10 104

10_103

10_105

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10 104 Quick Notes

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Knot presentations

Planar diagram presentation	X6271 X16,4,17,3 X18,9,19,10 X14,7,15,8 X20,13,1,14 X8,17,9,18 X10,19,11,20 X12,6,13,5 X4,12,5,11 X2,16,3,15
Gauss code	1, -10, 2, -9, 8, -1, 4, -6, 3, -7, 9, -8, 5, -4, 10, -2, 6, -3, 7, -5
Dowker-Thistlethwaite code	6 16 12 14 18 4 20 2 8 10
Conway Notation	[3:20:20]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 14.1071 A-Polynomial See Data:10 104/A-polynomial

[edit Notes for 10 104's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math] Topological 4 genus [math]\displaystyle{ 1 }[/math] Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant 0

[edit Notes for 10 104's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-4 t^3+9 t^2-15 t+19-15 t^{-1} +9 t^{-2} -4 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+4 z^6+5 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 77, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+3 q^4-6 q^3+10 q^2-12 q+13-12 q^{-1} +10 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-a^2 z^6-z^6 a^{-2} +6 z^6-4 a^2 z^4-4 z^4 a^{-2} +13 z^4-5 a^2 z^2-5 z^2 a^{-2} +11 z^2-a^2- a^{-2} +3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 a z^9+2 z^9 a^{-1} +4 a^2 z^8+5 z^8 a^{-2} +9 z^8+4 a^3 z^7+2 a z^7+3 z^7 a^{-1} +5 z^7 a^{-3} +3 a^4 z^6-5 a^2 z^6-11 z^6 a^{-2} +3 z^6 a^{-4} -22 z^6+a^5 z^5-5 a^3 z^5-6 a z^5-12 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4+3 a^2 z^4+12 z^4 a^{-2} -6 z^4 a^{-4} +27 z^4-2 a^5 z^3-a^3 z^3+4 a z^3+13 z^3 a^{-1} +8 z^3 a^{-3} -2 z^3 a^{-5} +3 a^4 z^2-4 a^2 z^2-6 z^2 a^{-2} +2 z^2 a^{-4} -15 z^2+a^5 z+a^3 z-2 a z-4 z a^{-1} -2 z a^{-3} +a^2+ a^{-2} +3 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{14}+q^{12}-2 q^{10}+2 q^8+q^6-q^4+3 q^2-3+3 q^{-2} - q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} - q^{-14} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-6 q^{70}-2 q^{68}+16 q^{66}-29 q^{64}+41 q^{62}-43 q^{60}+30 q^{58}-6 q^{56}-36 q^{54}+82 q^{52}-118 q^{50}+121 q^{48}-87 q^{46}+9 q^{44}+85 q^{42}-164 q^{40}+201 q^{38}-162 q^{36}+66 q^{34}+57 q^{32}-157 q^{30}+181 q^{28}-122 q^{26}+15 q^{24}+99 q^{22}-156 q^{20}+131 q^{18}-27 q^{16}-104 q^{14}+206 q^{12}-236 q^{10}+168 q^8-34 q^6-123 q^4+244 q^2-286+243 q^{-2} -119 q^{-4} -35 