10 51

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10 50.gif

10_50

10 52.gif

10_52

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10 51 Quick Notes


10 51 Further Notes and Views

Knot presentations

Planar diagram presentation X1425 X3849 X9,17,10,16 X5,15,6,14 X15,7,16,6 X13,1,14,20 X19,11,20,10 X11,19,12,18 X17,13,18,12 X7283
Gauss code -1, 10, -2, 1, -4, 5, -10, 2, -3, 7, -8, 9, -6, 4, -5, 3, -9, 8, -7, 6
Dowker-Thistlethwaite code 4 8 14 2 16 18 20 6 12 10
Conway Notation [32,21,2]

Three dimensional invariants

Symmetry type Reversible
Unknotting number [math]\displaystyle{ \{2,3\} }[/math]
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-2][-10]
Hyperbolic Volume 12.6314
A-Polynomial See Data:10 51/A-polynomial

[edit Notes for 10 51's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ [1,2] }[/math]
Topological 4 genus [math]\displaystyle{ [1,2] }[/math]
Concordance genus [math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant 2

[edit Notes for 10 51's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^3-7 t^2+15 t-19+15 t^{-1} -7 t^{-2} +2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 2 z^6+5 z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 67, 2 }
Jones polynomial [math]\displaystyle{ -q^8+2 q^7-5 q^6+8 q^5-10 q^4+12 q^3-10 q^2+9 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} +4 z^4 a^{-4} -z^4 a^{-6} -z^4+3 z^2 a^{-2} +7 z^2 a^{-4} -3 z^2 a^{-6} -2 z^2+ a^{-2} +4 a^{-4} -3 a^{-6} -1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +3 z^8 a^{-2} +6 z^8 a^{-4} +3 z^8 a^{-6} +4 z^7 a^{-1} +6 z^7 a^{-3} +5 z^7 a^{-5} +3 z^7 a^{-7} -z^6 a^{-2} -12 z^6 a^{-4} -6 z^6 a^{-6} +2 z^6 a^{-8} +3 z^6+a z^5-6 z^5 a^{-1} -16 z^5 a^{-3} -16 z^5 a^{-5} -6 z^5 a^{-7} +z^5 a^{-9} -6 z^4 a^{-2} +13 z^4 a^{-4} +9 z^4 a^{-6} -4 z^4 a^{-8} -6 z^4-2 a z^3+15 z^3 a^{-3} +21 z^3 a^{-5} +5 z^3 a^{-7} -3 z^3 a^{-9} +4 z^2 a^{-2} -8 z^2 a^{-4} -8 z^2 a^{-6} +z^2 a^{-8} +3 z^2+a z-5 z a^{-3} -9 z a^{-5} -3 z a^{-7} +2 z a^{-9} - a^{-2} +4 a^{-4} +3 a^{-6} -1 }[/math]
The A2 invariant [math]\displaystyle{ -q^6+q^4-q^2-1+2 q^{-2} -2 q^{-4} +3 q^{-6} + q^{-8} +2 q^{-10} +3 q^{-12} - q^{-14} +2 q^{-16} -2 q^{-18} -2 q^{-20} - q^{-24} }[/math]
The G2 invariant [math]\displaystyle{ q^{32}-2 q^{30}+5 q^{28}-8 q^{26}+8 q^{24}-6 q^{22}-3 q^{20}+16 q^{18}-31 q^{16}+42 q^{14}-44 q^{12}+24 q^{10}+7 q^8-48 q^6+87 q^4-101 q^2+88-42 q^{-2} -28 q^{-4} +91 q^{-6} -127 q^{-8} +120 q^{-10} -70 q^{-12} -2 q^{-14} +67 q^{-16} -98 q^{-18} +82 q^{-20} -25 q^{-22} -44 q^{-24} +92 q^{-26} -95 q^{-28} +44 q^{-30} +39 q^{-32} -116 q^{-34} +163 q^{-36} -147 q^{-38} +84 q^{-40} +20 q^{-42} -116 q^{-44} +180 q^{-46} -180 q^{-48} +130 