10 84

10_83

10_85

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10 84 Quick Notes

10 84 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X8,12,9,11 X20,15,1,16 X16,5,17,6 X12,18,13,17 X14,8,15,7 X18,14,19,13 X6,19,7,20 X2,10,3,9
Gauss code	1, -10, 2, -1, 5, -9, 7, -3, 10, -2, 3, -6, 8, -7, 4, -5, 6, -8, 9, -4
Dowker-Thistlethwaite code	4 10 16 14 2 8 18 20 12 6
Conway Notation	[.22.2]

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-2][-10] Hyperbolic Volume 14.7099 A-Polynomial See Data:10 84/A-polynomial

[edit Notes for 10 84'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{ 3 }[/math] Rasmussen s-Invariant -2

[edit Notes for 10 84's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-9 t^2+20 t-25+20 t^{-1} -9 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6+3 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^8+4 q^7-8 q^6+11 q^5-14 q^4+15 q^3-13 q^2+11 q-6+3 q^{-1} - q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +3 z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -z^4+5 z^2 a^{-2} -z^2 a^{-6} -2 z^2+4 a^{-2} -2 a^{-4} -1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^9 a^{-3} +2 z^9 a^{-5} +4 z^8 a^{-2} +10 z^8 a^{-4} +6 z^8 a^{-6} +4 z^7 a^{-1} +5 z^7 a^{-3} +8 z^7 a^{-5} +7 z^7 a^{-7} -2 z^6 a^{-2} -17 z^6 a^{-4} -8 z^6 a^{-6} +4 z^6 a^{-8} +3 z^6+a z^5-4 z^5 a^{-1} -11 z^5 a^{-3} -20 z^5 a^{-5} -13 z^5 a^{-7} +z^5 a^{-9} -5 z^4 a^{-2} +9 z^4 a^{-4} +2 z^4 a^{-6} -6 z^4 a^{-8} -6 z^4-2 a z^3-2 z^3 a^{-1} +4 z^3 a^{-3} +11 z^3 a^{-5} +6 z^3 a^{-7} -z^3 a^{-9} +7 z^2 a^{-2} +z^2 a^{-4} -z^2 a^{-6} +z^2 a^{-8} +4 z^2+a z+2 z a^{-1} +2 z a^{-3} -z a^{-7} -4 a^{-2} -2 a^{-4} -1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^6+q^4-q^2-1+4 q^{-2} - q^{-4} +4 q^{-6} + q^{-8} - q^{-10} + q^{-12} -4 q^{-14} +2 q^{-16} - q^{-18} - q^{-20} +2 q^{-22} - q^{-24} }[/math]
The G2 invariant	[math]\displaystyle{ q^{32}-2 q^{30}+5 q^{28}-8 q^{26}+8 q^{24}-7 q^{22}-2 q^{20}+15 q^{18}-30 q^{16}+43 q^{14}-49 q^{12}+39 q^{10}-14 q^8-31 q^6+88 q^4-135 q^2+155-130 q^{-2} +45 q^{-4} +73 q^{-6} -192 q^{-8} +270 q^{-10} -257 q^{-12} +158 q^{-14} +9 q^{-16} -171 q^{-18} +266 q^{-20} -245 q^{-22} +128 q^{-24} +45 q^{-26} -178 q^{-28} +213 q^{-30} -130 q^{-32} -29 q^{-34} +208 q^{-36} -308 q^{-38} +285 q^{-40} -137 q^{-42} -84 q^{-44} +291 q^{-46} -414 q^{-48} +398 q^{-50} -255 q^{-52} +33 q^{-54} +188 q^{-56} -340 q^{-58} +366 q^{-60} -262 q^{-62} +73 q^{-64} +112 q^{-66} -230 q^{-68} +224 q^{-70} -106 q^{-72} -61 q^{-74} +203 q^{-76} -245 q^{-78} +172 q^{-80} -12 q^{-82} -172 q^{-84} +292 q^{-86} -302 q^{-88} +206 q^{-90} -47 q^{-92} -113 q^{-94} +215 q^{-96} -231 q^{-98} +180 q^{-100} -86 q^{-102} -8 q^{-104} +69 q^{-106} -96 q^{-108} +82 q^{-110} -51 q^{-112} +23 q^{-114} +2 q^{-116} -13 q^{-118} +15 q^{-120} -13 q^{-122} +7 q^{-124} -3 q^{-126} + q^{-128} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_84.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^5+2 q^3-3 q+5 q^{-1} -2 q^{-3} +2 q^{-5} + q^{-7} -3 q^{-9} +3 q^{-11} -4 q^{-13} +3 q^{-15} - q^{-17} }[/math]