q^{-6} +168 q^{-8} -234 q^{-10} +212 q^{-12} -111 q^{-14} -17 q^{-16} +126 q^{-18} -158 q^{-20} +111 q^{-22} - q^{-24} -113 q^{-26} +180 q^{-28} -166 q^{-30} +68 q^{-32} +57 q^{-34} -166 q^{-36} +212 q^{-38} -177 q^{-40} +91 q^{-42} +12 q^{-44} -99 q^{-46} +137 q^{-48} -128 q^{-50} +84 q^{-52} -29 q^{-54} -16 q^{-56} +40 q^{-58} -47 q^{-60} +40 q^{-62} -25 q^{-64} +12 q^{-66} + q^{-68} -7 q^{-70} +7 q^{-72} -7 q^{-74} +4 q^{-76} -2 q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_104.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+2 q^9-3 q^7+4 q^5-2 q^3+q+ q^{-1} -2 q^{-3} +4 q^{-5} -3 q^{-7} +2 q^{-9} - q^{-11} }[/math]
2	[math]\displaystyle{ q^{32}-2 q^{30}-q^{28}+6 q^{26}-7 q^{24}-5 q^{22}+20 q^{20}-10 q^{18}-20 q^{16}+30 q^{14}-30 q^{10}+20 q^8+12 q^6-21 q^4+16-20 q^{-4} +12 q^{-6} +20 q^{-8} -30 q^{-10} + q^{-12} +30 q^{-14} -21 q^{-16} -10 q^{-18} +21 q^{-20} -5 q^{-22} -9 q^{-24} +6 q^{-26} -2 q^{-30} + q^{-32} }[/math]
3	[math]\displaystyle{ -q^{63}+2 q^{61}+q^{59}-2 q^{57}-3 q^{55}+4 q^{53}+5 q^{51}-12 q^{49}-9 q^{47}+23 q^{45}+25 q^{43}-37 q^{41}-58 q^{39}+43 q^{37}+106 q^{35}-25 q^{33}-152 q^{31}-25 q^{29}+187 q^{27}+83 q^{25}-184 q^{23}-144 q^{21}+150 q^{19}+186 q^{17}-100 q^{15}-196 q^{13}+42 q^{11}+182 q^9+15 q^7-148 q^5-63 q^3+110 q+105 q^{-1} -68 q^{-3} -143 q^{-5} +21 q^{-7} +176 q^{-9} +30 q^{-11} -195 q^{-13} -84 q^{-15} +195 q^{-17} +135 q^{-19} -162 q^{-21} -176 q^{-23} +106 q^{-25} +190 q^{-27} -44 q^{-29} -164 q^{-31} -16 q^{-33} +124 q^{-35} +46 q^{-37} -72 q^{-39} -50 q^{-41} +29 q^{-43} +36 q^{-45} -8 q^{-47} -20 q^{-49} +2 q^{-51} +8 q^{-53} -3 q^{-57} +2 q^{-61} - q^{-63} }[/math]
4	[math]\displaystyle{ q^{104}-2 q^{102}-q^{100}+2 q^{98}-q^{96}+6 q^{94}-4 q^{92}-q^{90}+6 q^{88}-13 q^{86}+3 q^{84}-17 q^{82}+14 q^{80}+59 q^{78}-2 q^{76}-37 q^{74}-139 q^{72}-39 q^{70}+201 q^{68}+216 q^{66}+68 q^{64}-404 q^{62}-455 q^{60}+99 q^{58}+619 q^{56}+712 q^{54}-286 q^{52}-1082 q^{50}-708 q^{48}+488 q^{46}+1553 q^{44}+648 q^{42}-1031 q^{40}-1655 q^{38}-494 q^{36}+1580 q^{34}+1633 q^{32}-98 q^{30}-1731 q^{28}-1431 q^{26}+730 q^{24}+1728 q^{22}+789 q^{20}-995 q^{18}-1542 q^{16}-151 q^{14}+1115 q^{12}+1070 q^{10}-204 q^8-1127 q^6-653 q^4+471 q^2+1058+374 q^{-2} -725 q^{-4} -1067 q^{-6} -73 q^{-8} +1107 q^{-10} +991 q^{-12} -316 q^{-14} -1535 q^{-16} -788 q^{-18} +970 q^{-20} +1658 q^{-22} +449 q^{-24} -1609 q^{-26} -1581 