q^{-50} -33 q^{-52} -57 q^{-54} +120 q^{-56} -126 q^{-58} +89 q^{-60} -17 q^{-62} -50 q^{-64} +83 q^{-66} -73 q^{-68} +18 q^{-70} +51 q^{-72} -103 q^{-74} +113 q^{-76} -78 q^{-78} +6 q^{-80} +62 q^{-82} -114 q^{-84} +123 q^{-86} -94 q^{-88} +37 q^{-90} +16 q^{-92} -60 q^{-94} +74 q^{-96} -66 q^{-98} +43 q^{-100} -15 q^{-102} -7 q^{-104} +19 q^{-106} -24 q^{-108} +20 q^{-110} -14 q^{-112} +8 q^{-114} - q^{-116} -3 q^{-118} +4 q^{-120} -4 q^{-122} +3 q^{-124} - q^{-126} + q^{-128} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_51.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^5+2 q^3-3 q+3 q^{-1} - q^{-3} +2 q^{-5} +2 q^{-7} -2 q^{-9} +3 q^{-11} -3 q^{-13} + q^{-15} - q^{-17} }[/math]
2 [math]\displaystyle{ q^{16}-2 q^{14}-q^{12}+7 q^{10}-6 q^8-8 q^6+17 q^4-4 q^2-19+20 q^{-2} +5 q^{-4} -22 q^{-6} +12 q^{-8} +12 q^{-10} -12 q^{-12} -2 q^{-14} +11 q^{-16} +5 q^{-18} -17 q^{-20} +5 q^{-22} +19 q^{-24} -21 q^{-26} -5 q^{-28} +21 q^{-30} -13 q^{-32} -9 q^{-34} +13 q^{-36} -3 q^{-38} -5 q^{-40} +4 q^{-42} - q^{-46} + q^{-48} }[/math]
3 [math]\displaystyle{ -q^{33}+2 q^{31}+q^{29}-3 q^{27}-4 q^{25}+6 q^{23}+11 q^{21}-11 q^{19}-21 q^{17}+11 q^{15}+37 q^{13}-4 q^{11}-59 q^9-10 q^7+77 q^5+35 q^3-85 q-72 q^{-1} +85 q^{-3} +101 q^{-5} -66 q^{-7} -124 q^{-9} +42 q^{-11} +130 q^{-13} -6 q^{-15} -122 q^{-17} -17 q^{-19} +99 q^{-21} +45 q^{-23} -69 q^{-25} -64 q^{-27} +34 q^{-29} +82 q^{-31} +5 q^{-33} -95 q^{-35} -36 q^{-37} +95 q^{-39} +75 q^{-41} -95 q^{-43} -102 q^{-45} +72 q^{-47} +118 q^{-49} -48 q^{-51} -121 q^{-53} +13 q^{-55} +111 q^{-57} +10 q^{-59} -84 q^{-61} -29 q^{-63} +57 q^{-65} +34 q^{-67} -31 q^{-69} -26 q^{-71} +12 q^{-73} +18 q^{-75} -4 q^{-77} -8 q^{-79} + q^{-81} +4 q^{-83} - q^{-85} - q^{-87} + q^{-91} - q^{-93} }[/math]
4 [math]\displaystyle{ q^{56}-2 q^{54}-q^{52}+3 q^{50}+4 q^{46}-9 q^{44}-6 q^{42}+12 q^{40}+6 q^{38}+16 q^{36}-32 q^{34}-34 q^{32}+24 q^{30}+38 q^{28}+69 q^{26}-62 q^{24}-122 q^{22}-23 q^{20}+85 q^{18}+233 q^{16}-5 q^{14}-255 q^{12}-233 q^{10}+14 q^8+468 q^6+275 q^4-234 q^2-548-326 q^{-2} +524 q^{-4} +672 q^{-6} +92 q^{-8} -666 q^{-10} -758 q^{-12} +238 q^{-14} +831 q^{-16} +522 q^{-18} -431 q^{-20} -920 q^{-22} -168 q^{-24} +621 q^{-26} +707 q^{-28} -54 q^{-30} -718 q^{-32} -418 q^{-34} +245 q^{-36} +613 q^{-38} +240 q^{-40} -361 q^{-42} -505 q^{-44} -105 q^{-46} +421 q^{-48} +458 q^{-50} +7 q^{-52} -554 q^{-54} -431 q^{-56} +210 q^{-58} +644 q^{-60} +385 q^{-62} -521 q^{-64} -729 q^{-66} -100 q^{-68} +692 q^{-70} +752 q^{-72} -277 q^{-74} -832 q^{-76} -471 q^{-78} +445 q^{-80} +895 q^{-82} +126 q^{-84} -580 q^{-86} -660 q^{-88} +12 q^{-90} +666 q^{-92} +376 q^{-94} -143 q^{-96} -496 q^{-98} -258 q^{-100} +255 q^{-102} +301 q^{-104} +122 q^{-106} -185 q^{-108} -219 q^{-110} +7 q^{-112} +98 q^{-114} +117 q^{-116} -12 q^{-118} -79 q^{-120} -26 q^{-122} +40 q^{-126} +10 q^{-128} -14 q^{-130} -3 q^{-132} -8 q^{-134} +7 q^{-136} +2 q^{-138} -3 q^{-140} +2 q^{-142} -2 q^{-144} + q^{-146} - q^{-150} + q^{-152} }[/math]
