10 47

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10 46.gif

10_46

10 48.gif

10_48

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10 47 Quick Notes


10 47 Further Notes and Views

Knot presentations

Planar diagram presentation X1425 X3849 X9,17,10,16 X5,15,6,14 X15,7,16,6 X11,19,12,18 X13,1,14,20 X17,11,18,10 X19,13,20,12 X7283
Gauss code -1, 10, -2, 1, -4, 5, -10, 2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -9, 7
Dowker-Thistlethwaite code 4 8 14 2 16 18 20 6 10 12
Conway Notation [5,21,2]

Three dimensional invariants

Symmetry type Reversible
Unknotting number [math]\displaystyle{ \{2,3\} }[/math]
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [0][-12]
Hyperbolic Volume 9.38519
A-Polynomial See Data:10 47/A-polynomial

[edit Notes for 10 47's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 2 }[/math]
Topological 4 genus [math]\displaystyle{ 2 }[/math]
Concordance genus [math]\displaystyle{ 4 }[/math]
Rasmussen s-Invariant 4

[edit Notes for 10 47's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-3 t^3+6 t^2-7 t+7-7 t^{-1} +6 t^{-2} -3 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+5 z^6+8 z^4+6 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 41, 4 }
Jones polynomial [math]\displaystyle{ -q^9+2 q^8-4 q^7+5 q^6-6 q^5+7 q^4-5 q^3+5 q^2-3 q+2- q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^8 a^{-4} -z^6 a^{-2} +7 z^6 a^{-4} -z^6 a^{-6} -5 z^4 a^{-2} +18 z^4 a^{-4} -5 z^4 a^{-6} -7 z^2 a^{-2} +21 z^2 a^{-4} -8 z^2 a^{-6} -3 a^{-2} +9 a^{-4} -5 a^{-6} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +5 z^8 a^{-4} +3 z^8 a^{-6} +z^7 a^{-1} -z^7 a^{-3} +z^7 a^{-5} +3 z^7 a^{-7} -10 z^6 a^{-2} -23 z^6 a^{-4} -10 z^6 a^{-6} +3 z^6 a^{-8} -5 z^5 a^{-1} -11 z^5 a^{-3} -14 z^5 a^{-5} -5 z^5 a^{-7} +3 z^5 a^{-9} +15 z^4 a^{-2} +35 z^4 a^{-4} +15 z^4 a^{-6} -3 z^4 a^{-8} +2 z^4 a^{-10} +7 z^3 a^{-1} +20 z^3 a^{-3} +19 z^3 a^{-5} +2 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} -9 z^2 a^{-2} -26 z^2 a^{-4} -15 z^2 a^{-6} +z^2 a^{-8} -z^2 a^{-10} -3 z a^{-1} -8 z a^{-3} -9 z a^{-5} -z a^{-7} +2 z a^{-9} -z a^{-11} +3 a^{-2} +9 a^{-4} +5 a^{-6} }[/math]
The A2 invariant [math]\displaystyle{ -q^2- q^{-2} + q^{-6} + q^{-8} +4 q^{-10} + q^{-12} +3 q^{-14} - q^{-18} - q^{-20} -2 q^{-22} - q^{-26} }[/math]
The G2 invariant [math]\displaystyle{ q^{12}-q^{10}+3 q^8-4 q^6+3 q^4-3 q^2-2+7 q^{-2} -14 q^{-4} +15 q^{-6} -14 q^{-8} +3 q^{-10} +7 q^{-12} -19 q^{-14} +23 q^{-16} -20 q^{-18} +9 q^{-20} +3 q^{-22} -15 q^{-24} +18 q^{-26} -12 q^{-28} +4 q^{-30} +8 q^{-32} -10 q^{-34} +11 q^{-36} -6 q^{-40} +16 q^{-42} -14 q^{-44} +15 q^{-46} -2 q^{-48} -5 q^{-50} +18 q^{-52} -20 q^{-54} +24 q^{-56} -12 q^{-58} +2 q^{-60} +11 q^{-62} -17 q^{-64} +19 q^{-66} -13 q^{-68} +5 q^{-70} +5 q^{-72} -10 q^{-74} +8 q^{-76} -4 q^{-78} -5 q^{-80} +8 q^{-82} -9 q^{-84} +3 q^{-88} -9 q^{-90} +8 q^{-92} -7 q^{-94} +2 q^{-96} - q^{-98} -4 q^{-100} +4 q^{-102} -7 q^{-104} +6 q^{-106} -6 q^{-108} +5 q^{-110} -3 q^{-112} -2 q^{-114} +6 q^{-116} -10 q^{-118} +11 q^{-120} -7 q^{-122} +3 q^{-124} + q^{-126} -5 q^{-128} +6 q^{-130} -6 q^{-132} +5 q^{-134} -2 q^{-136} + q^{-140} -2 q^{-142} +2 q^{-144} - q^{-146} + q^{-148} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_47.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^3+q- q^{-1} +2 q^{-3} +2 q^{-7} + q^{-9} - q^{-11} + q^{-13} -2 q^{-15} + q^{-17} - q^{-19} }[/math]
