10 12

10_11

10_13

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10 12 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X13,19,14,18 X5,15,6,14 X7,17,8,16 X15,7,16,6 X17,9,18,8 X11,1,12,20 X19,13,20,12 X9,2,10,3
Gauss code	-1, 10, -2, 1, -4, 6, -5, 7, -10, 2, -8, 9, -3, 4, -6, 5, -7, 3, -9, 8
Dowker-Thistlethwaite code	4 10 14 16 2 20 18 6 8 12
Conway Notation	[4312]

Minimum Braid Representative: Length is 11, width is 4. Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-2][-10] Hyperbolic Volume 9.8175 A-Polynomial See Data:10 12/A-polynomial

[edit Notes for 10 12'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 12's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-6 t^2+10 t-11+10 t^{-1} -6 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6+6 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 47, 2 }
Jones polynomial	[math]\displaystyle{ -q^8+2 q^7-4 q^6+6 q^5-7 q^4+8 q^3-7 q^2+6 q-3+2 q^{-1} - q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +4 z^4 a^{-2} +4 z^4 a^{-4} -z^4 a^{-6} -z^4+5 z^2 a^{-2} +5 z^2 a^{-4} -3 z^2 a^{-6} -3 z^2+2 a^{-2} +2 a^{-4} -2 a^{-6} -1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +4 z^8 a^{-4} +2 z^8 a^{-6} +2 z^7 a^{-1} -z^7 a^{-3} -z^7 a^{-5} +2 z^7 a^{-7} -5 z^6 a^{-2} -14 z^6 a^{-4} -5 z^6 a^{-6} +2 z^6 a^{-8} +2 z^6+a z^5-4 z^5 a^{-1} -z^5 a^{-3} -3 z^5 a^{-7} +z^5 a^{-9} +4 z^4 a^{-2} +23 z^4 a^{-4} +8 z^4 a^{-6} -5 z^4 a^{-8} -6 z^4-3 a z^3+5 z^3 a^{-3} +4 z^3 a^{-5} -z^3 a^{-7} -3 z^3 a^{-9} +2 z^2 a^{-2} -12 z^2 a^{-4} -8 z^2 a^{-6} +2 z^2 a^{-8} +4 z^2+a z+z a^{-1} -z a^{-3} -3 z a^{-5} +2 z a^{-9} -2 a^{-2} +2 a^{-4} +2 a^{-6} -1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^6+2 q^{-2} - q^{-4} +2 q^{-6} + q^{-8} + q^{-10} +2 q^{-12} - q^{-14} + q^{-16} - q^{-18} - q^{-20} - q^{-24} }[/math]
The G2 invariant	[math]\displaystyle{ q^{32}-q^{30}+2 q^{28}-3 q^{26}+2 q^{24}-2 q^{22}-2 q^{20}+6 q^{18}-9 q^{16}+9 q^{14}-10 q^{12}+6 q^{10}-10 q^6+17 q^4-23 q^2+23-15 q^{-2} + q^{-4} +14 q^{-6} -26 q^{-8} +38 q^{-10} -30 q^{-12} +14 q^{-14} +6 q^{-16} -23 q^{-18} +30 q^{-20} -21 q^{-22} +8 q^{-24} +12 q^{-26} -20 q^{-28} +21 q^{-30} -6 q^{-32} -14 q^{-34} +32 q^{-36} -39 q^{-38} +30 q^{-40} -9 q^{-42} -16 q^{-44} +40 q^{-46} -49 q^{-48} +48 q^{-50} -29 q^{-52} +3 q^{-54} +23 q^{-56} -40 q^{-58} +42 q^{-60} -27 q^{-62} +9 q^{-64} +13 q^{-66} -23 