2	[math]\displaystyle{ q^{16}-2 q^{14}-q^{12}+6 q^{10}-8 q^8-4 q^6+22 q^4-15 q^2-19+40 q^{-2} -7 q^{-4} -35 q^{-6} +33 q^{-8} +10 q^{-10} -30 q^{-12} +6 q^{-14} +19 q^{-16} -7 q^{-18} -23 q^{-20} +19 q^{-22} +19 q^{-24} -39 q^{-26} +8 q^{-28} +35 q^{-30} -33 q^{-32} -8 q^{-34} +31 q^{-36} -12 q^{-38} -12 q^{-40} +11 q^{-42} -3 q^{-46} + q^{-48} }[/math]
3	[math]\displaystyle{ -q^{33}+2 q^{31}+q^{29}-2 q^{27}-3 q^{25}+5 q^{23}+4 q^{21}-15 q^{19}-7 q^{17}+30 q^{15}+22 q^{13}-55 q^{11}-56 q^9+77 q^7+116 q^5-81 q^3-185 q+47 q^{-1} +263 q^{-3} +12 q^{-5} -297 q^{-7} -98 q^{-9} +294 q^{-11} +178 q^{-13} -248 q^{-15} -228 q^{-17} +171 q^{-19} +247 q^{-21} -81 q^{-23} -235 q^{-25} -9 q^{-27} +207 q^{-29} +85 q^{-31} -161 q^{-33} -167 q^{-35} +116 q^{-37} +228 q^{-39} -55 q^{-41} -284 q^{-43} -10 q^{-45} +310 q^{-47} +90 q^{-49} -299 q^{-51} -166 q^{-53} +253 q^{-55} +216 q^{-57} -172 q^{-59} -232 q^{-61} +80 q^{-63} +207 q^{-65} -5 q^{-67} -155 q^{-69} -34 q^{-71} +89 q^{-73} +47 q^{-75} -40 q^{-77} -36 q^{-79} +15 q^{-81} +16 q^{-83} -2 q^{-85} -7 q^{-87} +3 q^{-91} - q^{-93} }[/math]
4	[math]\displaystyle{ q^{56}-2 q^{54}-q^{52}+2 q^{50}-q^{48}+6 q^{46}-5 q^{44}+9 q^{40}-14 q^{38}-q^{36}-23 q^{34}+22 q^{32}+77 q^{30}-9 q^{28}-67 q^{26}-171 q^{24}-2 q^{22}+301 q^{20}+235 q^{18}-56 q^{16}-607 q^{14}-444 q^{12}+443 q^{10}+925 q^8+579 q^6-930 q^4-1509 q^2-245+1445 q^{-2} +1983 q^{-4} -232 q^{-6} -2323 q^{-8} -1791 q^{-10} +793 q^{-12} +3002 q^{-14} +1352 q^{-16} -1799 q^{-18} -2874 q^{-20} -729 q^{-22} +2579 q^{-24} +2432 q^{-26} -351 q^{-28} -2573 q^{-30} -1800 q^{-32} +1199 q^{-34} +2311 q^{-36} +854 q^{-38} -1459 q^{-40} -1961 q^{-42} -114 q^{-44} +1595 q^{-46} +1507 q^{-48} -353 q^{-50} -1761 q^{-52} -1176 q^{-54} +880 q^{-56} +2026 q^{-58} +701 q^{-60} -1513 q^{-62} -2235 q^{-64} -9 q^{-66} +2367 q^{-68} +1922 q^{-70} -797 q^{-72} -2995 q^{-74} -1313 q^{-76} +1876 q^{-78} +2831 q^{-80} +610 q^{-82} -2624 q^{-84} -2367 q^{-86} +390 q^{-88} +2514 q^{-90} +1840 q^{-92} -1100 q^{-94} -2163 q^{-96} -965 q^{-98} +1077 q^{-100} +1793 q^{-102} +276 q^{-104} -940 q^{-106} -1105 q^{-108} -115 q^{-110} +828 q^{-112} +531 q^{-114} -8 q^{-116} -475 q^{-118} -325 q^{-120} +130 q^{-122} +200 q^{-124} +148 q^{-126} -69 q^{-128} -119 q^{-130} -12 q^{-132} +15 q^{-134} +46 q^{-136} +2 q^{-138} -19 q^{-140} -2 q^{-142} -2 q^{-144} +7 q^{-146} -3 q^{-150} + q^{-152} }[/math]