2 [math]\displaystyle{ q^{12}-q^{10}-2 q^8+3 q^6-5 q^2+3+4 q^{-2} -5 q^{-4} +5 q^{-8} -2 q^{-10} - q^{-12} +5 q^{-14} + q^{-16} -2 q^{-18} +2 q^{-20} +3 q^{-22} -3 q^{-24} -2 q^{-26} +3 q^{-28} -5 q^{-32} +2 q^{-34} + q^{-36} -4 q^{-38} +2 q^{-40} - q^{-42} -2 q^{-44} +3 q^{-46} - q^{-50} + q^{-52} }[/math]
3 [math]\displaystyle{ -q^{27}+q^{25}+2 q^{23}-4 q^{19}-2 q^{17}+6 q^{15}+5 q^{13}-6 q^{11}-10 q^9+2 q^7+13 q^5+3 q^3-13 q-8 q^{-1} +8 q^{-3} +14 q^{-5} -3 q^{-7} -12 q^{-9} -4 q^{-11} +10 q^{-13} +9 q^{-15} -3 q^{-17} -11 q^{-19} + q^{-21} +11 q^{-23} +7 q^{-25} -9 q^{-27} -6 q^{-29} +10 q^{-31} +9 q^{-33} -10 q^{-35} -10 q^{-37} +9 q^{-39} +9 q^{-41} -5 q^{-43} -12 q^{-45} + q^{-47} +8 q^{-49} +6 q^{-51} -7 q^{-53} -11 q^{-55} -2 q^{-57} +14 q^{-59} +4 q^{-61} -11 q^{-63} -12 q^{-65} +6 q^{-67} +11 q^{-69} -7 q^{-73} -4 q^{-75} +6 q^{-77} +6 q^{-79} - q^{-81} -5 q^{-83} - q^{-85} +4 q^{-87} +2 q^{-89} -2 q^{-91} - q^{-93} + q^{-97} - q^{-99} }[/math]
4 [math]\displaystyle{ q^{48}-q^{46}-2 q^{44}+q^{40}+6 q^{38}-6 q^{34}-6 q^{32}-4 q^{30}+15 q^{28}+12 q^{26}-2 q^{24}-15 q^{22}-25 q^{20}+7 q^{18}+24 q^{16}+23 q^{14}+q^{12}-39 q^{10}-23 q^8+2 q^6+32 q^4+36 q^2-10-27 q^{-2} -33 q^{-4} -3 q^{-6} +34 q^{-8} +25 q^{-10} +14 q^{-12} -23 q^{-14} -38 q^{-16} -12 q^{-18} +15 q^{-20} +48 q^{-22} +25 q^{-24} -26 q^{-26} -48 q^{-28} -29 q^{-30} +40 q^{-32} +62 q^{-34} +11 q^{-36} -48 q^{-38} -57 q^{-40} +12 q^{-42} +62 q^{-44} +31 q^{-46} -30 q^{-48} -55 q^{-50} -3 q^{-52} +49 q^{-54} +27 q^{-56} -26 q^{-58} -44 q^{-60} -5 q^{-62} +44 q^{-64} +29 q^{-66} -21 q^{-68} -43 q^{-70} -23 q^{-72} +29 q^{-74} +46 q^{-76} +14 q^{-78} -24 q^{-80} -57 q^{-82} -27 q^{-84} +38 q^{-86} +65 q^{-88} +32 q^{-90} -58 q^{-92} -83 q^{-94} -13 q^{-96} +70 q^{-98} +87 q^{-100} -12 q^{-102} -86 q^{-104} -54 q^{-106} +29 q^{-108} +86 q^{-110} +23 q^{-112} -46 q^{-114} -51 q^{-116} -7 q^{-118} +52 q^{-120} +25 q^{-122} -14 q^{-124} -28 q^{-126} -16 q^{-128} +23 q^{-130} +14 q^{-132} + q^{-134} -11 q^{-136} -13 q^{-138} +7 q^{-140} +4 q^{-142} +3 q^{-144} -2 q^{-146} -5 q^{-148} +2 q^{-150} + q^{-154} - q^{-158} + q^{-160} }[/math]
5 [math]\displaystyle{ -q^{75}+q^{73}+2 q^{71}-q^{67}-3 q^{65}-4 q^{63}+8 q^{59}+8 q^{57}+2 q^{55}-7 q^{53}-16 q^{51}-14 q^{49}+4 q^{47}+26 q^{45}+28 q^{43}+10 q^{41}-21 q^{39}-47 q^{37}-38 q^{35}+5 q^{33}+53 q^{31}+64 q^{29}+32 q^{27}-31 q^{25}-81 q^{23}-74 q^{21}-12 q^{19}+64 q^{17}+95 q^{15}+65 q^{13}-14 q^{11}-83 q^9-97 q^7-48 q^5+34 q^3+84 q+86 q^{-1} +41 q^{-3} -30 q^{-5} -84 q^{-7} -91 q^{-9} -49 q^{-11} +19 q^{-13} +100 q^{-15} +127 q^{-17} +69 q^{-19} -48 q^{-21} -152 q^{-23} -164 q^{-25} -48 q^{-27} +135 q^{-29} +230 q^{-31} +153 q^{-33} -56 