q^{-68} +22 q^{-70} -8 q^{-72} -10 q^{-74} +24 q^{-76} -29 q^{-78} +14 q^{-80} +4 q^{-82} -23 q^{-84} +34 q^{-86} -34 q^{-88} +22 q^{-90} -7 q^{-92} -13 q^{-94} +22 q^{-96} -28 q^{-98} +23 q^{-100} -14 q^{-102} +4 q^{-104} +4 q^{-106} -11 q^{-108} +13 q^{-110} -11 q^{-112} +8 q^{-114} -3 q^{-116} - q^{-118} +3 q^{-120} -4 q^{-122} +3 q^{-124} - q^{-126} + q^{-128} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_12.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^5+q^3-q+3 q^{-1} - q^{-3} + q^{-5} + q^{-7} - q^{-9} +2 q^{-11} -2 q^{-13} + q^{-15} - q^{-17} }[/math]
2	[math]\displaystyle{ q^{16}-q^{14}-q^{12}+2 q^{10}-3 q^8-2 q^6+6 q^4-2 q^2-5+10 q^{-2} -9 q^{-6} +8 q^{-8} +3 q^{-10} -6 q^{-12} +2 q^{-14} +4 q^{-16} -6 q^{-20} +4 q^{-22} +5 q^{-24} -10 q^{-26} + q^{-28} +8 q^{-30} -8 q^{-32} -2 q^{-34} +7 q^{-36} -3 q^{-38} -2 q^{-40} +3 q^{-42} - q^{-44} - q^{-46} + q^{-48} }[/math]
3	[math]\displaystyle{ -q^{33}+q^{31}+q^{29}-q^{25}+q^{23}+q^{21}-3 q^{19}-2 q^{17}+4 q^{15}+3 q^{13}-8 q^{11}-7 q^9+7 q^7+15 q^5-6 q^3-19 q- q^{-1} +29 q^{-3} +9 q^{-5} -28 q^{-7} -18 q^{-9} +27 q^{-11} +27 q^{-13} -21 q^{-15} -26 q^{-17} +14 q^{-19} +25 q^{-21} -5 q^{-23} -21 q^{-25} -2 q^{-27} +16 q^{-29} +9 q^{-31} -11 q^{-33} -19 q^{-35} +6 q^{-37} +25 q^{-39} -32 q^{-43} -6 q^{-45} +31 q^{-47} +14 q^{-49} -29 q^{-51} -20 q^{-53} +21 q^{-55} +24 q^{-57} -13 q^{-59} -21 q^{-61} +3 q^{-63} +18 q^{-65} +2 q^{-67} -12 q^{-69} -3 q^{-71} +6 q^{-73} +3 q^{-75} -3 q^{-77} - q^{-79} +2 q^{-81} -2 q^{-85} + q^{-89} + q^{-91} - q^{-93} }[/math]
4	[math]\displaystyle{ q^{56}-q^{54}-q^{52}-q^{48}+3 q^{46}-q^{44}+q^{42}+2 q^{40}-5 q^{38}+2 q^{36}-3 q^{34}+4 q^{32}+11 q^{30}-7 q^{28}-5 q^{26}-16 q^{24}+2 q^{22}+27 q^{20}+6 q^{18}-2 q^{16}-40 q^{14}-23 q^{12}+29 q^{10}+37 q^8+40 q^6-41 q^4-70 q^2-21+40 q^{-2} +113 q^{-4} +16 q^{-6} -89 q^{-8} -99 q^{-10} -19 q^{-12} +143 q^{-14} +97 q^{-16} -40 q^{-18} -132 q^{-20} -92 q^{-22} +103 q^{-24} +124 q^{-26} +21 q^{-28} -96 q^{-30} -107 q^{-32} +35 q^{-34} +88 q^{-36} +50 q^{-38} -39 q^{-40} -79 q^{-42} -15 q^{-44} +44 q^{-46} +58 q^{-48} +5 q^{-50} -51 q^{-52} -63 q^{-54} +10 q^{-56} +78 q^{-58} +51 q^{-60} -32 q^{-62} -116 q^{-64} -33 q^{-66} +90 q^{-68} +106 q^{-70} +12 q^{-72} -143 q^{-74} -87 q^{-76} +55 q^{-78} +127 q^{-80} +78 q^{-82} -102 q^{-84} -112 q^{-86} -23 q^{-88} +81 q^{-90} +113 q^{-92} -21 q^{-94} -71 q^{-96} -65 q^{-98} +3 q^{-100} +78 q^{-102} +27 q^{-104} -4 q^{-106} -44 q^{-108} -33 q^{-110} +22 q^{-112} +15 q^{-114} +21 q^{-116} -6 q^{-118} -18 q^{-120} -6 q^{-124} +10 q^{-126} +4 q^{-128} -2 q^{-130} +3 q^{-132} -7 q^{-134} +4 q^{-142} - q^{-144} - q^{-148} - q^{-150} + q^{-152} }[/math]
