10 112

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10 111.gif

10_111

10 113.gif

10_113

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10 112 Quick Notes


10 112 Further Notes and Views

Knot presentations

Planar diagram presentation X6271 X8394 X18,11,19,12 X20,13,1,14 X2,16,3,15 X4,17,5,18 X12,19,13,20 X10,6,11,5 X14,7,15,8 X16,10,17,9
Gauss code 1, -5, 2, -6, 8, -1, 9, -2, 10, -8, 3, -7, 4, -9, 5, -10, 6, -3, 7, -4
Dowker-Thistlethwaite code 6 8 10 14 16 18 20 2 4 12
Conway Notation [8*3]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-9][-3]
Hyperbolic Volume 14.7559
A-Polynomial See Data:10 112/A-polynomial

[edit Notes for 10 112'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 2

[edit Notes for 10 112's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+5 t^3-11 t^2+17 t-19+17 t^{-1} -11 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-3 z^6-z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 87, -2 }
Jones polynomial [math]\displaystyle{ q^3-4 q^2+7 q-10+14 q^{-1} -14 q^{-2} +14 q^{-3} -11 q^{-4} +7 q^{-5} -4 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -a^2 z^8+a^4 z^6-5 a^2 z^6+z^6+3 a^4 z^4-7 a^2 z^4+3 z^4+a^4 z^2+z^2-2 a^4+4 a^2-1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 3 a^3 z^9+3 a z^9+7 a^4 z^8+13 a^2 z^8+6 z^8+8 a^5 z^7+4 a^3 z^7+4 z^7 a^{-1} +7 a^6 z^6-9 a^4 z^6-35 a^2 z^6+z^6 a^{-2} -18 z^6+4 a^7 z^5-8 a^5 z^5-17 a^3 z^5-16 a z^5-11 z^5 a^{-1} +a^8 z^4-7 a^6 z^4+3 a^4 z^4+28 a^2 z^4-2 z^4 a^{-2} +15 z^4-3 a^7 z^3-a^5 z^3+9 a^3 z^3+13 a z^3+6 z^3 a^{-1} +a^6 z^2+a^4 z^2-3 a^2 z^2-3 z^2+2 a^5 z+2 a^3 z-2 a^4-4 a^2-1 }[/math]
The A2 invariant [math]\displaystyle{ q^{20}-2 q^{18}+q^{16}-3 q^{14}-q^{12}+2 q^{10}-q^8+6 q^6-q^4+3 q^2-2 q^{-2} + q^{-4} -2 q^{-6} + q^{-8} }[/math]
The G2 invariant [math]\displaystyle{ q^{114}-3 q^{112}+6 q^{110}-10 q^{108}+9 q^{106}-6 q^{104}-2 q^{102}+18 q^{100}-32 q^{98}+47 q^{96}-50 q^{94}+34 q^{92}-6 q^{90}-34 q^{88}+75 q^{86}-104 q^{84}+117 q^{82}-101 q^{80}+50 q^{78}+25 q^{76}-110 q^{74}+182 q^{72}-205 q^{70}+161 q^{68}-63 q^{66}-68 q^{64}+177 q^{62}-218 q^{60}+166 q^{58}-45 q^{56}-99 q^{54}+184 q^{52}-176 q^{50}+58 q^{48}+108 q^{46}-239 q^{44}+275 q^{42}-192 q^{40}+16 q^{38}+182 q^{36}-322 q^{34}+358 q^{32}-271 q^{30}+102 q^{28}+103 q^{26}-251 q^{24}+316 q^{22}-264 q^{20}+137 q^{18}+29 q^{16}-166 q^{14}+221 q^{12}-170 q^{10}+47 q^8+109 q^6-214 q^4+212 q^2-106-64 q^{-2} +214 q^{-4} -286 q^{-6} +244 q^{-8} -112 q^{-10} -54 q^{-12} +184 q^{-14} -238 q^{-16} +206 q^{-18} -112 q^{-20} +3 q^{-22} +71 q^{-24} -104 q^{-26} +94 q^{-28} -58 q^{-30} +25 q^{-32} +5 q^{-34} -17 q^{-36} +17 q^{-38} -14 q^{-40} +7 q^{-42} -3 q^{-44} + q^{-46} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_112.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{15}-3 q^{13}+3 q^{11}-4 q^9+3 q^7+4 q-3 q^{-1} +3 q^{-3} -3 q^{-5} + q^{-7} }[/math]