5 [math]\displaystyle{ -q^{85}+2 q^{83}+q^{81}-3 q^{79}-q^{73}+4 q^{71}+5 q^{69}-8 q^{67}-9 q^{65}+2 q^{63}+9 q^{61}+17 q^{59}+9 q^{57}-21 q^{55}-51 q^{53}-24 q^{51}+46 q^{49}+93 q^{47}+76 q^{45}-41 q^{43}-179 q^{41}-201 q^{39}-6 q^{37}+282 q^{35}+401 q^{33}+181 q^{31}-317 q^{29}-706 q^{27}-557 q^{25}+209 q^{23}+1028 q^{21}+1132 q^{19}+211 q^{17}-1202 q^{15}-1891 q^{13}-1017 q^{11}+1049 q^9+2644 q^7+2157 q^5-384 q^3-3097 q-3513 q^{-1} -821 q^{-3} +3066 q^{-5} +4732 q^{-7} +2373 q^{-9} -2350 q^{-11} -5499 q^{-13} -4052 q^{-15} +1114 q^{-17} +5615 q^{-19} +5393 q^{-21} +456 q^{-23} -5006 q^{-25} -6196 q^{-27} -1970 q^{-29} +3889 q^{-31} +6262 q^{-33} +3182 q^{-35} -2507 q^{-37} -5732 q^{-39} -3854 q^{-41} +1140 q^{-43} +4764 q^{-45} +4068 q^{-47} +10 q^{-49} -3660 q^{-51} -3884 q^{-53} -874 q^{-55} +2552 q^{-57} +3565 q^{-59} +1528 q^{-61} -1627 q^{-63} -3235 q^{-65} -2069 q^{-67} +824 q^{-69} +3006 q^{-71} +2651 q^{-73} -67 q^{-75} -2927 q^{-77} -3322 q^{-79} -714 q^{-81} +2802 q^{-83} +4097 q^{-85} +1704 q^{-87} -2597 q^{-89} -4878 q^{-91} -2805 q^{-93} +2041 q^{-95} +5433 q^{-97} +4067 q^{-99} -1140 q^{-101} -5593 q^{-103} -5197 q^{-105} -139 q^{-107} +5156 q^{-109} +6002 q^{-111} +1594 q^{-113} -4116 q^{-115} -6198 q^{-117} -2986 q^{-119} +2618 q^{-121} +5729 q^{-123} +3923 q^{-125} -915 q^{-127} -4596 q^{-129} -4283 q^{-131} -600 q^{-133} +3131 q^{-135} +3936 q^{-137} +1626 q^{-139} -1575 q^{-141} -3105 q^{-143} -2061 q^{-145} +340 q^{-147} +2051 q^{-149} +1922 q^{-151} +436 q^{-153} -1051 q^{-155} -1462 q^{-157} -745 q^{-159} +343 q^{-161} +910 q^{-163} +686 q^{-165} +55 q^{-167} -443 q^{-169} -495 q^{-171} -183 q^{-173} +157 q^{-175} +276 q^{-177} +169 q^{-179} -19 q^{-181} -128 q^{-183} -103 q^{-185} -24 q^{-187} +43 q^{-189} +57 q^{-191} +20 q^{-193} -14 q^{-195} -17 q^{-197} -12 q^{-199} -4 q^{-201} +11 q^{-203} +6 q^{-205} -3 q^{-207} -3 q^{-213} + q^{-215} +2 q^{-217} - q^{-219} + q^{-223} - q^{-225} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^6+q^4-q^2-1+2 q^{-2} -2 q^{-4} +3 q^{-6} + q^{-8} +2 q^{-10} +3 q^{-12} - q^{-14} +2 q^{-16} -2 q^{-18} -2 q^{-20} - q^{-24} }[/math]