5	[math]\displaystyle{ -q^{85}+q^{83}+q^{81}+q^{77}-q^{75}-3 q^{73}-q^{71}+q^{69}+4 q^{65}+4 q^{63}-3 q^{61}-7 q^{59}-5 q^{57}-q^{55}+8 q^{53}+16 q^{51}+8 q^{49}-13 q^{47}-22 q^{45}-17 q^{43}+7 q^{41}+34 q^{39}+39 q^{37}+4 q^{35}-45 q^{33}-61 q^{31}-32 q^{29}+33 q^{27}+87 q^{25}+76 q^{23}-6 q^{21}-94 q^{19}-130 q^{17}-67 q^{15}+68 q^{13}+171 q^{11}+160 q^9+26 q^7-174 q^5-268 q^3-154 q+106 q^{-1} +332 q^{-3} +336 q^{-5} +34 q^{-7} -350 q^{-9} -478 q^{-11} -219 q^{-13} +263 q^{-15} +583 q^{-17} +426 q^{-19} -124 q^{-21} -595 q^{-23} -574 q^{-25} -54 q^{-27} +524 q^{-29} +655 q^{-31} +223 q^{-33} -403 q^{-35} -650 q^{-37} -324 q^{-39} +247 q^{-41} +566 q^{-43} +376 q^{-45} -109 q^{-47} -447 q^{-49} -364 q^{-51} +8 q^{-53} +312 q^{-55} +316 q^{-57} +64 q^{-59} -200 q^{-61} -265 q^{-63} -107 q^{-65} +115 q^{-67} +220 q^{-69} +142 q^{-71} -54 q^{-73} -208 q^{-75} -196 q^{-77} +4 q^{-79} +224 q^{-81} +261 q^{-83} +60 q^{-85} -241 q^{-87} -366 q^{-89} -142 q^{-91} +255 q^{-93} +465 q^{-95} +258 q^{-97} -220 q^{-99} -553 q^{-101} -399 q^{-103} +145 q^{-105} +589 q^{-107} +528 q^{-109} -5 q^{-111} -554 q^{-113} -618 q^{-115} -158 q^{-117} +429 q^{-119} +647 q^{-121} +316 q^{-123} -255 q^{-125} -579 q^{-127} -419 q^{-129} +45 q^{-131} +440 q^{-133} +455 q^{-135} +116 q^{-137} -260 q^{-139} -392 q^{-141} -224 q^{-143} +79 q^{-145} +283 q^{-147} +250 q^{-149} +45 q^{-151} -148 q^{-153} -207 q^{-155} -111 q^{-157} +37 q^{-159} +135 q^{-161} +121 q^{-163} +25 q^{-165} -60 q^{-167} -89 q^{-169} -54 q^{-171} +8 q^{-173} +51 q^{-175} +50 q^{-177} +16 q^{-179} -18 q^{-181} -32 q^{-183} -22 q^{-185} -3 q^{-187} +16 q^{-189} +20 q^{-191} +7 q^{-193} -4 q^{-195} -9 q^{-197} -10 q^{-199} -4 q^{-201} +6 q^{-203} +6 q^{-205} +2 q^{-207} +2 q^{-209} -2 q^{-211} -4 q^{-213} - q^{-215} + q^{-217} + q^{-221} + q^{-223} - q^{-225} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^6+2 q^{-2} - q^{-4} +2 q^{-6} + q^{-8} + q^{-10} +2 q^{-12} - q^{-14} + q^{-16} - q^{-18} - q^{-20} - q^{-24} }[/math]