2 [math]\displaystyle{ q^{42}-3 q^{40}+8 q^{36}-10 q^{34}+q^{32}+19 q^{30}-24 q^{28}-5 q^{26}+34 q^{24}-24 q^{22}-19 q^{20}+32 q^{18}-3 q^{16}-22 q^{14}+9 q^{12}+19 q^{10}-12 q^8-17 q^6+28 q^4+4 q^2-33+23 q^{-2} +18 q^{-4} -34 q^{-6} +5 q^{-8} +23 q^{-10} -17 q^{-12} -7 q^{-14} +12 q^{-16} - q^{-18} -3 q^{-20} + q^{-22} }[/math]
3 [math]\displaystyle{ q^{81}-3 q^{79}+5 q^{75}+2 q^{73}-6 q^{71}-6 q^{69}+12 q^{67}-5 q^{65}-17 q^{63}+17 q^{61}+41 q^{59}-34 q^{57}-85 q^{55}+41 q^{53}+149 q^{51}-16 q^{49}-209 q^{47}-40 q^{45}+238 q^{43}+117 q^{41}-225 q^{39}-182 q^{37}+154 q^{35}+224 q^{33}-74 q^{31}-218 q^{29}-21 q^{27}+189 q^{25}+96 q^{23}-142 q^{21}-152 q^{19}+96 q^{17}+191 q^{15}-48 q^{13}-219 q^{11}+q^9+241 q^7+53 q^5-238 q^3-119 q+221 q^{-1} +179 q^{-3} -157 q^{-5} -228 q^{-7} +78 q^{-9} +231 q^{-11} +10 q^{-13} -196 q^{-15} -81 q^{-17} +129 q^{-19} +109 q^{-21} -58 q^{-23} -95 q^{-25} +5 q^{-27} +62 q^{-29} +19 q^{-31} -30 q^{-33} -15 q^{-35} +7 q^{-37} +8 q^{-39} - q^{-41} -3 q^{-43} + q^{-45} }[/math]
4 [math]\displaystyle{ q^{132}-3 q^{130}+5 q^{126}-q^{124}+6 q^{122}-13 q^{120}-7 q^{118}+5 q^{116}-8 q^{114}+43 q^{112}+4 q^{110}-5 q^{108}-39 q^{106}-115 q^{104}+56 q^{102}+140 q^{100}+191 q^{98}-44 q^{96}-479 q^{94}-272 q^{92}+239 q^{90}+819 q^{88}+495 q^{86}-795 q^{84}-1234 q^{82}-420 q^{80}+1352 q^{78}+1806 q^{76}-107 q^{74}-1950 q^{72}-1927 q^{70}+623 q^{68}+2626 q^{66}+1469 q^{64}-1162 q^{62}-2728 q^{60}-1001 q^{58}+1753 q^{56}+2269 q^{54}+494 q^{52}-1872 q^{50}-1863 q^{48}+88 q^{46}+1660 q^{44}+1462 q^{42}-416 q^{40}-1605 q^{38}-1000 q^{36}+713 q^{34}+1600 q^{32}+521 q^{30}-1187 q^{28}-1500 q^{26}+174 q^{24}+1659 q^{22}+1136 q^{20}-980 q^{18}-2018 q^{16}-376 q^{14}+1728 q^{12}+1941 q^{10}-397 q^8-2381 q^6-1386 q^4+1078 q^2+2541+905 q^{-2} -1742 q^{-4} -2211 q^{-6} -471 q^{-8} +1943 q^{-10} +1957 q^{-12} -71 q^{-14} -1709 q^{-16} -1670 q^{-18} +288 q^{-20} +1549 q^{-22} +1148 q^{-24} -202 q^{-26} -1332 q^{-28} -801 q^{-30} +243 q^{-32} +887 q^{-34} +646 q^{-36} -264 q^{-38} -566 q^{-40} -377 q^{-42} +116 q^{-44} +400 q^{-46} +164 q^{-48} -55 q^{-50} -195 q^{-52} -101 q^{-54} +59 q^{-56} +62 q^{-58} +41 q^{-60} -20 q^{-62} -29 q^{-64} - q^{-66} +3 q^{-68} +8 q^{-70} - q^{-72} -3 q^{-74} + q^{-76} }[/math]