5	[math]\displaystyle{ -q^{85}+2 q^{83}+q^{81}-2 q^{79}+q^{77}-2 q^{75}-6 q^{73}+q^{71}+6 q^{69}+q^{67}+13 q^{65}+13 q^{63}-14 q^{61}-35 q^{59}-39 q^{57}-13 q^{55}+64 q^{53}+142 q^{51}+100 q^{49}-98 q^{47}-311 q^{45}-330 q^{43}-q^{41}+549 q^{39}+850 q^{37}+387 q^{35}-741 q^{33}-1659 q^{31}-1355 q^{29}+479 q^{27}+2674 q^{25}+3154 q^{23}+695 q^{21}-3376 q^{19}-5677 q^{17}-3348 q^{15}+2943 q^{13}+8388 q^{11}+7598 q^9-475 q^7-10214 q^5-12893 q^3-4415 q+9795 q^{-1} +17931 q^{-3} +11490 q^{-5} -6365 q^{-7} -21186 q^{-9} -19189 q^{-11} -93 q^{-13} +21141 q^{-15} +25898 q^{-17} +8431 q^{-19} -17601 q^{-21} -29735 q^{-23} -16708 q^{-25} +11076 q^{-27} +29878 q^{-29} +23196 q^{-31} -3242 q^{-33} -26528 q^{-35} -26612 q^{-37} -4145 q^{-39} +20728 q^{-41} +26751 q^{-43} +9861 q^{-45} -14068 q^{-47} -24316 q^{-49} -13301 q^{-51} +7817 q^{-53} +20429 q^{-55} +14786 q^{-57} -2721 q^{-59} -16324 q^{-61} -15064 q^{-63} -1184 q^{-65} +12810 q^{-67} +15086 q^{-69} +4250 q^{-71} -10052 q^{-73} -15511 q^{-75} -7310 q^{-77} +7856 q^{-79} +16703 q^{-81} +10795 q^{-83} -5503 q^{-85} -18203 q^{-87} -15247 q^{-89} +2294 q^{-91} +19488 q^{-93} +20227 q^{-95} +2311 q^{-97} -19372 q^{-99} -25126 q^{-101} -8404 q^{-103} +17105 q^{-105} +28708 q^{-107} +15262 q^{-109} -12185 q^{-111} -29724 q^{-113} -21717 q^{-115} +5021 q^{-117} +27387 q^{-119} +26228 q^{-121} +3107 q^{-123} -21689 q^{-125} -27471 q^{-127} -10525 q^{-129} +13702 q^{-131} +25036 q^{-133} +15499 q^{-135} -5220 q^{-137} -19530 q^{-139} -17038 q^{-141} -1869 q^{-143} +12466 q^{-145} +15290 q^{-147} +6291 q^{-149} -5787 q^{-151} -11362 q^{-153} -7627 q^{-155} +794 q^{-157} +6798 q^{-159} +6695 q^{-161} +1905 q^{-163} -3044 q^{-165} -4579 q^{-167} -2577 q^{-169} +609 q^{-171} +2454 q^{-173} +2134 q^{-175} +454 q^{-177} -998 q^{-179} -1275 q^{-181} -623 q^{-183} +198 q^{-185} +588 q^{-187} +465 q^{-189} +54 q^{-191} -220 q^{-193} -215 q^{-195} -82 q^{-197} +45 q^{-199} +86 q^{-201} +55 q^{-203} -12 q^{-205} -33 q^{-207} -12 q^{-209} +2 q^{-211} +5 q^{-213} +6 q^{-215} +2 q^{-217} -7 q^{-219} +3 q^{-223} - q^{-225} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^6+q^4-q^2-1+4 q^{-2} - q^{-4} +4 q^{-6} + q^{-8} - q^{-10} + q^{-12} -4 q^{-14} +2 q^{-16} - q^{-18} - q^{-20} +2 q^{-22} - q^{-24} }[/math]
1,1	[math]\displaystyle{ q^{20}-4 q^{18}+12 q^{16}-28 q^{14}+56 q^{12}-102 q^{10}+168 q^8-272 q^6+409 q^4-590 q^2+804-1024 q^{-2} +1229 q^{-4} -1340 q^{-6} +1342 q^{-8} -1158 q^{-10} +792 q^{-12} -252 q^{-14} -410 q^{-16} +1118 q^{-18} -1798 q^{-20} +2348 q^{-22} -2706 q^{-24} +2824 q^{-26} -2693 q^{-28} +2326 q^{-30} -1764 q^{-32} +1078 q^{-34} -351 q^{-36} -330 q^{-38} +882 q^{-40} -1262 q^{-42} +1445 q^{-44} -1444 q^{-46} +1300 q^{-48} -1064 q^{-50} +804 q^{-52} -560 q^{-54} +354 q^{-56} -204 q^{-58} +107 q^{-60} -50 q^{-62} +20 q^{-64} -6 q^{-66} + q^{-68} }[/math]