1,1 [math]\displaystyle{ q^{20}-4 q^{18}+12 q^{16}-28 q^{14}+56 q^{12}-98 q^{10}+160 q^8-240 q^6+325 q^4-414 q^2+488-532 q^{-2} +511 q^{-4} -436 q^{-6} +294 q^{-8} -86 q^{-10} -161 q^{-12} +436 q^{-14} -670 q^{-16} +884 q^{-18} -1004 q^{-20} +1056 q^{-22} -1000 q^{-24} +866 q^{-26} -664 q^{-28} +408 q^{-30} -160 q^{-32} -88 q^{-34} +284 q^{-36} -428 q^{-38} +496 q^{-40} -502 q^{-42} +467 q^{-44} -398 q^{-46} +310 q^{-48} -230 q^{-50} +161 q^{-52} -108 q^{-54} +66 q^{-56} -40 q^{-58} +25 q^{-60} -12 q^{-62} +6 q^{-64} -2 q^{-66} + q^{-68} }[/math]
2,0 [math]\displaystyle{ q^{18}-q^{16}-2 q^{14}+3 q^{12}+3 q^{10}-4 q^8-5 q^6+5 q^4+6 q^2-10-6 q^{-2} +10 q^{-4} +2 q^{-6} -10 q^{-8} +8 q^{-12} -2 q^{-14} -2 q^{-16} +8 q^{-18} +7 q^{-20} -3 q^{-22} +9 q^{-24} +9 q^{-26} -7 q^{-28} -3 q^{-30} +9 q^{-32} -12 q^{-36} -4 q^{-38} +5 q^{-40} -4 q^{-42} -11 q^{-44} - q^{-46} +4 q^{-48} - q^{-52} + q^{-54} +2 q^{-56} + q^{-58} + q^{-62} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{14}-2 q^{12}+q^{10}+4 q^8-8 q^6+2 q^4+8 q^2-16+4 q^{-2} +13 q^{-4} -19 q^{-6} +2 q^{-8} +14 q^{-10} -10 q^{-12} +13 q^{-16} +6 q^{-18} +2 q^{-20} +2 q^{-22} +14 q^{-24} -5 q^{-26} -16 q^{-28} +12 q^{-30} -6 q^{-32} -20 q^{-34} +13 q^{-36} -12 q^{-40} +8 q^{-42} + q^{-44} -4 q^{-46} +3 q^{-48} + q^{-50} - q^{-52} + q^{-54} }[/math]
1,0,0 [math]\displaystyle{ -q^7+q^5-2 q^3+q-2 q^{-1} +2 q^{-3} -2 q^{-5} +2 q^{-7} + q^{-9} +2 q^{-11} +3 q^{-13} +2 q^{-15} +4 q^{-17} - q^{-19} +2 q^{-21} -3 q^{-23} - q^{-25} -3 q^{-27} - q^{-31} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{16}-q^{14}+3 q^{10}-q^8-4 q^6+3 q^4+2 q^2-8-5 q^{-2} +8 q^{-4} +2 q^{-6} -15 q^{-8} +12 q^{-12} -9 q^{-14} -14 q^{-16} +12 q^{-18} +9 q^{-20} -4 q^{-22} +12 q^{-24} +26 q^{-26} +10 q^{-28} +3 q^{-30} +19 q^{-32} +7 q^{-34} -19 q^{-36} -7 q^{-38} +2 q^{-40} -19 q^{-42} -21 q^{-44} + q^{-46} +2 q^{-48} -10 q^{-50} -4 q^{-52} +8 q^{-54} +4 q^{-56} -3 q^{-58} +3 q^{-60} +4 q^{-62} + q^{-68} }[/math]
1,0,0,0 [math]\displaystyle{ -q^8+q^6-2 q^4-2 q^{-2} +2 q^{-4} -2 q^{-6} +2 q^{-8} +2 q^{-12} +2 q^{-14} +3 q^{-16} +3 q^{-18} +3 q^{-20} +4 q^{-22} - q^{-24} +2 q^{-26} -3 q^{-28} -2 q^{-30} -2 q^{-32} -3 q^{-34} - q^{-38} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{14}+2 q^{12}-5 q^{10}+8 q^8-12 q^6+16 q^4-20 q^2+20-20 q^{-2} +17 q^{-4} -9 q^{-6} +2 q^{-8} +10 q^{-10} -18 q^{-12} +30 q^{-14} -35 q^{-16} +40 q^{-18} -38 q^{-20} +36 q^{-22} -28 q^{-24} +19 q^{-26} -8 q^{-28} -2 q^{-30} +10 q^{-32} -16 q^{-34} +19 q^{-36} -20 q^{-38} +18 q^{-40} -16 q^{-42} +11 q^{-44} -8 q^{-46} +5 q^{-48} -3 q^{-50} + 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}-10 q^{10}-6 q^8+10 q^6+14 q^4-6 q^2-20-6 q^{-2} +19 q^{-4} +16 q^{-6} -13 q^{-8} -21 q^{-10} + q^{-12} +20 q^{-14} +6 q^{-16} -14 q^{-18} -9 q^{-20} +13 q^{-22} +13 q^{-24} -5 q^{-26} -10 q^{-28} +8 q^{-30} +16 q^{-32} + q^{-34} -13 q^{-36} +16 q^{-40} +6 q^{-42} -17 q^{-44} -14 q^{-46} +11 q^{-48} +17 