q^{-35} -248 q^{-37} -243 q^{-39} -39 q^{-41} +214 q^{-43} +300 q^{-45} +136 q^{-47} -148 q^{-49} -307 q^{-51} -207 q^{-53} +65 q^{-55} +284 q^{-57} +256 q^{-59} +6 q^{-61} -235 q^{-63} -261 q^{-65} -64 q^{-67} +176 q^{-69} +248 q^{-71} +90 q^{-73} -131 q^{-75} -211 q^{-77} -93 q^{-79} +96 q^{-81} +171 q^{-83} +77 q^{-85} -90 q^{-87} -150 q^{-89} -51 q^{-91} +95 q^{-93} +138 q^{-95} +45 q^{-97} -104 q^{-99} -157 q^{-101} -63 q^{-103} +94 q^{-105} +172 q^{-107} +115 q^{-109} -40 q^{-111} -179 q^{-113} -186 q^{-115} -53 q^{-117} +134 q^{-119} +250 q^{-121} +173 q^{-123} -47 q^{-125} -260 q^{-127} -294 q^{-129} -81 q^{-131} +231 q^{-133} +365 q^{-135} +203 q^{-137} -131 q^{-139} -380 q^{-141} -303 q^{-143} +38 q^{-145} +333 q^{-147} +334 q^{-149} +52 q^{-151} -250 q^{-153} -315 q^{-155} -99 q^{-157} +175 q^{-159} +257 q^{-161} +105 q^{-163} -115 q^{-165} -188 q^{-167} -89 q^{-169} +74 q^{-171} +139 q^{-173} +62 q^{-175} -54 q^{-177} -97 q^{-179} -43 q^{-181} +38 q^{-183} +67 q^{-185} +32 q^{-187} -27 q^{-189} -49 q^{-191} -26 q^{-193} +17 q^{-195} +31 q^{-197} +18 q^{-199} -5 q^{-201} -19 q^{-203} -14 q^{-205} + q^{-207} +12 q^{-209} +6 q^{-211} + q^{-213} -3 q^{-215} -5 q^{-217} +3 q^{-221} + q^{-223} - q^{-229} + q^{-233} - q^{-235} }[/math]
6 [math]\displaystyle{ q^{108}-q^{106}-2 q^{104}+q^{100}+3 q^{98}+q^{96}+4 q^{94}-2 q^{92}-10 q^{90}-7 q^{88}-2 q^{86}+8 q^{84}+10 q^{82}+21 q^{80}+8 q^{78}-16 q^{76}-30 q^{74}-32 q^{72}-11 q^{70}+8 q^{68}+59 q^{66}+64 q^{64}+30 q^{62}-23 q^{60}-78 q^{58}-93 q^{56}-82 q^{54}+23 q^{52}+108 q^{50}+147 q^{48}+110 q^{46}+11 q^{44}-106 q^{42}-213 q^{40}-164 q^{38}-53 q^{36}+109 q^{34}+216 q^{32}+229 q^{30}+127 q^{28}-85 q^{26}-209 q^{24}-263 q^{22}-171 q^{20}-13 q^{18}+167 q^{16}+265 q^{14}+206 q^{12}+103 q^{10}-67 q^8-176 q^6-243 q^4-193 q^2-75+42 q^{-2} +200 q^{-4} +278 q^{-6} +284 q^{-8} +122 q^{-10} -113 q^{-12} -346 q^{-14} -484 q^{-16} -361 q^{-18} -17 q^{-20} +434 q^{-22} +678 q^{-24} +597 q^{-26} +161 q^{-28} -465 q^{-30} -888 q^{-32} -837 q^{-34} -251 q^{-36} +504 q^{-38} +1045 q^{-40} +1027 q^{-42} +359 q^{-44} -566 q^{-46} -1199 q^{-48} -1118 q^{-50} -396 q^{-52} +602 q^{-54} +1296 q^{-56} +1197 q^{-58} +367 q^{-60} -691 q^{-62} -1316 q^{-64} -1180 q^{-66} -317 q^{-68} +776 q^{-70} +1356 q^{-72} +1090 q^{-74} +161 q^{-76} -834 q^{-78} -1323 q^{-80} -971 q^{-82} +26 q^{-84} +949 q^{-86} +1228 q^{-88} +723 q^{-90} -212 q^{-92} -997 q^{-94} -1099 q^{-96} -439 q^{-98} +443 q^{-100} +965 q^{-102} +820 q^{-104} +153 q^{-106} -578 q^{-108} -861 q^{-110} -491 q^{-112} +154 q^{-114} +598 q^{-116} +578 q^{-118} +153 q^{-120} -336 q^{-122} -521 q^{-124} -253 q^{-126} +185 q^{-128} +419 q^{-130} +298 q^{-132} -74 q^{-134} -403 q^{-136} -425 q^{-138} -90 q^{-140} +350 q^{-142} +558 q^{-144} +391 q^{-146} -60 q^{-148} -510 q^{-150} -679 q^{-152} -444 q^{-154} +100 q^{-156} +627 q^{-158} +836 q^{-160} +576 q^{-162} -44 q^{-164} -716 q^{-166} -1066 q^{-168} -791 q^{-170} -10 q^{-172} +862 q^{-174} +1295 q^{-176} +997 q^{-178} +52 q^{-180} -1039 q^{-182} -1516 q^{-184} -1090 