q^{-28} +226 q^{-30} +1774 q^{-32} +1368 q^{-34} -824 q^{-36} -1739 q^{-38} -798 q^{-40} +938 q^{-42} +1594 q^{-44} +274 q^{-46} -928 q^{-48} -1106 q^{-50} -124 q^{-52} +874 q^{-54} +645 q^{-56} +2 q^{-58} -577 q^{-60} -442 q^{-62} +123 q^{-64} +301 q^{-66} +244 q^{-68} -75 q^{-70} -200 q^{-72} -65 q^{-74} +21 q^{-76} +100 q^{-78} +24 q^{-80} -34 q^{-82} -16 q^{-84} -14 q^{-86} +18 q^{-88} +6 q^{-90} -6 q^{-92} + q^{-94} -3 q^{-96} +3 q^{-98} -2 q^{-102} + q^{-104} }[/math]
5	[math]\displaystyle{ -q^{155}+2 q^{153}+q^{151}-2 q^{149}+q^{147}-2 q^{145}-6 q^{143}+7 q^{139}+4 q^{137}+12 q^{135}+10 q^{133}-17 q^{131}-36 q^{129}-32 q^{127}-3 q^{125}+56 q^{123}+116 q^{121}+87 q^{119}-66 q^{117}-240 q^{115}-284 q^{113}-73 q^{111}+348 q^{109}+680 q^{107}+494 q^{105}-287 q^{103}-1156 q^{101}-1335 q^{99}-316 q^{97}+1447 q^{95}+2588 q^{93}+1730 q^{91}-1032 q^{89}-3792 q^{87}-4020 q^{85}-680 q^{83}+4223 q^{81}+6723 q^{79}+3881 q^{77}-3075 q^{75}-8851 q^{73}-8085 q^{71}-149 q^{69}+9311 q^{67}+12280 q^{65}+5034 q^{63}-7480 q^{61}-15006 q^{59}-10508 q^{57}+3365 q^{55}+15430 q^{53}+15145 q^{51}+1960 q^{49}-13320 q^{47}-17683 q^{45}-7211 q^{43}+9314 q^{41}+17762 q^{39}+11109 q^{37}-4611 q^{35}-15671 q^{33}-12973 q^{31}+295 q^{29}+12259 q^{27}+12965 q^{25}+2839 q^{23}-8578 q^{21}-11604 q^{19}-4646 q^{17}+5353 q^{15}+9734 q^{13}+5462 q^{11}-2959 q^9-8065 q^7-5830 q^5+1317 q^3+6996 q+6351 q^{-1} -16 q^{-3} -6591 q^{-5} -7451 q^{-7} -1423 q^{-9} +6534 q^{-11} +9215 q^{-13} +3487 q^{-15} -6274 q^{-17} -11394 q^{-19} -6417 q^{-21} +5223 q^{-23} +13382 q^{-25} +10045 q^{-27} -2878 q^{-29} -14348 q^{-31} -13827 q^{-33} -870 q^{-35} +13628 q^{-37} +16794 q^{-39} +5499 q^{-41} -10758 q^{-43} -17996 q^{-45} -10100 q^{-47} +6084 q^{-49} +16818 q^{-51} +13399 q^{-53} -605 q^{-55} -13207 q^{-57} -14458 q^{-59} -4393 q^{-61} +8092 q^{-63} +13040 q^{-65} +7534 q^{-67} -2798 q^{-69} -9589 q^{-71} -8378 q^{-73} -1344 q^{-75} +5448 q^{-77} +7124 q^{-79} +3503 q^{-81} -1776 q^{-83} -4729 q^{-85} -3816 q^{-87} -556 q^{-89} +2305 q^{-91} +2921 q^{-93} +1441 q^{-95} -578 q^{-97} -1659 q^{-99} -1356 q^{-101} -269 q^{-103} +680 q^{-105} +875 q^{-107} +431 q^{-109} -133 q^{-111} -412 q^{-113} -323 q^{-115} -59 q^{-117} +155 q^{-119} +169 q^{-121} +60 q^{-123} -35 q^{-125} -62 q^{-127} -40 q^{-129} + q^{-131} +28 q^{-133} +13 q^{-135} -5 q^{-137} -3 q^{-139} -2 q^{-141} -3 q^{-143} +2 q^{-145} +3 q^{-147} -3 q^{-149} +2 q^{-153} - q^{-155} }[/math]