q^{-50} -7 q^{-52} -24 q^{-54} -8 q^{-56} +17 q^{-58} +14 q^{-60} -9 q^{-62} -17 q^{-64} -2 q^{-66} +12 q^{-68} +6 q^{-70} -4 q^{-72} -6 q^{-74} +4 q^{-78} +2 q^{-80} - q^{-82} - 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}-10 q^8+10 q^6-14 q^4+15 q^2-18+14 q^{-2} -15 q^{-4} +14 q^{-6} -10 q^{-8} +2 q^{-10} + q^{-12} -4 q^{-14} +13 q^{-16} -19 q^{-18} +24 q^{-20} -21 q^{-22} +37 q^{-24} -24 q^{-26} +35 q^{-28} -22 q^{-30} +32 q^{-32} -17 q^{-34} +13 q^{-36} -15 q^{-38} - q^{-40} - q^{-42} -12 q^{-44} +4 q^{-46} -17 q^{-48} +15 q^{-50} -15 q^{-52} +14 q^{-54} -15 q^{-56} +13 q^{-58} -9 q^{-60} +7 q^{-62} -6 q^{-64} +5 q^{-66} -2 q^{-68} +2 q^{-70} - 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}-6 q^{22}-3 q^{20}+16 q^{18}-31 q^{16}+42 q^{14}-44 q^{12}+24 q^{10}+7 q^8-48 q^6+87 q^4-101 q^2+88-42 q^{-2} -28 q^{-4} +91 q^{-6} -127 q^{-8} +120 q^{-10} -70 q^{-12} -2 q^{-14} +67 q^{-16} -98 q^{-18} +82 q^{-20} -25 q^{-22} -44 q^{-24} +92 q^{-26} -95 q^{-28} +44 q^{-30} +39 q^{-32} -116 q^{-34} +163 q^{-36} -147 q^{-38} +84 q^{-40} +20 q^{-42} -116 q^{-44} +180 q^{-46} -180 q^{-48} +130 q^{-50} -33 q^{-52} -57 q^{-54} +120 q^{-56} -126 q^{-58} +89 q^{-60} -17 q^{-62} -50 q^{-64} +83 q^{-66} -73 q^{-68} +18 q^{-70} +51 q^{-72} -103 q^{-74} +113 q^{-76} -78 q^{-78} +6 q^{-80} +62 q^{-82} -114 q^{-84} +123 q^{-86} -94 q^{-88} +37 q^{-90} +16 q^{-92} -60 q^{-94} +74 q^{-96} -66 q^{-98} +43 q^{-100} -15 q^{-102} -7 q^{-104} +19 q^{-106} -24 q^{-108} +20 q^{-110} -14 q^{-112} +8 q^{-114} - q^{-116} -3 q^{-118} +4 q^{-120} -4 q^{-122} +3 q^{-124} - 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 51"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^3-7 t^2+15 t-19+15 t^{-1} -7 t^{-2} +2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 2 z^6+5 z^4+5 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]= { 67, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^8+2 q^7-5 q^6+8 q^5-10 q^4+12 q^3-10 q^2+9 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} +4 z^4 a^{-4} -z^4 a^{-6} -z^4+3 z^2 a^{-2} +7 z^2 a^{-4} -3 z^2 a^{-6} -2 z^2+ a^{-2} +4 a^{-4} -3 a^{-6} -1 }[/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} +3 z^8 a^{-2} +6 z^8 a^{-4} +3 z^8 a^{-6} +4 z^7 a^{-1} +6 z^7 a^{-3} +5 z^7 a^{-5} +3 z^7 a^{-7} -z^6 a^{-2} -12 z^6 a^{-4} -6 z^6 a^{-6} +2 z^6 a^{-8} +3 z^6+a z^5-6 z^5 a^{-1} -16 z^5 a^{-3} -16 z^5 a^{-5} -6 z^5 a^{-7} +z^5 a^{-9} -6 z^4 a^{-2} +13 z^4 a^{-4} +9 z^4 a^{-6} -4 z^4 a^{-8} -6 z^4-2 a z^3+15 z^3 a^{-3} +21 z^3 a^{-5} +5 z^3 a^{-7} -3 z^3 a^{-9} +4 z^2 a^{-2} -8 z^2 a^{-4} -8 z^2 a^{-6} +z^2 a^{-8} +3 z^2+a z-5 z a^{-3} -9 z a^{-5} -3 z a^{-7} +2 z a^{-9} - a^{-2} +4 a^{-4} +3 a^{-6} -1 }[/math]