q^{-186} +34 q^{-188} +1155 q^{-190} +1607 q^{-192} +1067 q^{-194} -184 q^{-196} -1246 q^{-198} -1514 q^{-200} -831 q^{-202} +302 q^{-204} +1199 q^{-206} +1278 q^{-208} +515 q^{-210} -424 q^{-212} -997 q^{-214} -864 q^{-216} -262 q^{-218} +432 q^{-220} +722 q^{-222} +468 q^{-224} +42 q^{-226} -324 q^{-228} -374 q^{-230} -217 q^{-232} +52 q^{-234} +186 q^{-236} +114 q^{-238} +37 q^{-240} -39 q^{-242} -21 q^{-244} -9 q^{-246} +32 q^{-248} +6 q^{-250} -67 q^{-252} -54 q^{-254} -25 q^{-256} +49 q^{-258} +61 q^{-260} +66 q^{-262} +11 q^{-264} -61 q^{-266} -58 q^{-268} -40 q^{-270} +15 q^{-272} +30 q^{-274} +47 q^{-276} +22 q^{-278} -16 q^{-280} -20 q^{-282} -24 q^{-284} -3 q^{-286} +2 q^{-288} +17 q^{-290} +10 q^{-292} -2 q^{-294} -2 q^{-296} -8 q^{-298} - q^{-300} -2 q^{-302} +4 q^{-304} +2 q^{-306} - q^{-308} + q^{-310} -3 q^{-312} + q^{-318} - q^{-322} + q^{-324} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^2- q^{-2} + q^{-6} + q^{-8} +4 q^{-10} + q^{-12} +3 q^{-14} - q^{-18} - q^{-20} -2 q^{-22} - q^{-26} }[/math]
1,1 [math]\displaystyle{ q^{12}-2 q^{10}+6 q^8-12 q^6+21 q^4-34 q^2+48-62 q^{-2} +65 q^{-4} -72 q^{-6} +58 q^{-8} -42 q^{-10} +14 q^{-12} +16 q^{-14} -38 q^{-16} +78 q^{-18} -80 q^{-20} +108 q^{-22} -94 q^{-24} +102 q^{-26} -84 q^{-28} +60 q^{-30} -48 q^{-32} +16 q^{-34} -9 q^{-36} -14 q^{-38} +14 q^{-40} -20 q^{-42} +17 q^{-44} -12 q^{-46} +12 q^{-48} -14 q^{-50} +19 q^{-52} -20 q^{-54} +22 q^{-56} -28 q^{-58} +26 q^{-60} -22 q^{-62} +18 q^{-64} -14 q^{-66} +11 q^{-68} -6 q^{-70} +4 q^{-72} -2 q^{-74} + q^{-76} }[/math]
2,0 [math]\displaystyle{ q^{10}-q^6+q^2-1-3 q^{-2} +2 q^{-6} -2 q^{-8} -3 q^{-10} + q^{-12} + q^{-14} - q^{-16} + q^{-18} +5 q^{-20} +3 q^{-22} +5 q^{-24} +5 q^{-26} +5 q^{-28} + q^{-30} +4 q^{-32} +4 q^{-34} -2 q^{-36} -3 q^{-38} -2 q^{-40} -4 q^{-42} -8 q^{-44} -6 q^{-46} -3 q^{-48} - q^{-50} - q^{-52} +2 q^{-56} +2 q^{-58} +2 q^{-60} + q^{-62} + q^{-66} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^6-q^4+q^2+1-3 q^{-2} -3 q^{-6} -6 q^{-8} - q^{-10} - q^{-12} +9 q^{-16} +8 q^{-18} +7 q^{-20} +11 q^{-22} +6 q^{-24} -2 q^{-26} - q^{-28} -4 q^{-30} -5 q^{-32} -7 q^{-34} -2 q^{-36} -5 q^{-40} +3 q^{-44} -3 q^{-46} - q^{-48} +5 q^{-50} -2 q^{-52} -2 q^{-54} +3 q^{-56} - q^{-60} + q^{-62} }[/math]
1,0,0 [math]\displaystyle{ -q-2 q^{-3} -2 q^{-7} +2 q^{-9} + q^{-11} +4 q^{-13} +4 q^{-15} +4 q^{-17} +3 q^{-19} -4 q^{-25} - q^{-27} -3 q^{-29} - q^{-33} }[/math]
1,0,1 [math]\displaystyle{ q^{12}-2 q^{10}+5 q^8-5 q^6+4 q^4+2 q^2-10+20 q^{-2} -22 q^{-4} +18 q^{-6} -12 q^{-8} -11 q^{-10} +7 q^{-12} -35 q^{-14} +17 q^{-16} -39 q^{-18} +23 q^{-20} -18 q^{-22} +31 q^{-24} +22 q^{-26} +22 q^{-28} +61 q^{-30} -9 q^{-32} +66 q^{-34} -34 q^{-36} +24 q^{-38} -38 q^{-40} -20 q^{-42} -19 q^{-44} -44 q^{-46} +18 q^{-48} -51 q^{-50} +34 q^{-52} -20 q^{-54} +7 q^{-56} +14 q^{-58} -6 q^{-60} +14 q^{-62} -5 q^{-64} +8 q^{-66} -6 q^{-68} +10 q^{-70} -12 q^{-72} +8 q^{-74} -3 q^{-76} -10 q^{-78} +17 q^{-80} -15 q^{-82} +5 q^{-84} +5 q^{-86} -9 q^{-88} +9 q^{-90} -3 q^{-92} - q^{-94} +3 