8 18

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+5 t^2-10 t+13-10 t^{-1} +5 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6-z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \left\{t^2-t+1\right\} }[/math]
Determinant and Signature	{ 45, 0 }
Jones polynomial	[math]\displaystyle{ q^4-4 q^3+6 q^2-7 q+9-7 q^{-1} +6 q^{-2} -4 q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6+a^2 z^4+z^4 a^{-2} -3 z^4+a^2 z^2+z^2 a^{-2} -z^2-a^2- a^{-2} +3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 3 a z^7+3 z^7 a^{-1} +6 a^2 z^6+6 z^6 a^{-2} +12 z^6+4 a^3 z^5+3 a z^5+3 z^5 a^{-1} +4 z^5 a^{-3} +a^4 z^4-9 a^2 z^4-9 z^4 a^{-2} +z^4 a^{-4} -20 z^4-4 a^3 z^3-9 a z^3-9 z^3 a^{-1} -4 z^3 a^{-3} +3 a^2 z^2+3 z^2 a^{-2} +6 z^2+a z+z a^{-1} +a^2+ a^{-2} +3 }[/math]
The A2 invariant	[math]\displaystyle{ q^{12}-2 q^{10}-q^6-q^4+4 q^2+1+4 q^{-2} - q^{-4} - q^{-6} -2 q^{-10} + q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{66}-3 q^{64}+6 q^{62}-10 q^{60}+8 q^{58}-4 q^{56}-5 q^{54}+23 q^{52}-36 q^{50}+48 q^{48}-38 q^{46}+7 q^{44}+28 q^{42}-67 q^{40}+84 q^{38}-71 q^{36}+29 q^{34}+17 q^{32}-58 q^{30}+77 q^{28}-56 q^{26}+8 q^{24}+34 q^{22}-59 q^{20}+45 q^{18}-6 q^{16}-45 q^{14}+81 q^{12}-81 q^{10}+64 q^8-11 q^6-48 q^4+97 q^2-111+97 q^{-2} -48 q^{-4} -11 q^{-6} +64 q^{-8} -81 q^{-10} +81 q^{-12} -45 q^{-14} -6 q^{-16} +45 q^{-18} -59 q^{-20} +34 q^{-22} +8 q^{-24} -56 q^{-26} +77 q^{-28} -58 q^{-30} +17 q^{-32} +29 q^{-34} -71 q^{-36} +84 q^{-38} -67 q^{-40} +28 q^{-42} +7 q^{-44} -38 q^{-46} +48 q^{-48} -36 q^{-50} +23 q^{-52} -5 q^{-54} -4 q^{-56} +8 q^{-58} -10 q^{-60} +6 q^{-62} -3 q^{-64} + q^{-66} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_18.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^9-3 q^7+2 q^5-q^3+2 q+2 q^{-1} - q^{-3} +2 q^{-5} -3 q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{26}-3 q^{24}-q^{22}+11 q^{20}-6 q^{18}-12 q^{16}+16 q^{14}-q^{12}-17 q^{10}+11 q^8+6 q^6-10 q^4+2 q^2+9+2 q^{-2} -10 q^{-4} +6 q^{-6} +11 q^{-8} -17 q^{-10} - q^{-12} +16 q^{-14} -12 q^{-16} -6 q^{-18} +11 q^{-20} - q^{-22} -3 q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ q^{51}-3 q^{49}-q^{47}+8 q^{45}+6 q^{43}-14 q^{41}-26 q^{39}+20 q^{37}+48 q^{35}-7 q^{33}-70 q^{31}-17 q^{29}+86 q^{27}+44 q^{25}-82 q^{23}-71 q^{21}+65 