Vassiliev invariants

V2 and V3: (5, 8)
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{ 20 }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ \frac{1126}{3} }[/math] [math]\displaystyle{ \frac{170}{3} }[/math] [math]\displaystyle{ 1280 }[/math] [math]\displaystyle{ \frac{6304}{3} }[/math] [math]\displaystyle{ \frac{1024}{3} }[/math] [math]\displaystyle{ 320 }[/math] [math]\displaystyle{ \frac{4000}{3} }[/math] [math]\displaystyle{ 2048 }[/math] [math]\displaystyle{ \frac{22520}{3} }[/math] [math]\displaystyle{ \frac{3400}{3} }[/math] [math]\displaystyle{ \frac{75871}{6} }[/math] [math]\displaystyle{ \frac{274}{3} }[/math] [math]\displaystyle{ \frac{46574}{9} }[/math] [math]\displaystyle{ \frac{2149}{18} }[/math] [math]\displaystyle{ \frac{4063}{6} }[/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 51. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 -3-2-101234567χ
17          1-1
15         1 1
13        41 -3
11       41  3
9      64   -2
7     64    2
5    46     2
3   56      -1
1  25       3
-1 14        -3
-3 2         2
-51          -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, 51]]
Out[2]=  
10
In[3]:=
PD[Knot[10, 51]]
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[13, 1, 14, 20], X[19, 11, 20, 10], 