q^{-96} -2 q^{-98} + q^{-100} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^4+1+2 q^{-2} + q^{-4} - q^{-6} - q^{-8} -5 q^{-10} -8 q^{-12} -10 q^{-14} -11 q^{-16} -7 q^{-18} - q^{-20} +6 q^{-22} +11 q^{-24} +21 q^{-26} +24 q^{-28} +22 q^{-30} +13 q^{-32} +14 q^{-34} +4 q^{-36} -8 q^{-38} -10 q^{-40} -10 q^{-42} -16 q^{-44} -14 q^{-46} -6 q^{-48} -8 q^{-50} -6 q^{-52} +3 q^{-56} -2 q^{-58} +5 q^{-62} +2 q^{-64} -2 q^{-66} +2 q^{-68} +3 q^{-70} + q^{-76} }[/math]
1,0,0,0 [math]\displaystyle{ -1-2 q^{-4} - q^{-6} -2 q^{-8} - q^{-10} + q^{-12} + q^{-14} +5 q^{-16} +4 q^{-18} +7 q^{-20} +4 q^{-22} +4 q^{-24} - q^{-28} -3 q^{-30} -4 q^{-32} -2 q^{-34} -3 q^{-36} - q^{-40} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^6+q^4-3 q^2+3-5 q^{-2} +6 q^{-4} -7 q^{-6} +6 q^{-8} -5 q^{-10} +5 q^{-12} + q^{-16} +6 q^{-18} -5 q^{-20} +11 q^{-22} -10 q^{-24} +12 q^{-26} -11 q^{-28} +10 q^{-30} -9 q^{-32} +5 q^{-34} -4 q^{-36} + q^{-40} -4 q^{-42} +5 q^{-44} -5 q^{-46} +5 q^{-48} -5 q^{-50} +4 q^{-52} -4 q^{-54} +3 q^{-56} -2 q^{-58} + q^{-60} - q^{-62} }[/math]
1,0 [math]\displaystyle{ q^{12}-q^8-q^6+2 q^4+2 q^2-2-4 q^{-2} +4 q^{-6} -7 q^{-10} -5 q^{-12} +3 q^{-14} +5 q^{-16} - q^{-18} -7 q^{-20} +7 q^{-24} +8 q^{-26} - q^{-30} +3 q^{-32} +8 q^{-34} +4 q^{-36} +4 q^{-42} + q^{-44} -4 q^{-46} -4 q^{-48} + q^{-50} +2 q^{-52} -4 q^{-54} -7 q^{-56} -2 q^{-58} +4 q^{-60} -5 q^{-64} -4 q^{-66} +2 q^{-68} +4 q^{-70} -4 q^{-74} -3 q^{-76} +2 q^{-78} +5 q^{-80} + q^{-82} -3 q^{-84} -3 q^{-86} + q^{-88} +3 q^{-90} + q^{-92} - q^{-94} - q^{-96} + q^{-100} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^6-q^4+2 q^2-2+4 q^{-2} -5 q^{-4} +3 q^{-6} -8 q^{-8} +2 q^{-10} -10 q^{-12} - q^{-14} -7 q^{-16} +2 q^{-18} + q^{-20} +4 q^{-22} +11 q^{-24} +8 q^{-26} +18 q^{-28} +4 q^{-30} +16 q^{-32} -4 q^{-34} +10 q^{-36} -12 q^{-38} +3 q^{-40} -14 q^{-42} -11 q^{-46} -5 q^{-50} - q^{-52} -3 q^{-56} +2 q^{-58} -3 q^{-60} +4 q^{-62} -4 q^{-64} +3 q^{-66} -3 q^{-68} +5 q^{-70} -3 q^{-72} +2 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} + q^{-82} - q^{-84} + q^{-86} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{12}-q^{10}+3 q^8-4 q^6+3 q^4-3 q^2-2+7 q^{-2} -14 q^{-4} +15 q^{-6} -14 q^{-8} +3 q^{-10} +7 q^{-12} -19 q^{-14} +23 q^{-16} -20 q^{-18} +9 q^{-20} +3 q^{-22} -15 q^{-24} +18 q^{-26} -12 q^{-28} +4 q^{-30} +8 q^{-32} -10 q^{-34} +11 q^{-36} -6 q^{-40} +16 q^{-42} -14 q^{-44} +15 q^{-46} -2 q^{-48} -5 q^{-50} +18 q^{-52} -20 q^{-54} +24 q^{-56} -12 q^{-58} +2 q^{-60} +11 q^{-62} -17 q^{-64} +19 q^{-66} -13 q^{-68} +5 q^{-70} +5 q^{-72} -10 q^{-74} +8 q^{-76} -4 q^{-78} -5 q^{-80} +8 q^{-82} -9 q^{-84} +3 q^{-88} -9 q^{-90} +8 q^{-92} -7 q^{-94} +2 q^{-96} - q^{-98} -4 q^{-100} +4 q^{-102} -7 q^{-104} +6 q^{-106} -6 q^{-108} +5 q^{-110} -3 q^{-112} -2 q^{-114} +6 q^{-116} -10 q^{-118} +11 q^{-120} -7 q^{-122} +3 q^{-124} + q^{-126} -5 q^{-128} +6 q^{-130} -6 q^{-132} +5 q^{-134} -2 q^{-136} + q^{-140} -2 q^{-142} +2 q^{-144} - q^{-146} + q^{-148} }[/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.