6	[math]\displaystyle{ q^{216}-2 q^{214}-q^{212}+2 q^{210}-q^{208}+2 q^{206}+2 q^{204}+10 q^{202}-6 q^{200}-17 q^{198}-3 q^{196}-13 q^{194}-q^{192}+18 q^{190}+64 q^{188}+32 q^{186}-30 q^{184}-46 q^{182}-110 q^{180}-107 q^{178}-16 q^{176}+221 q^{174}+288 q^{172}+191 q^{170}+22 q^{168}-402 q^{166}-719 q^{164}-663 q^{162}+120 q^{160}+965 q^{158}+1509 q^{156}+1455 q^{154}+40 q^{152}-2039 q^{150}-3613 q^{148}-2902 q^{146}-184 q^{144}+3784 q^{142}+7091 q^{140}+6335 q^{138}+887 q^{136}-7324 q^{134}-12778 q^{132}-11978 q^{130}-2741 q^{128}+11913 q^{126}+22565 q^{124}+21343 q^{122}+5211 q^{120}-17867 q^{118}-36180 q^{116}-35286 q^{114}-10285 q^{112}+26717 q^{110}+54694 q^{108}+52634 q^{106}+17382 q^{104}-37270 q^{102}-77786 q^{100}-74642 q^{98}-23577 q^{96}+50689 q^{94}+102814 q^{92}+98204 q^{90}+28659 q^{88}-67117 q^{86}-130261 q^{84}-117497 q^{82}-28938 q^{80}+84770 q^{78}+155408 q^{76}+130572 q^{74}+22279 q^{72}-105255 q^{70}-171854 q^{68}-132481 q^{66}-9272 q^{64}+123887 q^{62}+178331 q^{60}+121446 q^{58}-11339 q^{56}-134837 q^{54}-171214 q^{52}-99656 q^{50}+33358 q^{48}+137568 q^{46}+150892 q^{44}+68755 q^{42}-50367 q^{40}-129519 q^{38}-121518 q^{36}-36909 q^{34}+61371 q^{32}+112138 q^{30}+86762 q^{28}+10286 q^{26}-64462 q^{24}-89710 q^{22}-54145 q^{20}+10755 q^{18}+61572 q^{16}+65831 q^{14}+27076 q^{12}-26133 q^{10}-56386 q^8-45327 q^6-3919 q^4+38752 q^2+51908+28086 q^{-2} -17504 q^{-4} -51727 q^{-6} -50221 q^{-8} -10989 q^{-10} +40669 q^{-12} +67436 q^{-14} +48458 q^{-16} -9260 q^{-18} -67262 q^{-20} -85269 q^{-22} -43241 q^{-24} +35261 q^{-26} +98116 q^{-28} +100004 q^{-30} +31346 q^{-32} -66062 q^{-34} -129418 q^{-36} -108427 q^{-38} -11799 q^{-40} +100315 q^{-42} +153682 q^{-44} +107507 q^{-46} -12011 q^{-48} -131484 q^{-50} -168070 q^{-52} -97266 q^{-54} +37826 q^{-56} +151378 q^{-58} +169818 q^{-60} +81856 q^{-62} -58799 q^{-64} -158959 q^{-66} -159612 q^{-68} -62701 q^{-70} +68653 q^{-72} +152982 q^{-74} +141879 q^{-76} +45428 q^{-78} -69353 q^{-80} -135727 q^{-82} -117868 q^{-84} -34108 q^{-86} +61500 q^{-88} +112599 q^{-90} +93187 q^{-92} +25736 q^{-94} -48170 q^{-96} -85693 q^{-98} -71766 q^{-100} -20451 q^{-102} +34132 q^{-104} +60737 q^{-106} +52052 q^{-108} +17061 q^{-110} -20425 q^{-112} -40974 q^{-114} -35931 q^{-116} -13594 q^{-118} +10413 q^{-120} +25035 q^{-122} +23690 