1,1	[math]\displaystyle{ q^{20}-2 q^{18}+4 q^{16}-8 q^{14}+15 q^{12}-22 q^{10}+28 q^8-42 q^6+55 q^4-70 q^2+78-96 q^{-2} +105 q^{-4} -98 q^{-6} +90 q^{-8} -56 q^{-10} +29 q^{-12} +30 q^{-14} -76 q^{-16} +128 q^{-18} -171 q^{-20} +202 q^{-22} -216 q^{-24} +214 q^{-26} -199 q^{-28} +162 q^{-30} -118 q^{-32} +64 q^{-34} -19 q^{-36} -32 q^{-38} +70 q^{-40} -92 q^{-42} +100 q^{-44} -102 q^{-46} +96 q^{-48} -80 q^{-50} +66 q^{-52} -54 q^{-54} +40 q^{-56} -28 q^{-58} +19 q^{-60} -12 q^{-62} +6 q^{-64} -2 q^{-66} + q^{-68} }[/math]
2,0	[math]\displaystyle{ q^{18}-q^{14}+q^{10}-2 q^8-4 q^6+q^4+3 q^2-2-2 q^{-2} +5 q^{-4} +2 q^{-6} -3 q^{-8} + q^{-10} +4 q^{-12} +5 q^{-18} +4 q^{-20} -2 q^{-22} +4 q^{-24} +4 q^{-26} -3 q^{-28} -3 q^{-30} +2 q^{-32} - q^{-34} -6 q^{-36} -3 q^{-38} + q^{-40} - q^{-42} -3 q^{-44} + q^{-46} +2 q^{-48} + q^{-62} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{14}-q^{12}+q^8-3 q^6+2 q^2-6+7 q^{-4} -7 q^{-6} +2 q^{-8} +10 q^{-10} -2 q^{-12} + q^{-14} +7 q^{-16} + q^{-18} - q^{-20} + q^{-22} +4 q^{-24} -2 q^{-26} -6 q^{-28} +5 q^{-30} - q^{-32} -10 q^{-34} +5 q^{-36} + q^{-38} -7 q^{-40} +3 q^{-42} + q^{-44} -3 q^{-46} +2 q^{-48} + q^{-50} - q^{-52} + q^{-54} }[/math]
1,0,0	[math]\displaystyle{ -q^7-q^3+q- q^{-1} +2 q^{-3} - q^{-5} +2 q^{-7} + q^{-9} +2 q^{-11} +2 q^{-13} + q^{-15} +2 q^{-17} - q^{-19} + q^{-21} -2 q^{-23} -2 q^{-27} - q^{-31} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{14}+q^{12}-2 q^{10}+3 q^8-5 q^6+6 q^4-8 q^2+8-8 q^{-2} +9 q^{-4} -5 q^{-6} +4 q^{-8} +2 q^{-10} -4 q^{-12} +11 q^{-14} -13 q^{-16} +17 q^{-18} -17 q^{-20} +17 q^{-22} -16 q^{-24} +12 q^{-26} -8 q^{-28} +3 q^{-30} + q^{-32} -4 q^{-34} +7 q^{-36} -9 q^{-38} +9 q^{-40} -9 q^{-42} +7 q^{-44} -5 q^{-46} +4 q^{-48} -3 q^{-50} + q^{-52} - q^{-54} }[/math]
1,0	[math]\displaystyle{ q^{24}-q^{20}-q^{18}+q^{16}+2 q^{14}-q^{12}-4 q^{10}-2 q^8+3 q^6+5 q^4-2 q^2-8-3 q^{-2} +7 q^{-4} +9 q^{-6} -4 q^{-8} -9 q^{-10} - q^{-12} +10 q^{-14} +6 q^{-16} -4 q^{-18} -5 q^{-20} +5 q^{-22} +7 q^{-24} -5 q^{-28} + q^{-30} +6 q^{-32} +2 q^{-34} -5 q^{-36} -3 q^{-38} +5 q^{-40} +4 q^{-42} -5 q^{-44} -7 q^{-46} +3 q^{-48} +8 q^{-50} - q^{-52} -10 q^{-54} -6 q^{-56} +7 q^{-58} +8 q^{-60} -3 q^{-62} -9 q^{-64} -3 q^{-66} +6 q^{-68} +4 q^{-70} -2 q^{-72} -4 q^{-74} - q^{-76} +3 q^{-78} +2 q^{-80} - q^{-82} - q^{-84} + q^{-88} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{18}-q^{16}+q^{14}-2 q^{12}+3 q^{10}-4 q^8+3 q^6-6 