5 [math]\displaystyle{ q^{195}-3 q^{193}+5 q^{189}-q^{187}+3 q^{185}-q^{183}-14 q^{181}-14 q^{179}+8 q^{177}+26 q^{175}+39 q^{173}+27 q^{171}-44 q^{169}-119 q^{167}-111 q^{165}+46 q^{163}+233 q^{161}+297 q^{159}+89 q^{157}-396 q^{155}-732 q^{153}-406 q^{151}+522 q^{149}+1377 q^{147}+1244 q^{145}-325 q^{143}-2358 q^{141}-2807 q^{139}-538 q^{137}+3226 q^{135}+5261 q^{133}+2790 q^{131}-3383 q^{129}-8363 q^{127}-6776 q^{125}+1819 q^{123}+11098 q^{121}+12328 q^{119}+2363 q^{117}-12050 q^{115}-18353 q^{113}-9116 q^{111}+9844 q^{109}+22819 q^{107}+17308 q^{105}-3982 q^{103}-23854 q^{101}-24722 q^{99}-4508 q^{97}+20466 q^{95}+29011 q^{93}+13444 q^{91}-13220 q^{89}-28708 q^{87}-20448 q^{85}+4095 q^{83}+24112 q^{81}+23630 q^{79}+4488 q^{77}-16577 q^{75}-22887 q^{73}-10738 q^{71}+8556 q^{69}+19145 q^{67}+13802 q^{65}-1675 q^{63}-14248 q^{61}-14346 q^{59}-2957 q^{57}+9853 q^{55}+13501 q^{53}+5484 q^{51}-6920 q^{49}-12671 q^{47}-6700 q^{45}+5592 q^{43}+12784 q^{41}+7707 q^{39}-5350 q^{37}-14066 q^{35}-9442 q^{33}+5142 q^{31}+16192 q^{29}+12420 q^{27}-3966 q^{25}-18267 q^{23}-16515 q^{21}+1026 q^{19}+19110 q^{17}+21012 q^{15}+3968 q^{13}-17673 q^{11}-24717 q^9-10372 q^7+13293 q^5+26080 q^3+17093 q-6169 q^{-1} -24159 q^{-3} -22158 q^{-5} -2447 q^{-7} +18441 q^{-9} +24012 q^{-11} +10687 q^{-13} -10037 q^{-15} -21692 q^{-17} -16237 q^{-19} +756 q^{-21} +15621 q^{-23} +17721 q^{-25} +6925 q^{-27} -7576 q^{-29} -15018 q^{-31} -11105 q^{-33} -52 q^{-35} +9478 q^{-37} +11239 q^{-39} +5213 q^{-41} -3369 q^{-43} -8312 q^{-45} -6957 q^{-47} -1275 q^{-49} +4158 q^{-51} +5888 q^{-53} +3506 q^{-55} -666 q^{-57} -3504 q^{-59} -3462 q^{-61} -1239 q^{-63} +1192 q^{-65} +2281 q^{-67} +1653 q^{-69} +133 q^{-71} -1007 q^{-73} -1162 q^{-75} -552 q^{-77} +181 q^{-79} +572 q^{-81} +454 q^{-83} +82 q^{-85} -172 q^{-87} -215 q^{-89} -108 q^{-91} +16 q^{-93} +82 q^{-95} +55 q^{-97} + q^{-99} -19 q^{-101} -15 q^{-103} -5 q^{-105} +3 q^{-107} +8 q^{-109} - q^{-111} -3 q^{-113} + q^{-115} }[/math]
6 [math]\displaystyle{ q^{270}-3 q^{268}+5 q^{264}-q^{262}+3 q^{260}-4 q^{258}-2 q^{256}-21 q^{254}-11 q^{252}+42 q^{250}+28 q^{248}+36 q^{246}-3 q^{244}-55 q^{242}-157 q^{240}-127 q^{238}+107 q^{236}+225 q^{234}+326 q^{232}+190 q^{230}-160 q^{228}-734 q^{226}-821 q^{224}-93 q^{222}+758 q^{220}+1583 q^{218}+1449 q^{216}+33 q^{214}-2441 q^{212}-3745 q^{210}-2161 q^{208}+1426 q^{206}+5785 q^{204}+7359 q^{202}+3530 q^{200}-5400 q^{198}-13415 q^{196}-13431 q^{194}-3589 q^{192}+13408 q^{190}+26609 q^{188}+24074 q^{186}+1478 