q^{-124} +10983 q^{-126} -4816 q^{-128} -14087 q^{-130} -14363 q^{-132} -7978 q^{-134} +1259 q^{-136} +7438 q^{-138} +8505 q^{-140} +4808 q^{-142} +131 q^{-144} -3439 q^{-146} -4580 q^{-148} -2905 q^{-150} -329 q^{-152} +1697 q^{-154} +2045 q^{-156} +1509 q^{-158} +338 q^{-160} -756 q^{-162} -989 q^{-164} -644 q^{-166} -59 q^{-168} +225 q^{-170} +398 q^{-172} +291 q^{-174} +5 q^{-176} -134 q^{-178} -137 q^{-180} -47 q^{-182} -25 q^{-184} +44 q^{-186} +61 q^{-188} +9 q^{-190} -14 q^{-192} -16 q^{-194} +5 q^{-196} -10 q^{-198} - q^{-200} +11 q^{-202} -2 q^{-206} -3 q^{-208} +3 q^{-210} -2 q^{-214} + q^{-216} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{14}+q^{12}-2 q^{10}+2 q^8+q^6-q^4+3 q^2-3+3 q^{-2} - q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} - q^{-14} }[/math]
1,1	[math]\displaystyle{ q^{44}-4 q^{42}+12 q^{40}-28 q^{38}+54 q^{36}-96 q^{34}+154 q^{32}-236 q^{30}+341 q^{28}-470 q^{26}+610 q^{24}-728 q^{22}+816 q^{20}-832 q^{18}+740 q^{16}-528 q^{14}+208 q^{12}+180 q^{10}-618 q^8+1040 q^6-1381 q^4+1618 q^2-1706+1654 q^{-2} -1453 q^{-4} +1130 q^{-6} -728 q^{-8} +288 q^{-10} +134 q^{-12} -476 q^{-14} +726 q^{-16} -856 q^{-18} +868 q^{-20} -788 q^{-22} +654 q^{-24} -510 q^{-26} +365 q^{-28} -238 q^{-30} +148 q^{-32} -90 q^{-34} +48 q^{-36} -22 q^{-38} +10 q^{-40} -4 q^{-42} + q^{-44} }[/math]
2,0	[math]\displaystyle{ q^{38}-q^{36}-q^{34}+2 q^{32}-3 q^{30}-4 q^{28}+5 q^{26}+3 q^{24}-5 q^{22}-2 q^{20}+10 q^{18}+6 q^{16}-14 q^{14}+2 q^{12}+10 q^{10}-8 q^8-7 q^6+7 q^4+3 q^2-8+4 q^{-2} +8 q^{-4} -4 q^{-6} -6 q^{-8} +12 q^{-10} +2 q^{-12} -13 q^{-14} +7 q^{-16} +9 q^{-18} -4 q^{-20} -7 q^{-22} +3 q^{-24} +4 q^{-26} -5 q^{-28} -3 q^{-30} +3 q^{-32} - q^{-36} + q^{-38} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-2 q^{32}+q^{30}+4 q^{28}-8 q^{26}+2 q^{24}+11 q^{22}-17 q^{20}+2 q^{18}+19 q^{16}-24 q^{14}+q^{12}+20 q^{10}-16 q^8-2 q^6+13 q^4+q^2-4+13 q^{-4} -3 q^{-6} -16 q^{-8} +19 q^{-10} + q^{-12} -24 q^{-14} +19 q^{-16} +2 q^{-18} -17 q^{-20} +12 q^{-22} +2 q^{-24} -7 q^{-26} +4 q^{-28} -2 q^{-32} + q^{-34} }[/math]
1,0,0	[math]\displaystyle{ -q^{17}+q^{15}-3 q^{13}+3 q^{11}-2 q^9+3 q^7-q^5+2 q^3+2 q^{-3} - q^{-5} +3 q^{-7} -2 q^{-9} +3 q^{-11} -3 q^{-13} + q^{-15} - q^{-17} }[/math]