2,0	[math]\displaystyle{ q^{18}-q^{16}-2 q^{14}+3 q^{12}+2 q^{10}-7 q^8-5 q^6+9 q^4+6 q^2-14-5 q^{-2} +22 q^{-4} +5 q^{-6} -15 q^{-8} +5 q^{-10} +16 q^{-12} -3 q^{-14} -12 q^{-16} +6 q^{-18} +4 q^{-20} -14 q^{-22} +4 q^{-24} +8 q^{-26} -10 q^{-28} -4 q^{-30} +16 q^{-32} -18 q^{-36} + q^{-38} +15 q^{-40} - q^{-42} -16 q^{-44} +6 q^{-46} +11 q^{-48} -4 q^{-50} -6 q^{-52} +5 q^{-56} - q^{-58} -2 q^{-60} + q^{-62} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{14}-2 q^{12}+q^{10}+4 q^8-9 q^6+q^4+12 q^2-20+27 q^{-4} -25 q^{-6} +3 q^{-8} +34 q^{-10} -17 q^{-12} -3 q^{-14} +18 q^{-16} -7 q^{-18} -12 q^{-20} -4 q^{-22} +11 q^{-24} -5 q^{-26} -19 q^{-28} +24 q^{-30} +5 q^{-32} -30 q^{-34} +23 q^{-36} +7 q^{-38} -26 q^{-40} +16 q^{-42} +4 q^{-44} -12 q^{-46} +7 q^{-48} + q^{-50} -3 q^{-52} + q^{-54} }[/math]
1,0,0	[math]\displaystyle{ -q^7+q^5-2 q^3+q-2 q^{-1} +4 q^{-3} - q^{-5} +5 q^{-7} +2 q^{-9} +2 q^{-11} -2 q^{-15} -4 q^{-19} +2 q^{-21} -2 q^{-23} +2 q^{-25} -2 q^{-27} +2 q^{-29} - q^{-31} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{14}+2 q^{12}-5 q^{10}+8 q^8-13 q^6+19 q^4-26 q^2+32-34 q^{-2} +35 q^{-4} -27 q^{-6} +19 q^{-8} -2 q^{-10} -13 q^{-12} +33 q^{-14} -48 q^{-16} +61 q^{-18} -68 q^{-20} +68 q^{-22} -63 q^{-24} +49 q^{-26} -33 q^{-28} +14 q^{-30} +3 q^{-32} -18 q^{-34} +29 q^{-36} -35 q^{-38} +36 q^{-40} -34 q^{-42} +28 q^{-44} -20 q^{-46} +13 q^{-48} -7 q^{-50} +3 q^{-52} - q^{-54} }[/math]
1,0	[math]\displaystyle{ q^{24}-2 q^{20}-2 q^{18}+3 q^{16}+6 q^{14}-q^{12}-11 q^{10}-8 q^8+10 q^6+19 q^4-2 q^2-28-15 q^{-2} +24 q^{-4} +33 q^{-6} -9 q^{-8} -37 q^{-10} -7 q^{-12} +37 q^{-14} +24 q^{-16} -20 q^{-18} -26 q^{-20} +13 q^{-22} +28 q^{-24} -3 q^{-26} -27 q^{-28} -4 q^{-30} +22 q^{-32} +6 q^{-34} -23 q^{-36} -15 q^{-38} +19 q^{-40} +19 q^{-42} -17 q^{-44} -27 q^{-46} +11 q^{-48} +35 q^{-50} +4 q^{-52} -36 q^{-54} -21 q^{-56} +28 q^{-58} +34 q^{-60} -11 q^{-62} -35 q^{-64} -8 q^{-66} +25 q^{-68} +17 q^{-70} -11 q^{-72} -16 q^{-74} +10 q^{-78} +4 q^{-80} -3 q^{-82} -3 q^{-84} + q^{-88} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{18}-2 q^{16}+3 q^{14}-4 q^{12}+7 q^{10}-11 q^8+11 q^6-16 q^4+20 q^2-27+23 q^{-2} -26 q^{-4} +30 q^{-6} -23 q^{-8} +20 q^{-10} -7 q^{-12} +13 q^{-14} +13 q^{-16} -15 q^{-18} +26 q^{-20} -33 q^{-22} +45 q^{-24} -53 q^{-26} +45 q^{-28} -57 q^{-30} +51 q^{-32} -46 q^{-34} +37 q^{-36} -34 q^{-38} +22 q^{-40} -6 