q^{184}-29679 q^{182}-48096 q^{180}-37772 q^{178}+4186 q^{176}+55610 q^{174}+81157 q^{172}+54349 q^{170}-17776 q^{168}-93662 q^{166}-122736 q^{164}-71743 q^{162}+39507 q^{160}+145090 q^{158}+169711 q^{156}+83551 q^{154}-71025 q^{152}-202254 q^{150}-215345 q^{148}-87509 q^{146}+112381 q^{144}+257605 q^{142}+248375 q^{140}+79265 q^{138}-154324 q^{136}-301417 q^{134}-263819 q^{132}-57139 q^{130}+189978 q^{128}+322701 q^{126}+256764 q^{124}+29559 q^{122}-212564 q^{120}-320019 q^{118}-227969 q^{116}-1781 q^{114}+216152 q^{112}+293743 q^{110}+188732 q^{108}-21342 q^{106}-204151 q^{104}-250290 q^{102}-145830 q^{100}+36003 q^{98}+180150 q^{96}+202898 q^{94}+104568 q^{92}-45655 q^{90}-151559 q^{88}-157267 q^{86}-67675 q^{84}+52872 q^{82}+127501 q^{80}+115629 q^{78}+31904 q^{76}-62677 q^{74}-108267 q^{72}-76318 q^{70}+5389 q^{68}+78347 q^{66}+90646 q^{64}+34071 q^{62}-48065 q^{60}-95505 q^{58}-68472 q^{56}+14805 q^{54}+95245 q^{52}+107948 q^{50}+36161 q^{48}-72932 q^{46}-140329 q^{44}-108342 q^{42}+8598 q^{40}+135384 q^{38}+176663 q^{36}+93826 q^{34}-65353 q^{32}-193987 q^{30}-197546 q^{28}-65014 q^{26}+124891 q^{24}+241972 q^{22}+202838 q^{20}+25026 q^{18}-178074 q^{16}-272696 q^{14}-194117 q^{12}+14895 q^{10}+218361 q^8+286782 q^6+174267 q^4-47186 q^2-239543-284538 q^{-2} -153010 q^{-4} +67717 q^{-6} +243003 q^{-8} +267322 q^{-10} +134083 q^{-12} -72869 q^{-14} -229033 q^{-16} -243101 q^{-18} -118385 q^{-20} +66287 q^{-22} +199949 q^{-24} +213825 q^{-26} +107454 q^{-28} -49879 q^{-30} -164671 q^{-32} -180650 q^{-34} -97700 q^{-36} +27443 q^{-38} +126436 q^{-40} +147194 q^{-42} +88688 q^{-44} -7493 q^{-46} -88419 q^{-48} -113621 q^{-50} -79365 q^{-52} -8326 q^{-54} +55890 q^{-56} +83122 q^{-58} +66022 q^{-60} +19031 q^{-62} -29776 q^{-64} -57423 q^{-66} -51503 q^{-68} -22737 q^{-70} +12115 q^{-72} +35303 q^{-74} +37728 q^{-76} +21839 q^{-78} -2141 q^{-80} -19462 q^{-82} -24867 q^{-84} -17354 q^{-86} -3304 q^{-88} +9548 q^{-90} +15028 q^{-92} +11770 q^{-94} +4397 q^{-96} -3622 q^{-98} -7855 q^{-100} -7459 q^{-102} -3406 q^{-104} +1041 q^{-106} +3498 q^{-108} +3919 q^{-110} +2199 q^{-112} +41 q^{-114} -1561 q^{-116} -1718 q^{-118} -1042 q^{-120} -227 q^{-122} +523 q^{-124} +687 q^{-126} +471 q^{-128} +48 q^{-130} -147 q^{-132} -200 q^{-134} -164 q^{-136} -25 q^{-138} +53 q^{-140} +74 q^{-142} +15 q^{-144} +2 q^{-146} -5 q^{-148} -19 q^{-150} -5 q^{-152} +3 q^{-154} +8 q^{-156} - q^{-158} -3 q^{-160} + q^{-162} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{20}-2 q^{18}+q^{16}-3 q^{14}-q^{12}+2 q^{10}-q^8+6 q^6-q^4+3 q^2-2 q^{-2} + q^{-4} -2 q^{-6} + q^{-8} }[/math]