q^4+5 q^2-8+5 q^{-2} -7 q^{-4} +7 q^{-6} -5 q^{-8} +4 q^{-10} + q^{-12} +4 q^{-14} +6 q^{-16} -4 q^{-18} +10 q^{-20} -7 q^{-22} +15 q^{-24} -11 q^{-26} +14 q^{-28} -12 q^{-30} +15 q^{-32} -10 q^{-34} +9 q^{-36} -10 q^{-38} +3 q^{-40} -3 q^{-42} -2 q^{-44} - q^{-46} -6 q^{-48} +5 q^{-50} -7 q^{-52} +6 q^{-54} -8 q^{-56} +7 q^{-58} -6 q^{-60} +4 q^{-62} -4 q^{-64} +4 q^{-66} -2 q^{-68} +2 q^{-70} - q^{-72} + q^{-74} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{32}-q^{30}+2 q^{28}-3 q^{26}+2 q^{24}-2 q^{22}-2 q^{20}+6 q^{18}-9 q^{16}+9 q^{14}-10 q^{12}+6 q^{10}-10 q^6+17 q^4-23 q^2+23-15 q^{-2} + q^{-4} +14 q^{-6} -26 q^{-8} +38 q^{-10} -30 q^{-12} +14 q^{-14} +6 q^{-16} -23 q^{-18} +30 q^{-20} -21 q^{-22} +8 q^{-24} +12 q^{-26} -20 q^{-28} +21 q^{-30} -6 q^{-32} -14 q^{-34} +32 q^{-36} -39 q^{-38} +30 q^{-40} -9 q^{-42} -16 q^{-44} +40 q^{-46} -49 q^{-48} +48 q^{-50} -29 q^{-52} +3 q^{-54} +23 q^{-56} -40 q^{-58} +42 q^{-60} -27 q^{-62} +9 q^{-64} +13 q^{-66} -23 q^{-68} +22 q^{-70} -8 q^{-72} -10 q^{-74} +24 q^{-76} -29 q^{-78} +14 q^{-80} +4 q^{-82} -23 q^{-84} +34 q^{-86} -34 q^{-88} +22 q^{-90} -7 q^{-92} -13 q^{-94} +22 q^{-96} -28 q^{-98} +23 q^{-100} -14 q^{-102} +4 q^{-104} +4 q^{-106} -11 q^{-108} +13 q^{-110} -11 q^{-112} +8 q^{-114} -3 q^{-116} - q^{-118} +3 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 12"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^3-6 t^2+10 t-11+10 t^{-1} -6 t^{-2} +2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^6+6 z^4+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{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 47, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^8+2 q^7-4 q^6+6 q^5-7 q^4+8 q^3-7 q^2+6 q-3+2 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} +4 z^4 a^{-2} +4 z^4 a^{-4} -z^4 a^{-6} -z^4+5 z^2 a^{-2} +5 z^2 a^{-4} -3 z^2 a^{-6} -3 z^2+2 a^{-2} +2 a^{-4} -2 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} +2 z^8 a^{-2} +4 z^8 a^{-4} +2 z^8 a^{-6} +2 z^7 a^{-1} -z^7 a^{-3} -z^7 a^{-5} +2 z^7 a^{-7} -5 z^6 a^{-2} -14 z^6 a^{-4} -5 z^6 a^{-6} +2 z^6 a^{-8} +2 z^6+a z^5-4 z^5 a^{-1} -z^5 a^{-3} -3 z^5 a^{-7} +z^5 a^{-9} +4 z^4 a^{-2} +23 z^4 a^{-4} +8 z^4 a^{-6} -5 z^4 a^{-8} -6 z^4-3 a z^3+5 z^3 a^{-3} +4 z^3 a^{-5} -z^3 a^{-7} -3 z^3 a^{-9} +2 z^2 a^{-2} -12 z^2 a^{-4} -8 z^2 a^{-6} +2 z^2 a^{-8} +4 z^2+a z+z a^{-1} -z a^{-3} -3 z a^{-5} +2 z a^{-9} -2 a^{-2} +2 a^{-4} +2 a^{-6} -1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_54, ...}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {...}