q^{-42} + q^{-44} +5 q^{-46} -17 q^{-48} +25 q^{-50} -26 q^{-52} +27 q^{-54} -30 q^{-56} +28 q^{-58} -22 q^{-60} +19 q^{-62} -16 q^{-64} +11 q^{-66} -6 q^{-68} +4 q^{-70} -3 q^{-72} + q^{-74} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{32}-2 q^{30}+5 q^{28}-8 q^{26}+8 q^{24}-7 q^{22}-2 q^{20}+15 q^{18}-30 q^{16}+43 q^{14}-49 q^{12}+39 q^{10}-14 q^8-31 q^6+88 q^4-135 q^2+155-130 q^{-2} +45 q^{-4} +73 q^{-6} -192 q^{-8} +270 q^{-10} -257 q^{-12} +158 q^{-14} +9 q^{-16} -171 q^{-18} +266 q^{-20} -245 q^{-22} +128 q^{-24} +45 q^{-26} -178 q^{-28} +213 q^{-30} -130 q^{-32} -29 q^{-34} +208 q^{-36} -308 q^{-38} +285 q^{-40} -137 q^{-42} -84 q^{-44} +291 q^{-46} -414 q^{-48} +398 q^{-50} -255 q^{-52} +33 q^{-54} +188 q^{-56} -340 q^{-58} +366 q^{-60} -262 q^{-62} +73 q^{-64} +112 q^{-66} -230 q^{-68} +224 q^{-70} -106 q^{-72} -61 q^{-74} +203 q^{-76} -245 q^{-78} +172 q^{-80} -12 q^{-82} -172 q^{-84} +292 q^{-86} -302 q^{-88} +206 q^{-90} -47 q^{-92} -113 q^{-94} +215 q^{-96} -231 q^{-98} +180 q^{-100} -86 q^{-102} -8 q^{-104} +69 q^{-106} -96 q^{-108} +82 q^{-110} -51 q^{-112} +23 q^{-114} +2 q^{-116} -13 q^{-118} +15 q^{-120} -13 q^{-122} +7 q^{-124} -3 q^{-126} + q^{-128} }[/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 84"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^3-9 t^2+20 t-25+20 t^{-1} -9 t^{-2} +2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^6+3 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^8+4 q^7-8 q^6+11 q^5-14 q^4+15 q^3-13 q^2+11 q-6+3 q^{-1} - q^{-2} }[/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^6 a^{-4} +3 z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -z^4+5 z^2 a^{-2} -z^2 a^{-6} -2 z^2+4 a^{-2} -2 a^{-4} -1 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 z^9 a^{-3} +2 z^9 a^{-5} +4 z^8 a^{-2} +10 z^8 a^{-4} +6 z^8 a^{-6} +4 z^7 a^{-1} +5 z^7 a^{-3} +8 z^7 a^{-5} +7 z^7 a^{-7} -2 z^6 a^{-2} -17 z^6 a^{-4} -8 z^6 a^{-6} +4 z^6 a^{-8} +3 z^6+a z^5-4 z^5 a^{-1} -11 z^5 a^{-3} -20 z^5 a^{-5} -13 z^5 a^{-7} +z^5 a^{-9} -5 z^4 a^{-2} +9 z^4 a^{-4} +2 z^4 a^{-6} -6 z^4 a^{-8} -6 z^4-2 a z^3-2 z^3 a^{-1} +4 z^3 a^{-3} +11 z^3 a^{-5} +6 z^3 a^{-7} -z^3 a^{-9} +7 z^2 a^{-2} +z^2 a^{-4} -z^2 a^{-6} +z^2 a^{-8} +4 z^2+a z+2 z a^{-1} +2 z a^{-3} -z a^{-7} -4 a^{-2} -2 a^{-4} -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{124}{3} }[/math] [math]\displaystyle{ -\frac{28}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{448}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{992}{3} }[/math] [math]\displaystyle{ -\frac{224}{3} }[/math] [math]\displaystyle{ \frac{7951}{15} }[/math] [math]\displaystyle{ \frac{92}{5} }[/math] [math]\displaystyle{ \frac{2044}{45} }[/math] [math]\displaystyle{ \frac{65}{9} }[/math] [math]\displaystyle{ -\frac{449}{15} }[/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]2 is the signature of 10 84. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-3	-2	-1	0	1	2	3	4	5	6	7	χ