1,1 [math]\displaystyle{ q^{60}-6 q^{58}+18 q^{56}-38 q^{54}+69 q^{52}-120 q^{50}+190 q^{48}-268 q^{46}+357 q^{44}-460 q^{42}+576 q^{40}-686 q^{38}+790 q^{36}-878 q^{34}+906 q^{32}-846 q^{30}+671 q^{28}-372 q^{26}-56 q^{24}+558 q^{22}-1086 q^{20}+1554 q^{18}-1926 q^{16}+2140 q^{14}-2158 q^{12}+2014 q^{10}-1680 q^8+1236 q^6-688 q^4+132 q^2+388-822 q^{-2} +1101 q^{-4} -1220 q^{-6} +1182 q^{-8} -1040 q^{-10} +827 q^{-12} -590 q^{-14} +384 q^{-16} -226 q^{-18} +117 q^{-20} -52 q^{-22} +20 q^{-24} -6 q^{-26} + q^{-28} }[/math]
2,0 [math]\displaystyle{ q^{52}-2 q^{50}-q^{48}+3 q^{46}-3 q^{44}+2 q^{42}+6 q^{40}-q^{38}-3 q^{36}+2 q^{34}+8 q^{32}-11 q^{30}-15 q^{28}+7 q^{26}+q^{24}-15 q^{22}+2 q^{20}+17 q^{18}-q^{16}-2 q^{14}+9 q^{12}+6 q^{10}-7 q^8+q^6+14 q^4-9 q^2-6+11 q^{-2} -13 q^{-6} - q^{-8} +9 q^{-10} - q^{-12} -7 q^{-14} +2 q^{-16} +6 q^{-18} -2 q^{-20} -2 q^{-22} + q^{-24} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{48}-3 q^{46}+8 q^{42}-9 q^{40}-3 q^{38}+18 q^{36}-13 q^{34}-11 q^{32}+28 q^{30}-12 q^{28}-18 q^{26}+24 q^{24}-11 q^{22}-18 q^{20}+7 q^{18}+3 q^{16}-q^{14}-3 q^{12}+19 q^{10}+14 q^8-19 q^6+11 q^4+14 q^2-28+4 q^{-2} +15 q^{-4} -21 q^{-6} +7 q^{-8} +9 q^{-10} -10 q^{-12} +5 q^{-14} + q^{-16} -3 q^{-18} + q^{-20} }[/math]
1,0,0 [math]\displaystyle{ q^{25}-2 q^{23}+2 q^{21}-5 q^{19}+q^{17}-4 q^{15}+2 q^{13}+q^{11}+4 q^9+4 q^7+q^5+3 q^3-3 q+2 q^{-1} -4 q^{-3} +2 q^{-5} -2 q^{-7} + q^{-9} }[/math]
1,0,1 [math]\displaystyle{ q^{78}-6 q^{76}+15 q^{74}-17 q^{72}-3 q^{70}+45 q^{68}-86 q^{66}+84 q^{64}-10 q^{62}-112 q^{60}+210 q^{58}-204 q^{56}+69 q^{54}+154 q^{52}-343 q^{50}+362 q^{48}-147 q^{46}-237 q^{44}+591 q^{42}-689 q^{40}+425 q^{38}+131 q^{36}-695 q^{34}+991 q^{32}-892 q^{30}+417 q^{28}+108 q^{26}-527 q^{24}+596 q^{22}-431 q^{20}+217 q^{18}-87 q^{16}+239 q^{14}-424 q^{12}+582 q^{10}-413 q^8-31 q^6+577 q^4-987 q^2+1018-706 q^{-2} +167 q^{-4} +339 q^{-6} -639 q^{-8} +659 q^{-10} -445 q^{-12} +140 q^{-14} +113 q^{-16} -223 q^{-18} +197 q^{-20} -104 q^{-22} +17 q^{-24} +26 q^{-26} -29 q^{-28} +17 q^{-30} -6 q^{-32} + q^{-34} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{58}-2 q^{56}-2 q^{54}+6 q^{52}-2 q^{50}-9 q^{48}+11 q^{46}+10 q^{44}-17 q^{42}+q^{40}+20 q^{38}-5 q^{36}-19 q^{34}+9 q^{32}+7 q^{30}-28 q^{28}-15 q^{26}+17 q^{24}-12 q^{22}-20 q^{20}+34 q^{18}+19 q^{16}-11 q^{14}+20 q^{12}+30 q^{10}-13 q^8-14 q^6+8 q^4-21-4 q^{-2} +13 q^{-4} -5 q^{-6} -3 q^{-8} +9 q^{-10} -5 q^{-14} +4 q^{-16} - q^{-18} -2 q^{-20} + q^{-22} }[/math]