q^{19}+81 q^{17}-42 q^{15}-87 q^{13}+21 q^{11}+73 q^9+6 q^7-58 q^5-21 q^3+42 q+42 q^{-1} -21 q^{-3} -58 q^{-5} +6 q^{-7} +73 q^{-9} +21 q^{-11} -87 q^{-13} -42 q^{-15} +81 q^{-17} +65 q^{-19} -71 q^{-21} -82 q^{-23} +44 q^{-25} +86 q^{-27} -17 q^{-29} -70 q^{-31} -7 q^{-33} +48 q^{-35} +20 q^{-37} -26 q^{-39} -14 q^{-41} +6 q^{-43} +8 q^{-45} - q^{-47} -3 q^{-49} + q^{-51} }[/math]
4	[math]\displaystyle{ q^{84}-3 q^{82}-q^{80}+8 q^{78}+3 q^{76}-2 q^{74}-28 q^{72}-16 q^{70}+41 q^{68}+56 q^{66}+40 q^{64}-103 q^{62}-153 q^{60}-2 q^{58}+172 q^{56}+269 q^{54}-41 q^{52}-349 q^{50}-286 q^{48}+88 q^{46}+541 q^{44}+295 q^{42}-291 q^{40}-582 q^{38}-260 q^{36}+507 q^{34}+589 q^{32}+29 q^{30}-565 q^{28}-529 q^{26}+216 q^{24}+566 q^{22}+282 q^{20}-309 q^{18}-511 q^{16}-45 q^{14}+349 q^{12}+338 q^{10}-61 q^8-348 q^6-197 q^4+133 q^2+323+133 q^{-2} -197 q^{-4} -348 q^{-6} -61 q^{-8} +338 q^{-10} +349 q^{-12} -45 q^{-14} -511 q^{-16} -309 q^{-18} +282 q^{-20} +566 q^{-22} +216 q^{-24} -529 q^{-26} -565 q^{-28} +29 q^{-30} +589 q^{-32} +507 q^{-34} -260 q^{-36} -582 q^{-38} -291 q^{-40} +295 q^{-42} +541 q^{-44} +88 q^{-46} -286 q^{-48} -349 q^{-50} -41 q^{-52} +269 q^{-54} +172 q^{-56} -2 q^{-58} -153 q^{-60} -103 q^{-62} +40 q^{-64} +56 q^{-66} +41 q^{-68} -16 q^{-70} -28 q^{-72} -2 q^{-74} +3 q^{-76} +8 q^{-78} - q^{-80} -3 q^{-82} + q^{-84} }[/math]
5	[math]\displaystyle{ q^{125}-3 q^{123}-q^{121}+8 q^{119}+3 q^{117}-5 q^{115}-16 q^{113}-18 q^{111}+5 q^{109}+55 q^{107}+75 q^{105}+5 q^{103}-117 q^{101}-199 q^{99}-116 q^{97}+135 q^{95}+429 q^{93}+425 q^{91}-31 q^{89}-637 q^{87}-905 q^{85}-422 q^{83}+649 q^{81}+1500 q^{79}+1200 q^{77}-277 q^{75}-1879 q^{73}-2201 q^{71}-621 q^{69}+1832 q^{67}+3129 q^{65}+1860 q^{63}-1222 q^{61}-3614 q^{59}-3154 q^{57}+104 q^{55}+3538 q^{53}+4145 q^{51}+1191 q^{49}-2871 q^{47}-4563 q^{45}-2378 q^{43}+1862 q^{41}+4404 q^{39}+3127 q^{37}-751 q^{35}-3793 q^{33}-3415 q^{31}-164 q^{29}+2928 q^{27}+3251 q^{25}+827 q^{23}-2050 q^{21}-2874 q^{19}-1163 q^{17}+1281 q^{15}+2369 q^{13}+1365 q^{11}-676 q^9-1958 q^7-1474 q^5+218 q^3+1650 q+1650 q^{-1} +218 q^{-3} -1474 q^{-5} -1958 q^{-7} -676 q^{-9} +1365 q^{-11} +2369 q^{-13} +1281 q^{-15} -1163 q^{-17} -2874 q^{-19} -2050 q^{-21} +827 q^{-23} +3251 q^{-25} +2928 q^{-27} -164 q^{-29} -3415 q^{-31} -3793 