a^3 z^3+13 a z^3+7 z^3 a^{-1} -a^8 z^2+4 a^6 z^2+10 a^4 z^2+9 a^2 z^2+3 z^2 a^{-2} +7 z^2+a^7 z-2 a^3 z-2 a z-z a^{-1} -a^6-2 a^4-2 a^2- a^{-2} -1}