(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 47"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-3 t^3+6 t^2-7 t+7-7 t^{-1} +6 t^{-2} -3 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+5 z^6+8 z^4+6 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]= { 41, 4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^9+2 q^8-4 q^7+5 q^6-6 q^5+7 q^4-5 q^3+5 q^2-3 q+2- q^{-1} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^8 a^{-4} -z^6 a^{-2} +7 z^6 a^{-4} -z^6 a^{-6} -5 z^4 a^{-2} +18 z^4 a^{-4} -5 z^4 a^{-6} -7 z^2 a^{-2} +21 z^2 a^{-4} -8 z^2 a^{-6} -3 a^{-2} +9 a^{-4} -5 a^{-6} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +5 z^8 a^{-4} +3 z^8 a^{-6} +z^7 a^{-1} -z^7 a^{-3} +z^7 a^{-5} +3 z^7 a^{-7} -10 z^6 a^{-2} -23 z^6 a^{-4} -10 z^6 a^{-6} +3 z^6 a^{-8} -5 z^5 a^{-1} -11 z^5 a^{-3} -14 z^5 a^{-5} -5 z^5 a^{-7} +3 z^5 a^{-9} +15 z^4 a^{-2} +35 z^4 a^{-4} +15 z^4 a^{-6} -3 z^4 a^{-8} +2 z^4 a^{-10} +7 z^3 a^{-1} +20 z^3 a^{-3} +19 z^3 a^{-5} +2 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} -9 z^2 a^{-2} -26 z^2 a^{-4} -15 z^2 a^{-6} +z^2 a^{-8} -z^2 a^{-10} -3 z a^{-1} -8 z a^{-3} -9 z a^{-5} -z a^{-7} +2 z a^{-9} -z a^{-11} +3 a^{-2} +9 a^{-4} +5 a^{-6} }[/math]