q^{-33} -751 q^{-35} +3127 q^{-37} +4404 q^{-39} +1862 q^{-41} -2378 q^{-43} -4563 q^{-45} -2871 q^{-47} +1191 q^{-49} +4145 q^{-51} +3538 q^{-53} +104 q^{-55} -3154 q^{-57} -3614 q^{-59} -1222 q^{-61} +1860 q^{-63} +3129 q^{-65} +1832 q^{-67} -621 q^{-69} -2201 q^{-71} -1879 q^{-73} -277 q^{-75} +1200 q^{-77} +1500 q^{-79} +649 q^{-81} -422 q^{-83} -905 q^{-85} -637 q^{-87} -31 q^{-89} +425 q^{-91} +429 q^{-93} +135 q^{-95} -116 q^{-97} -199 q^{-99} -117 q^{-101} +5 q^{-103} +75 q^{-105} +55 q^{-107} +5 q^{-109} -18 q^{-111} -16 q^{-113} -5 q^{-115} +3 q^{-117} +8 q^{-119} - q^{-121} -3 q^{-123} + q^{-125} }[/math]
6	[math]\displaystyle{ q^{174}-3 q^{172}-q^{170}+8 q^{168}+3 q^{166}-5 q^{164}-19 q^{162}-6 q^{160}+3 q^{158}+19 q^{156}+74 q^{154}+46 q^{152}-37 q^{150}-165 q^{148}-191 q^{146}-117 q^{144}+105 q^{142}+493 q^{140}+615 q^{138}+334 q^{136}-418 q^{134}-1106 q^{132}-1471 q^{130}-907 q^{128}+794 q^{126}+2478 q^{124}+3133 q^{122}+1676 q^{120}-1190 q^{118}-4531 q^{116}-5858 q^{114}-3284 q^{112}+2081 q^{110}+7610 q^{108}+9275 q^{106}+5639 q^{104}-3099 q^{102}-11670 q^{100}-14037 q^{98}-7817 q^{96}+4735 q^{94}+16040 q^{92}+19384 q^{90}+9942 q^{88}-7128 q^{86}-21493 q^{84}-23749 q^{82}-10860 q^{80}+9930 q^{78}+26865 q^{76}+27036 q^{74}+10046 q^{72}-14232 q^{70}-30534 q^{68}-27888 q^{66}-7581 q^{64}+18756 q^{62}+32595 q^{60}+25896 q^{58}+2664 q^{56}-21877 q^{54}-31950 q^{52}-21520 q^{50}+2933 q^{48}+23687 q^{46}+28502 q^{44}+14738 q^{42}-7366 q^{40}-23259 q^{38}-23130 q^{36}-7741 q^{34}+10599 q^{32}+20653 q^{30}+16332 q^{28}+2223 q^{26}-11946 q^{24}-16899 q^{22}-10125 q^{20}+2028 q^{18}+11760 q^{16}+12663 q^{14}+5442 q^{12}-4872 q^{10}-11043 q^8-9364 q^6-1663 q^4+7039 q^2+10561+7039 q^{-2} -1663 q^{-4} -9364 q^{-6} -11043 q^{-8} -4872 q^{-10} +5442 q^{-12} +12663 q^{-14} +11760 q^{-16} +2028 q^{-18} -10125 q^{-20} -16899 q^{-22} -11946 q^{-24} +2223 q^{-26} +16332 q^{-28} +20653 q^{-30} +10599 q^{-32} -7741 q^{-34} -23130 q^{-36} -23259 q^{-38} -7366 q^{-40} +14738 q^{-42} +28502 q^{-44} +23687 q^{-46} +2933 q^{-48} -21520 q^{-50} -31950 q^{-52} -21877 q^{-54} +2664 q^{-56} +25896 q^{-58} +32595 q^{-60} +18756 q^{-62} -7581 q^{-64} -27888 q^{-66} -30534 q^{-68} -14232 q^{-70} +10046 q^{-72} +27036 q^{-74} +26865 q^{-76} +9930 q^{-78} -10860 