Vassiliev invariants

V2 and V3:

(4, 6)

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{ 16 }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{680}{3} }[/math] [math]\displaystyle{ \frac{40}{3} }[/math] [math]\displaystyle{ 768 }[/math] [math]\displaystyle{ 1152 }[/math] [math]\displaystyle{ 192 }[/math] [math]\displaystyle{ 80 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 1152 }[/math] [math]\displaystyle{ \frac{10880}{3} }[/math] [math]\displaystyle{ \frac{640}{3} }[/math] [math]\displaystyle{ \frac{92822}{15} }[/math] [math]\displaystyle{ \frac{12232}{15} }[/math] [math]\displaystyle{ \frac{59048}{45} }[/math] [math]\displaystyle{ \frac{538}{9} }[/math] [math]\displaystyle{ \frac{422}{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 12. 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                   1   1 13                 3 1   -2 11               3 1     2 9             4 3       -1 7           4 3         1 5         3 4           1 3       3 4             -1 1     1 4               3 -1   1 2                 -1 -3   1                   1 -5 1                     -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

[math]\displaystyle{ n }[/math]	[math]\displaystyle{ J_n }[/math]
2	[math]\displaystyle{ q^{23}-2 q^{22}+5 q^{20}-7 q^{19}-q^{18}+15 q^{17}-16 q^{16}-7 q^{15}+31 q^{14}-23 q^{13}-18 q^{12}+46 q^{11}-24 q^{10}-28 q^9+52 q^8-20 q^7-30 q^6+44 q^5-11 q^4-25 q^3+27 q^2-2 q-15+12 q^{-1} + q^{-2} -7 q^{-3} +4 q^{-4} -2 q^{-6} + q^{-7} }[/math]
3	[math]\displaystyle{ -q^{45}+2 q^{44}-q^{42}-3 q^{41}+4 q^{40}+2 q^{39}-4 q^{38}-5 q^{37}+10 q^{36}+5 q^{35}-13 q^{34}-14 q^{33}+24 q^{32}+21 q^{31}-28 q^{30}-38 q^{29}+32 q^{28}+58 q^{27}-31 q^{26}-79 q^{25}+23 q^{24}+101 q^{23}-14 q^{22}-116 q^{21}-3 q^{20}+133 q^{19}+11 q^{18}-135 q^{17}-28 q^{16}+141 q^{15}+31 q^{14}-128 q^{13}-46 q^{12}+122 q^{11}+47 q^{10}-98 q^9-57 q^8+82 q^7+52 q^6-50 q^5-57 q^4+37 q^3+42 q^2-13 q-37+7 q^{-1} +24 q^{-2} -16 q^{-4} - q^{-5} +10 q^{-6} - q^{-7} -5 q^{-8} +4 q^{-10} -2 q^{-11} - q^{-12} +2 q^{-14} - q^{-15} }[/math]