Vassiliev invariants

V2 and V3: (6, 11)
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{ 24 }[/math] [math]\displaystyle{ 88 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 540 }[/math] [math]\displaystyle{ 76 }[/math] [math]\displaystyle{ 2112 }[/math] [math]\displaystyle{ \frac{10096}{3} }[/math] [math]\displaystyle{ \frac{1696}{3} }[/math] [math]\displaystyle{ 440 }[/math] [math]\displaystyle{ 2304 }[/math] [math]\displaystyle{ 3872 }[/math] [math]\displaystyle{ 12960 }[/math] [math]\displaystyle{ 1824 }[/math] [math]\displaystyle{ \frac{110351}{5} }[/math] [math]\displaystyle{ \frac{11108}{15} }[/math] [math]\displaystyle{ \frac{40988}{5} }[/math] [math]\displaystyle{ 155 }[/math] [math]\displaystyle{ \frac{5151}{5} }[/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]4 is the signature of 10 47. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-101234567χ
19          1-1
17         1 1
15        31 -2
13       21  1
11      43   -1
9     32    1
7    24     2
5   33      0
3  13       2
1 12        -1
-1 1         1
-31          -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ i=5 }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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, 47]]
Out[2]=  
10
In[3]:=
PD[Knot[10, 47]]
Out[3]=  
PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[9, 17, 10, 16], X[5, 15, 6, 14], 