17											1	-1
15										3		3
13									5	1		-4
11								6	3			3
9							8	5				-3
7						7	6					1
5					6	8						2
3				5	7							-2
1			2	7								5
-1		1	4									-3
-3		2										2
-5	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, 84]]
Out[2]=	10
In[3]:=	PD[Knot[10, 84]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[8, 12, 9, 11], X[20, 15, 1, 16], X[16, 5, 17, 6], X[12, 18, 13, 17], X[14, 8, 15, 7], X[18, 14, 19, 13], X[6, 19, 7, 20], X[2, 10, 3, 9]]
In[4]:=	GaussCode[Knot[10, 84]]
Out[4]=	GaussCode[1, -10, 2, -1, 5, -9, 7, -3, 10, -2, 3, -6, 8, -7, 4, -5, 6, -8, 9, -4]
In[5]:=	BR[Knot[10, 84]]
Out[5]=	BR[4, {1, 1, 1, 2, -1, -3, 2, 2, -3, 2, -3}]
In[6]:=	alex = Alexander[Knot[10, 84]][t]
Out[6]=	2 9 20 2 3 -25 + -- - -- + -- + 20 t - 9 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 84]][z]
Out[7]=	2 4 6 1 + 2 z + 3 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 84], Knot[11, Alternating, 46], Knot[11, NonAlternating, 184]}
In[9]:=	{KnotDet[Knot[10, 84]], KnotSignature[Knot[10, 84]]}
Out[9]=	{87, 2}
In[10]:=	J=Jones[Knot[10, 84]][q]
Out[10]=	-2 3 2 3 4 5 6 7 8 -6 - q + - + 11 q - 13 q + 15 q - 14 q + 11 q - 8 q + 4 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 84]}
In[12]:=	A2Invariant[Knot[10, 84]][q]
Out[12]=	-6 -4 -2 2 4 6 8 10 12 14 -1 - q + q - q + 4 q - q + 4 q + q - q + q - 4 q + 16 18 20 22 24 2 q - q - q + 2 q - q
In[13]:=	Kauffman[Knot[10, 84]][a, z]
Out[13]=	2 2 2 2 3 2 4 z 2 z 2 z 2 z z z 7 z z -1 - -- - -- - -- + --- + --- + a z + 4 z + -- - -- + -- + ---- - -- + 4 2 7 3 a 8 6 4 2 9 a a a a a a a a a 3 3 3 3 4 4 4 6 z 11 z 4 z 2 z 3 4 6 z 2 z 9 z ---- + ----- + ---- - ---- - 2 a z - 6 z - ---- + ---- + ---- - 7 5 3 a 8 6 4 a a a a a a 4 5 5 5 5 5 6 5 z z 13 z 20 z 11 z 4 z 5 6 4 z ---- + -- - ----- - ----- - ----- - ---- + a z + 3 z + ---- - 2 9 7 5 3 a 8 a a a a a a 6 6 6 7 7 7 7 8 8 8 z 17 z 2 z 7 z 8 z 5 z 4 z 6 z 10 z ---- - ----- - ---- + ---- + ---- + ---- + ---- + ---- + ----- + 6 4 2 7 5 3 a 6 4 a a a a a a a a 8 9 9 4 z 2 z 2 z ---- + ---- + ---- 2 5 3 a a a
In[14]:=	{Vassiliev[2][Knot[10, 84]], Vassiliev[3][Knot[10, 84]]}
Out[14]=	{0, 2}
In[15]:=	Kh[Knot[10, 84]][q, t]
Out[15]=	3 1 2 1 4 2 q 3 5 7 q + 5 q + ----- + ----- + ---- + --- + --- + 7 q t + 6 q t + 5 3 3 2 2 q t t q t q t q t 5 2 7 2 7 3 9 3 9 4 11 4 8 q t + 7 q t + 6 q t + 8 q t + 5 q t + 6 q t + 11 5 13 5 13 6 15 6 17 7 3 q t + 5 q t + q t + 3 q t + q t