1,0,1	[math]\displaystyle{ q^{56}-4 q^{54}+10 q^{52}-14 q^{50}+7 q^{48}+18 q^{46}-57 q^{44}+82 q^{42}-65 q^{40}-20 q^{38}+149 q^{36}-257 q^{34}+246 q^{32}-61 q^{30}-242 q^{28}+540 q^{26}-622 q^{24}+402 q^{22}+63 q^{20}-586 q^{18}+887 q^{16}-836 q^{14}+431 q^{12}+105 q^{10}-503 q^8+600 q^6-388 q^4+105 q^2+46+67 q^{-2} -326 q^{-4} +506 q^{-6} -412 q^{-8} +21 q^{-10} +497 q^{-12} -860 q^{-14} +894 q^{-16} -552 q^{-18} +18 q^{-20} +456 q^{-22} -679 q^{-24} +576 q^{-26} -275 q^{-28} -51 q^{-30} +247 q^{-32} -269 q^{-34} +170 q^{-36} -38 q^{-38} -49 q^{-40} +72 q^{-42} -54 q^{-44} +20 q^{-46} +4 q^{-48} -10 q^{-50} +8 q^{-52} -4 q^{-54} + q^{-56} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+2 q^{32}-5 q^{30}+8 q^{28}-12 q^{26}+18 q^{24}-23 q^{22}+27 q^{20}-28 q^{18}+25 q^{16}-18 q^{14}+9 q^{12}+4 q^{10}-18 q^8+32 q^6-43 q^4+51 q^2-54+52 q^{-2} -43 q^{-4} +33 q^{-6} -18 q^{-8} +5 q^{-10} +9 q^{-12} -18 q^{-14} +25 q^{-16} -28 q^{-18} +27 q^{-20} -24 q^{-22} +18 q^{-24} -13 q^{-26} +8 q^{-28} -4 q^{-30} +2 q^{-32} - q^{-34} }[/math]
1,0	[math]\displaystyle{ q^{56}-2 q^{52}-2 q^{50}+3 q^{48}+6 q^{46}-q^{44}-10 q^{42}-7 q^{40}+10 q^{38}+17 q^{36}-3 q^{34}-24 q^{32}-11 q^{30}+22 q^{28}+24 q^{26}-12 q^{24}-30 q^{22}-3 q^{20}+28 q^{18}+13 q^{16}-20 q^{14}-17 q^{12}+13 q^{10}+19 q^8-6 q^6-18 q^4+3 q^2+20+3 q^{-2} -18 q^{-4} -6 q^{-6} +18 q^{-8} +13 q^{-10} -17 q^{-12} -20 q^{-14} +12 q^{-16} +28 q^{-18} -2 q^{-20} -30 q^{-22} -13 q^{-24} +24 q^{-26} +23 q^{-28} -11 q^{-30} -25 q^{-32} -3 q^{-34} +18 q^{-36} +10 q^{-38} -7 q^{-40} -10 q^{-42} +6 q^{-46} +2 q^{-48} -2 q^{-50} -2 q^{-52} + q^{-56} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{46}-2 q^{44}+3 q^{42}-4 q^{40}+7 q^{38}-10 q^{36}+11 q^{34}-14 q^{32}+19 q^{30}-22 q^{28}+20 q^{26}-21 q^{24}+21 q^{22}-18 q^{20}+8 q^{18}-6 q^{16}+11 q^{12}-21 q^{10}+25 q^8-28 q^6+42 q^4-38 q^2+42-38 q^{-2} +42 q^{-4} -29 q^{-6} +25 q^{-8} -22 q^{-10} +11 q^{-12} - q^{-14} -6 q^{-16} +7 q^{-18} -18 q^{-20} +21 q^{-22} -21 q^{-24} +21 q^{-26} -22 q^{-28} +20 q^{-30} -14 q^{-32} +12 q^{-34} -10 q^{-36} +7 q^{-38} -4 q^{-40} +2 q^{-42} -2 q^{-44} + q^{-46} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-6 q^{70}-2 q^{68}+16 q^{66}-29 q^{64}+41 q^{62}-43 q^{60}+30 q^{58}-6 q^{56}-36 q^{54}+82 q^{52}-118 q^{50}+121 q^{48}-87 q^{46}+9 q^{44}+85 q^{42}-164 q^{40}+201 q^{38}-162 q^{36}+66 q^{34}+57 q^{32}-157 q^{30}+181 q^{28}-122 q^{26}+15 q^{24}+99 q^{22}-156 q^{20}+131 q^{18}-27 q^{16}-104 q^{14}+206 q^{12}-236 q^{10}+168 q^8-34 q^6-123 q^4+244 