 X[15, 7, 16, 6], X[11, 19, 12, 18], X[13, 1, 14, 20], 



  X[17, 11, 18, 10], X[19, 13, 20, 12], X[7, 2, 8, 3]]
In[4]:=
GaussCode[Knot[10, 47]]
Out[4]=  
GaussCode[-1, 10, -2, 1, -4, 5, -10, 2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 
  6, -9, 7]
In[5]:=
BR[Knot[10, 47]]
Out[5]=  
BR[3, {1, 1, 1, 1, 1, -2, 1, 1, -2, -2}]
In[6]:=
alex = Alexander[Knot[10, 47]][t]
Out[6]=  
     -4   3    6    7            2      3    4

7 + t   - -- + -- - - - 7 t + 6 t  - 3 t  + t



          3    2   t


          t    t
In[7]:=
Conway[Knot[10, 47]][z]
Out[7]=  
       2      4      6    8
1 + 6 z  + 8 z  + 5 z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[10, 47]}
In[9]:=
{KnotDet[Knot[10, 47]], KnotSignature[Knot[10, 47]]}
Out[9]=  
{41, 4}
In[10]:=
J=Jones[Knot[10, 47]][q]
Out[10]=  
    1            2      3      4      5      6      7      8    9

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


    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[10, 47]}
In[12]:=
A2Invariant[Knot[10, 47]][q]
Out[12]=  
  -2    2    6    8      10    12      14    18    20      22    26
-q   - q  + q  + q  + 4 q   + q   + 3 q   - q   - q   - 2 q   - q
In[13]:=
Kauffman[Knot[10, 47]][a, z]
Out[13]=  
                                                   2     2       2

5    9    3     z    2 z   z    9 z   8 z   3 z   z     z    15 z
-- + -- + -- - --- + --- - -- - --- - --- - --- - --- + -- - ----- - 



6    4    2    11    9     7    5     3     a     10    8     6



a    a    a    a     a     a    a     a           a     a     a



     2      2    3       3      3       3       3      3      4
 26 z    9 z    z     3 z    2 z    19 z    20 z    7 z    2 z
 ----- - ---- + --- - ---- + ---- + ----- + ----- + ---- + ---- - 
   4       2     11     9      7      5       3      a      10
  a       a     a      a      a      a       a             a

    4       4       4       4      5      5       5       5      5
 3 z    15 z    35 z    15 z    3 z    5 z    14 z    11 z    5 z
 ---- + ----- + ----- + ----- + ---- - ---- - ----- - ----- - ---- + 
   8      6       4       2       9      7      5       3      a
  a      a       a       a       a      a      a       a

    6       6       6       6      7    7    7    7      8      8
 3 z    10 z    23 z    10 z    3 z    z    z    z    3 z    5 z
 ---- - ----- - ----- - ----- + ---- + -- - -- + -- + ---- + ---- + 
   8      6       4       2       7     5    3   a      6      4
  a      a       a       a       a     a    a          a      a

    8    9    9
 2 z    z    z
 ---- + -- + --
   2     5    3


   a     a    a
In[14]:=
{Vassiliev[2][Knot[10, 47]], Vassiliev[3][Knot[10, 47]]}
Out[14]=  
{0, 11}
In[15]:=
Kh[Knot[10, 47]][q, t]
Out[15]=  
                                         3

  3      5     1      1     q    2 q   q       5        7



3 q  + 3 q  + ----- + ---- + -- + --- + -- + 3 q  t + 2 q  t + 



              3  3      2    2    t    t
             q  t    q t    t

    7  2      9  2      9  3      11  3      11  4      13  4
 4 q  t  + 3 q  t  + 2 q  t  + 4 q   t  + 3 q   t  + 2 q   t  + 

  13  5      15  5    15  6    17  6    19  7


  q   t  + 3 q   t  + q   t  + q   t  + q   t
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