1,0,0,0 [math]\displaystyle{ q^{30}-2 q^{28}+2 q^{26}-4 q^{24}-q^{22}-2 q^{20}-3 q^{18}+2 q^{16}+6 q^{12}+2 q^{10}+6 q^8+3 q^4-2 q^2-1-3 q^{-4} +2 q^{-6} -2 q^{-8} + q^{-10} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{48}-3 q^{46}+6 q^{44}-10 q^{42}+15 q^{40}-21 q^{38}+26 q^{36}-31 q^{34}+33 q^{32}-32 q^{30}+24 q^{28}-14 q^{26}-2 q^{24}+17 q^{22}-34 q^{20}+49 q^{18}-59 q^{16}+67 q^{14}-63 q^{12}+59 q^{10}-44 q^8+31 q^6-11 q^4-4 q^2+18-28 q^{-2} +33 q^{-4} -35 q^{-6} +31 q^{-8} -27 q^{-10} +20 q^{-12} -13 q^{-14} +7 q^{-16} -3 q^{-18} + q^{-20} }[/math]
1,0 [math]\displaystyle{ q^{78}-3 q^{74}-3 q^{72}+3 q^{70}+9 q^{68}+2 q^{66}-12 q^{64}-12 q^{62}+8 q^{60}+22 q^{58}+6 q^{56}-23 q^{54}-22 q^{52}+13 q^{50}+34 q^{48}+3 q^{46}-33 q^{44}-19 q^{42}+23 q^{40}+25 q^{38}-15 q^{36}-31 q^{34}+q^{32}+26 q^{30}+3 q^{28}-24 q^{26}-9 q^{24}+22 q^{22}+15 q^{20}-13 q^{18}-13 q^{16}+19 q^{14}+24 q^{12}-8 q^{10}-29 q^8+q^6+33 q^4+15 q^2-30-32 q^{-2} +13 q^{-4} +35 q^{-6} +3 q^{-8} -31 q^{-10} -17 q^{-12} +18 q^{-14} +21 q^{-16} -5 q^{-18} -15 q^{-20} -2 q^{-22} +9 q^{-24} +4 q^{-26} -3 q^{-28} -3 q^{-30} + q^{-34} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{66}-3 q^{64}+3 q^{62}-4 q^{60}+9 q^{58}-12 q^{56}+12 q^{54}-15 q^{52}+22 q^{50}-23 q^{48}+21 q^{46}-25 q^{44}+29 q^{42}-21 q^{40}+15 q^{38}-13 q^{36}+3 q^{34}+6 q^{32}-20 q^{30}+18 q^{28}-39 q^{26}+41 q^{24}-46 q^{22}+51 q^{20}-48 q^{18}+56 q^{16}-35 q^{14}+45 q^{12}-27 q^{10}+23 q^8-9 q^6+q^4+3 q^2-18+20 q^{-2} -26 q^{-4} +25 q^{-6} -28 q^{-8} +28 q^{-10} -23 q^{-12} +20 q^{-14} -16 q^{-16} +12 q^{-18} -7 q^{-20} +4 q^{-22} -3 q^{-24} + q^{-26} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{114}-3 q^{112}+6 q^{110}-10 q^{108}+9 q^{106}-6 q^{104}-2 q^{102}+18 q^{100}-32 q^{98}+47 q^{96}-50 q^{94}+34 q^{92}-6 q^{90}-34 q^{88}+75 q^{86}-104 q^{84}+117 q^{82}-101 q^{80}+50 q^{78}+25 q^{76}-110 q^{74}+182 q^{72}-205 q^{70}+161 q^{68}-63 q^{66}-68 q^{64}+177 q^{62}-218 q^{60}+166 q^{58}-45 q^{56}-99 q^{54}+184 q^{52}-176 q^{50}+58 q^{48}+108 q^{46}-239 q^{44}+275 q^{42}-192 q^{40}+16 q^{38}+182 q^{36}-322 q^{34}+358 q^{32}-271 q^{30}+102 q^{28}+103 q^{26}-251 q^{24}+316 q^{22}-264 q^{20}+137 q^{18}+29 q^{16}-166 q^{14}+221 q^{12}-170 q^{10}+47 q^8+109 q^6-214 q^4+212 q^2-106-64 q^{-2} +214 q^{-4} -286 q^{-6} +244 q^{-8} -112 q^{-10} -54 q^{-12} +184 q^{-14} -238 q^{-16} +206 q^{-18} -112 q^{-20} +3 q^{-22} +71 q^{-24} -104 q^{-26} +94 q^{-28} -58 q^{-30} +25 q^{-32} +5 q^{-34} -17 q^{-36} +17 q^{-38} -14 q^{-40} +7 q^{-42} -3 q^{-44} + q^{-46} }[/math]