q^2-286+243 q^{-2} -119 q^{-4} -35 q^{-6} +168 q^{-8} -234 q^{-10} +212 q^{-12} -111 q^{-14} -17 q^{-16} +126 q^{-18} -158 q^{-20} +111 q^{-22} - q^{-24} -113 q^{-26} +180 q^{-28} -166 q^{-30} +68 q^{-32} +57 q^{-34} -166 q^{-36} +212 q^{-38} -177 q^{-40} +91 q^{-42} +12 q^{-44} -99 q^{-46} +137 q^{-48} -128 q^{-50} +84 q^{-52} -29 q^{-54} -16 q^{-56} +40 q^{-58} -47 q^{-60} +40 q^{-62} -25 q^{-64} +12 q^{-66} + q^{-68} -7 q^{-70} +7 q^{-72} -7 q^{-74} +4 q^{-76} -2 q^{-78} + q^{-80} }[/math]

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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.

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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["10 104"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-4 t^3+9 t^2-15 t+19-15 t^{-1} +9 t^{-2} -4 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+4 z^6+5 z^4+z^2+1 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 77, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^5+3 q^4-6 q^3+10 q^2-12 q+13-12 q^{-1} +10 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^8-a^2 z^6-z^6 a^{-2} +6 z^6-4 a^2 z^4-4 z^4 a^{-2} +13 z^4-5 a^2 z^2-5 z^2 a^{-2} +11 z^2-a^2- a^{-2} +3 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 a z^9+2 z^9 a^{-1} +4 a^2 z^8+5 z^8 a^{-2} +9 z^8+4 a^3 z^7+2 a z^7+3 z^7 a^{-1} +5 z^7 a^{-3} +3 a^4 z^6-5 a^2 z^6-11 z^6 a^{-2} +3 z^6 a^{-4} -22 z^6+a^5 z^5-5 a^3 z^5-6 a z^5-12 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -6 a^4 z^4+3 a^2 z^4+12 z^4 a^{-2} -6 z^4 a^{-4} +27 z^4-2 a^5 z^3-a^3 z^3+4 a z^3+13 z^3 a^{-1} +8 z^3 a^{-3} -2 z^3 a^{-5} +3 a^4 z^2-4 a^2 z^2-6 z^2 a^{-2} +2 z^2 a^{-4} -15 z^2+a^5 z+a^3 z-2 a z-4 z a^{-1} -2 z a^{-3} +a^2+ a^{-2} +3 }[/math]

Vassiliev invariants

V2 and V3:

(1, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ 4 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{82}{3} }[/math] [math]\displaystyle{ -\frac{110}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{328}{3} }[/math] [math]\displaystyle{ -\frac{440}{3} }[/math] [math]\displaystyle{ -\frac{4769}{30} }[/math] [math]\displaystyle{ \frac{338}{15} }[/math] [math]\displaystyle{ -\frac{7138}{45} }[/math] [math]\displaystyle{ -\frac{895}{18} }[/math] [math]\displaystyle{ -\frac{1409}{30} }[/math]

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of 10 104. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-5	-4	-3	-2	-1	0	1	2	3	4	5	χ