q^{-80} -23749 q^{-82} -21493 q^{-84} -7128 q^{-86} +9942 q^{-88} +19384 q^{-90} +16040 q^{-92} +4735 q^{-94} -7817 q^{-96} -14037 q^{-98} -11670 q^{-100} -3099 q^{-102} +5639 q^{-104} +9275 q^{-106} +7610 q^{-108} +2081 q^{-110} -3284 q^{-112} -5858 q^{-114} -4531 q^{-116} -1190 q^{-118} +1676 q^{-120} +3133 q^{-122} +2478 q^{-124} +794 q^{-126} -907 q^{-128} -1471 q^{-130} -1106 q^{-132} -418 q^{-134} +334 q^{-136} +615 q^{-138} +493 q^{-140} +105 q^{-142} -117 q^{-144} -191 q^{-146} -165 q^{-148} -37 q^{-150} +46 q^{-152} +74 q^{-154} +19 q^{-156} +3 q^{-158} -6 q^{-160} -19 q^{-162} -5 q^{-164} +3 q^{-166} +8 q^{-168} - q^{-170} -3 q^{-172} + q^{-174} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{12}-2 q^{10}-q^6-q^4+4 q^2+1+4 q^{-2} - q^{-4} - q^{-6} -2 q^{-10} + q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{36}-6 q^{34}+18 q^{32}-38 q^{30}+71 q^{28}-124 q^{26}+188 q^{24}-246 q^{22}+300 q^{20}-322 q^{18}+298 q^{16}-236 q^{14}+111 q^{12}+36 q^{10}-204 q^8+360 q^6-482 q^4+578 q^2-598+578 q^{-2} -482 q^{-4} +360 q^{-6} -204 q^{-8} +36 q^{-10} +111 q^{-12} -236 q^{-14} +298 q^{-16} -322 q^{-18} +300 q^{-20} -246 q^{-22} +188 q^{-24} -124 q^{-26} +71 q^{-28} -38 q^{-30} +18 q^{-32} -6 q^{-34} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{32}-2 q^{30}-2 q^{28}+5 q^{26}+2 q^{24}-3 q^{22}-2 q^{20}+5 q^{18}+2 q^{16}-11 q^{14}-2 q^{12}+3 q^{10}-6 q^8-4 q^6+7 q^4+8 q^2+4+8 q^{-2} +7 q^{-4} -4 q^{-6} -6 q^{-8} +3 q^{-10} -2 q^{-12} -11 q^{-14} +2 q^{-16} +5 q^{-18} -2 q^{-20} -3 q^{-22} +2 q^{-24} +5 q^{-26} -2 q^{-28} -2 q^{-30} + q^{-32} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{28}-3 q^{26}+7 q^{22}-8 q^{20}+2 q^{18}+11 q^{16}-14 q^{14}-q^{12}+7 q^{10}-12 q^8-2 q^6+9 q^4+4 q^2+4+4 q^{-2} +9 q^{-4} -2 q^{-6} -12 q^{-8} +7 q^{-10} - q^{-12} -14 q^{-14} +11 q^{-16} +2 q^{-18} -8 q^{-20} +7 q^{-22} -3 q^{-26} + q^{-28} }[/math]
1,0,0	[math]\displaystyle{ q^{15}-2 q^{13}+q^{11}-3 q^9-q^5+3 q^3+3 q+3 q^{-1} +3 q^{-3} - q^{-5} -3 q^{-9} + q^{-11} -2 q^{-13} + q^{-15} }[/math]