  X[11, 19, 12, 18], X[17, 13, 18, 12], X[7, 2, 8, 3]]
In[4]:=
GaussCode[Knot[10, 51]]
Out[4]=  
GaussCode[-1, 10, -2, 1, -4, 5, -10, 2, -3, 7, -8, 9, -6, 4, -5, 3, -9, 
  8, -7, 6]
In[5]:=
BR[Knot[10, 51]]
Out[5]=  
BR[4, {1, 1, 2, -1, 2, 2, -3, 2, 2, -3, -3}]
In[6]:=
alex = Alexander[Knot[10, 51]][t]
Out[6]=  
      2    7    15             2      3

-19 + -- - -- + -- + 15 t - 7 t  + 2 t



      3    2   t


      t    t
In[7]:=
Conway[Knot[10, 51]][z]
Out[7]=  
       2      4      6
1 + 5 z  + 5 z  + 2 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[10, 51]}
In[9]:=
{KnotDet[Knot[10, 51]], KnotSignature[Knot[10, 51]]}
Out[9]=  
{67, 2}
In[10]:=
J=Jones[Knot[10, 51]][q]
Out[10]=  
      -2   3             2       3       4      5      6      7    8

-6 - q   + - + 9 q - 10 q  + 12 q  - 10 q  + 8 q  - 5 q  + 2 q  - q


           q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[10, 51]}
In[12]:=
A2Invariant[Knot[10, 51]][q]
Out[12]=  
      -6    -4    -2      2      4      6    8      10      12    14

-1 - q   + q   - q   + 2 q  - 2 q  + 3 q  + q  + 2 q   + 3 q   - q   + 



    16      18      20    24


  2 q   - 2 q   - 2 q   - q
In[13]:=
Kauffman[Knot[10, 51]][a, z]
Out[13]=  
                                                           2      2

    3    4     -2   2 z   3 z   9 z   5 z            2   z    8 z



-1 + -- + -- - a   + --- - --- - --- - --- + a z + 3 z  + -- - ---- - 



     6    4          9     7     5     3                  8     6
    a    a          a     a     a     a                  a     a

    2      2      3      3       3       3                      4
 8 z    4 z    3 z    5 z    21 z    15 z         3      4   4 z
 ---- + ---- - ---- + ---- + ----- + ----- - 2 a z  - 6 z  - ---- + 
   4      2      9      7      5       3                       8
  a      a      a      a      a       a                       a

    4       4      4    5      5       5       5      5
 9 z    13 z    6 z    z    6 z    16 z    16 z    6 z       5
 ---- + ----- - ---- + -- - ---- - ----- - ----- - ---- + a z  + 
   6      4       2     9     7      5       3      a
  a      a       a     a     a      a       a

           6      6       6    6      7      7      7      7      8
    6   2 z    6 z    12 z    z    3 z    5 z    6 z    4 z    3 z
 3 z  + ---- - ---- - ----- - -- + ---- + ---- + ---- + ---- + ---- + 
          8      6      4      2     7      5      3     a       6
         a      a      a      a     a      a      a             a

    8      8    9    9
 6 z    3 z    z    z
 ---- + ---- + -- + --
   4      2     5    3


   a      a     a    a
In[14]:=
{Vassiliev[2][Knot[10, 51]], Vassiliev[3][Knot[10, 51]]}
Out[14]=  
{0, 8}
In[15]:=
Kh[Knot[10, 51]][q, t]
Out[15]=  
         3     1       2      1      4    2 q      3        5

5 q + 5 q  + ----- + ----- + ---- + --- + --- + 6 q  t + 4 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    11  5
 6 q  t  + 6 q  t  + 4 q  t  + 6 q  t  + 4 q  t  + 4 q   t  + q   t  + 

    13  5    13  6    15  6    17  7


  4 q   t  + q   t  + q   t  + q   t
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