.

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["10 112"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+5 t^3-11 t^2+17 t-19+17 t^{-1} -11 t^{-2} +5 t^{-3} - t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^8-3 z^6-z^4+2 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]= { 87, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^3-4 q^2+7 q-10+14 q^{-1} -14 q^{-2} +14 q^{-3} -11 q^{-4} +7 q^{-5} -4 q^{-6} + q^{-7} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -a^2 z^8+a^4 z^6-5 a^2 z^6+z^6+3 a^4 z^4-7 a^2 z^4+3 z^4+a^4 z^2+z^2-2 a^4+4 a^2-1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 3 a^3 z^9+3 a z^9+7 a^4 z^8+13 a^2 z^8+6 z^8+8 a^5 z^7+4 a^3 z^7+4 z^7 a^{-1} +7 a^6 z^6-9 a^4 z^6-35 a^2 z^6+z^6 a^{-2} -18 z^6+4 a^7 z^5-8 a^5 z^5-17 a^3 z^5-16 a z^5-11 z^5 a^{-1} +a^8 z^4-7 a^6 z^4+3 a^4 z^4+28 a^2 z^4-2 z^4 a^{-2} +15 z^4-3 a^7 z^3-a^5 z^3+9 a^3 z^3+13 a z^3+6 z^3 a^{-1} +a^6 z^2+a^4 z^2-3 a^2 z^2-3 z^2+2 a^5 z+2 a^3 z-2 a^4-4 a^2-1 }[/math]