1,0,1	[math]\displaystyle{ q^{46}-6 q^{44}+15 q^{42}-17 q^{40}-3 q^{38}+48 q^{36}-95 q^{34}+100 q^{32}-28 q^{30}-107 q^{28}+237 q^{26}-277 q^{24}+195 q^{22}+11 q^{20}-243 q^{18}+384 q^{16}-393 q^{14}+209 q^{12}+14 q^{10}-210 q^8+257 q^6-153 q^4+47 q^2+43+47 q^{-2} -153 q^{-4} +257 q^{-6} -210 q^{-8} +14 q^{-10} +209 q^{-12} -393 q^{-14} +384 q^{-16} -243 q^{-18} +11 q^{-20} +195 q^{-22} -277 q^{-24} +237 q^{-26} -107 q^{-28} -28 q^{-30} +100 q^{-32} -95 q^{-34} +48 q^{-36} -3 q^{-38} -17 q^{-40} +15 q^{-42} -6 q^{-44} + q^{-46} }[/math]

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{38}-3 q^{36}+3 q^{34}-4 q^{32}+8 q^{30}-10 q^{28}+10 q^{26}-9 q^{24}+11 q^{22}-9 q^{20}+3 q^{18}-6 q^{16}+2 q^{12}-11 q^{10}+9 q^8-10 q^6+21 q^4-13 q^2+22-13 q^{-2} +21 q^{-4} -10 q^{-6} +9 q^{-8} -11 q^{-10} +2 q^{-12} -6 q^{-16} +3 q^{-18} -9 q^{-20} +11 q^{-22} -9 q^{-24} +10 q^{-26} -10 q^{-28} +8 q^{-30} -4 q^{-32} +3 q^{-34} -3 q^{-36} + q^{-38} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{66}-3 q^{64}+6 q^{62}-10 q^{60}+8 q^{58}-4 q^{56}-5 q^{54}+23 q^{52}-36 q^{50}+48 q^{48}-38 q^{46}+7 q^{44}+28 q^{42}-67 q^{40}+84 q^{38}-71 q^{36}+29 q^{34}+17 q^{32}-58 q^{30}+77 q^{28}-56 q^{26}+8 q^{24}+34 q^{22}-59 q^{20}+45 q^{18}-6 q^{16}-45 q^{14}+81 q^{12}-81 q^{10}+64 q^8-11 q^6-48 q^4+97 q^2-111+97 q^{-2} -48 q^{-4} -11 q^{-6} +64 q^{-8} -81 q^{-10} +81 q^{-12} -45 q^{-14} -6 q^{-16} +45 q^{-18} -59 q^{-20} +34 q^{-22} +8 q^{-24} -56 q^{-26} +77 q^{-28} -58 q^{-30} +17 q^{-32} +29 q^{-34} -71 q^{-36} +84 q^{-38} -67 q^{-40} +28 q^{-42} +7 q^{-44} -38 q^{-46} +48 q^{-48} -36 q^{-50} +23 q^{-52} -5 q^{-54} -4 q^{-56} +8 q^{-58} -10 q^{-60} +6 q^{-62} -3 q^{-64} + q^{-66} }[/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["8 18"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+5 t^2-10 t+13-10 t^{-1} +5 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6-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{ \left\{t^2-t+1\right\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 45, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^4-4 q^3+6 q^2-7 q+9-7 q^{-1} +6 q^{-2} -4 q^{-3} + q^{-4} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^6+a^2 z^4+z^4 a^{-2} -3 z^4+a^2 z^2+z^2 a^{-2} -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{ 3 a z^7+3 z^7 a^{-1} +6 a^2 z^6+6 z^6 a^{-2} +12 z^6+4 a^3 z^5+3 a z^5+3 z^5 a^{-1} +4 z^5 a^{-3} +a^4 z^4-9 a^2 z^4-9 z^4 a^{-2} +z^4 a^{-4} -20 z^4-4 a^3 z^3-9 a z^3-9 z^3 a^{-1} -4 z^3 a^{-3} +3 a^2 z^2+3 z^2 a^{-2} +6 z^2+a z+z a^{-1} +a^2+ a^{-2} +3 }[/math]