11											1	-1
9										2		2
7									4	1		-3
5								6	2			4
3							6	4				-2
1						7	6					1
-1					6	7						1
-3				4	6							-2
-5			2	6								4
-7		1	4									-3
-9		2										2
-11	1											-1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 104]]
Out[2]=	10
In[3]:=	PD[Knot[10, 104]]
Out[3]=	PD[X[6, 2, 7, 1], X[16, 4, 17, 3], X[18, 9, 19, 10], X[14, 7, 15, 8], X[20, 13, 1, 14], X[8, 17, 9, 18], X[10, 19, 11, 20], X[12, 6, 13, 5], X[4, 12, 5, 11], X[2, 16, 3, 15]]
In[4]:=	GaussCode[Knot[10, 104]]
Out[4]=	GaussCode[1, -10, 2, -9, 8, -1, 4, -6, 3, -7, 9, -8, 5, -4, 10, -2, 6, -3, 7, -5]
In[5]:=	BR[Knot[10, 104]]
Out[5]=	BR[3, {-1, -1, -1, 2, 2, -1, 2, -1, 2, 2}]
In[6]:=	alex = Alexander[Knot[10, 104]][t]
Out[6]=	-4 4 9 15 2 3 4 19 + t - -- + -- - -- - 15 t + 9 t - 4 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 104]][z]
Out[7]=	2 4 6 8 1 + z + 5 z + 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 104]}
In[9]:=	{KnotDet[Knot[10, 104]], KnotSignature[Knot[10, 104]]}
Out[9]=	{77, 0}
In[10]:=	J=Jones[Knot[10, 104]][q]
Out[10]=	-5 3 6 10 12 2 3 4 5 13 - q + -- - -- + -- - -- - 12 q + 10 q - 6 q + 3 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 71], Knot[10, 104]}
In[12]:=	A2Invariant[Knot[10, 104]][q]
Out[12]=	-14 -12 2 2 -6 -4 3 2 4 6 8 -3 - q + q - --- + -- + q - q + -- + 3 q - q + q + 2 q - 10 8 2 q q q 10 12 14 2 q + q - q
In[13]:=	Kauffman[Knot[10, 104]][a, z]
Out[13]=	2 2 -2 2 2 z 4 z 3 5 2 2 z 6 z 3 + a + a - --- - --- - 2 a z + a z + a z - 15 z + ---- - ---- - 3 a 4 2 a a a 3 3 3 2 2 4 2 2 z 8 z 13 z 3 3 3 5 3 4 a z + 3 a z - ---- + ---- + ----- + 4 a z - a z - 2 a z + 5 3 a a a 4 4 5 5 5 4 6 z 12 z 2 4 4 4 z 11 z 12 z 27 z - ---- + ----- + 3 a z - 6 a z + -- - ----- - ----- - 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 3 z 11 z 2 6 4 6 6 a z - 5 a z + a z - 22 z + ---- - ----- - 5 a z + 3 a z + 4 2 a a 7 7 8 9 5 z 3 z 7 3 7 8 5 z 2 8 2 z 9 ---- + ---- + 2 a z + 4 a z + 9 z + ---- + 4 a z + ---- + 2 a z 3 a 2 a a a
In[14]:=	{Vassiliev[2][Knot[10, 104]], Vassiliev[3][Knot[10, 104]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 104]][q, t]
Out[15]=	7 1 2 1 4 2 6 4 - + 7 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 6 6 3 3 2 5 2 5 3 7 3 ---- + --- + 6 q t + 6 q t + 4 q t + 6 q t + 2 q t + 4 q t + 3 q t q t 7 4 9 4 11 5 q t + 2 q t + q t