Vassiliev invariants

V2 and V3: (2, -2)
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{ 8 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{220}{3} }[/math] [math]\displaystyle{ \frac{68}{3} }[/math] [math]\displaystyle{ -128 }[/math] [math]\displaystyle{ -\frac{736}{3} }[/math] [math]\displaystyle{ -\frac{160}{3} }[/math] [math]\displaystyle{ -48 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{1760}{3} }[/math] [math]\displaystyle{ \frac{544}{3} }[/math] [math]\displaystyle{ \frac{14431}{15} }[/math] [math]\displaystyle{ -\frac{604}{15} }[/math] [math]\displaystyle{ \frac{21724}{45} }[/math] [math]\displaystyle{ \frac{545}{9} }[/math] [math]\displaystyle{ \frac{1471}{15} }[/math]

V2,1 through V6,9 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]-2 is the signature of 10 112. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 -6-5-4-3-2-101234χ
7          11
5         3 -3
3        41 3
1       63  -3
-1      84   4
-3     77    0
-5    77     0
-7   47      3
-9  37       -4
-11 14        3
-13 3         -3
-151          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, 112]]
Out[2]=  
10
In[3]:=
PD[Knot[10, 112]]
Out[3]=  
PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[18, 11, 19, 12], X[20, 13, 1, 14], 

 X[2, 16, 3, 15], X[4, 17, 5, 18], X[12, 19, 13, 20], X[10, 6, 11, 5], 



  X[14, 7, 15, 8], X[16, 10, 17, 9]]
In[4]:=
GaussCode[Knot[10, 112]]
Out[4]=  
GaussCode[1, -5, 2, -6, 8, -1, 9, -2, 10, -8, 3, -7, 4, -9, 5, -10, 6, 
  -3, 7, -4]
In[5]:=
BR[Knot[10, 112]]
Out[5]=  
BR[3, {-1, -1, -1, 2, -1, 2, -1, 2, -1, 2}]
In[6]:=
alex = Alexander[Knot[10, 112]][t]
Out[6]=  
       -4   5    11   17              2      3    4

-19 - t   + -- - -- + -- + 17 t - 11 t  + 5 t  - t



            3    2   t


            t    t
In[7]:=
Conway[Knot[10, 112]][z]
Out[7]=  
       2    4      6    8
1 + 2 z  - z  - 3 z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[10, 112], Knot[11, Alternating, 184]}
In[9]:=
{KnotDet[Knot[10, 112]], KnotSignature[Knot[10, 112]]}
Out[9]=  
{87, -2}
In[10]:=
J=Jones[Knot[10, 112]][q]
Out[10]=  
       -7   4    7    11   14   14   14            2    3

-10 + q   - -- + -- - -- + -- - -- + -- + 7 q - 4 q  + q



            6    5    4    3    2   q


            q    q    q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[10, 112]}
In[12]:=
A2Invariant[Knot[10, 112]][q]
Out[12]=  
 -20    2     -16    3     -12    2     -8   6     -4   3       2

q    - --- + q    - --- - q    + --- - q   + -- - q   + -- - 2 q  + 



       18           14           10          6          2
      q            q            q           q          q

  4      6    8


  q  - 2 q  + q
In[13]:=
Kauffman[Knot[10, 112]][a, z]
Out[13]=  
        2      4      3        5        2      2  2    4  2    6  2

-1 - 4 a  - 2 a  + 2 a  z + 2 a  z - 3 z  - 3 a  z  + a  z  + a  z  + 



    3                                                    4
 6 z          3      3  3    5  3      7  3       4   2 z
 ---- + 13 a z  + 9 a  z  - a  z  - 3 a  z  + 15 z  - ---- + 
  a                                                     2
                                                       a

                                            5
     2  4      4  4      6  4    8  4   11 z          5       3  5
 28 a  z  + 3 a  z  - 7 a  z  + a  z  - ----- - 16 a z  - 17 a  z  - 
                                          a

                              6
    5  5      7  5       6   z        2  6      4  6      6  6
 8 a  z  + 4 a  z  - 18 z  + -- - 35 a  z  - 9 a  z  + 7 a  z  + 
                              2
                             a

    7
 4 z       3  7      5  7      8       2  8      4  8        9
 ---- + 4 a  z  + 8 a  z  + 6 z  + 13 a  z  + 7 a  z  + 3 a z  + 
  a

    3  9


  3 a  z
In[14]:=
{Vassiliev[2][Knot[10, 112]], Vassiliev[3][Knot[10, 112]]}
Out[14]=  
{0, -2}
In[15]:=
Kh[Knot[10, 112]][q, t]
Out[15]=  
7    8     1        3        1        4        3       7       4

-- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 



3   q    15  6    13  5    11  5    11  4    9  4    9  3    7  3



q        q   t    q   t    q   t    q   t    q  t    q  t    q  t



   7       7      7      7     4 t                2      3  2
 ----- + ----- + ---- + ---- + --- + 6 q t + 3 q t  + 4 q  t  + 
  7  2    5  2    5      3      q
 q  t    q  t    q  t   q  t

  3  3      5  3    7  4


  q  t  + 3 q  t  + q  t
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