4	[math]\displaystyle{ q^{74}-2 q^{73}+q^{71}-q^{70}+6 q^{69}-6 q^{68}+q^{66}-8 q^{65}+16 q^{64}-11 q^{63}+6 q^{62}+7 q^{61}-24 q^{60}+22 q^{59}-29 q^{58}+18 q^{57}+34 q^{56}-30 q^{55}+29 q^{54}-84 q^{53}+7 q^{52}+74 q^{51}+q^{50}+80 q^{49}-159 q^{48}-61 q^{47}+68 q^{46}+51 q^{45}+214 q^{44}-191 q^{43}-165 q^{42}-21 q^{41}+61 q^{40}+394 q^{39}-142 q^{38}-237 q^{37}-163 q^{36}+5 q^{35}+549 q^{34}-48 q^{33}-253 q^{32}-286 q^{31}-78 q^{30}+633 q^{29}+35 q^{28}-226 q^{27}-354 q^{26}-151 q^{25}+645 q^{24}+91 q^{23}-173 q^{22}-368 q^{21}-210 q^{20}+581 q^{19}+131 q^{18}-84 q^{17}-330 q^{16}-263 q^{15}+439 q^{14}+142 q^{13}+33 q^{12}-227 q^{11}-284 q^{10}+244 q^9+102 q^8+125 q^7-90 q^6-238 q^5+82 q^4+22 q^3+135 q^2+15 q-141+9 q^{-1} -39 q^{-2} +86 q^{-3} +44 q^{-4} -60 q^{-5} +6 q^{-6} -47 q^{-7} +34 q^{-8} +27 q^{-9} -22 q^{-10} +14 q^{-11} -26 q^{-12} +9 q^{-13} +9 q^{-14} -11 q^{-15} +12 q^{-16} -8 q^{-17} +2 q^{-18} +2 q^{-19} -6 q^{-20} +5 q^{-21} - q^{-22} + q^{-23} -2 q^{-25} + q^{-26} }[/math]
5	Not Available
6	Not Available
7	Not Available

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[10, 12]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[13, 19, 14, 18], X[5, 15, 6, 14], X[7, 17, 8, 16], X[15, 7, 16, 6], X[17, 9, 18, 8], X[11, 1, 12, 20], X[19, 13, 20, 12], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[10, 12]]
Out[3]=	GaussCode[-1, 10, -2, 1, -4, 6, -5, 7, -10, 2, -8, 9, -3, 4, -6, 5, -7, 3, -9, 8]
In[4]:=	DTCode[Knot[10, 12]]
Out[4]=	DTCode[4, 10, 14, 16, 2, 20, 18, 6, 8, 12]
In[5]:=	br = BR[Knot[10, 12]]
Out[5]=	BR[4, {1, 1, 1, 1, 1, 2, -1, -3, 2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 12]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 12]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 12]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 12]][t]
Out[10]=	2 6 10 2 3 -11 + -- - -- + -- + 10 t - 6 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 12]][z]
Out[11]=	2 4 6 1 + 4 z + 6 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 12], Knot[10, 54]}
In[13]:=	{KnotDet[Knot[10, 12]], KnotSignature[Knot[10, 12]]}
Out[13]=	{47, 2}
In[14]:=	Jones[Knot[10, 12]][q]
Out[14]=	-2 2 2 3 4 5 6 7 8 -3 - q + - + 6 q - 7 q + 8 q - 7 q + 6 q - 4 q + 2 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 12]}
In[16]:=	A2Invariant[Knot[10, 12]][q]
Out[16]=	-6 2 4 6 8 10 12 14 16 18 20 24 -q + 2 q - q + 2 q + q + q + 2 q - q + q - q - q - q
In[17]:=	HOMFLYPT[Knot[10, 12]][a, z]
Out[17]=	2 2 2 4 4 4 2 2 2 2 3 z 5 z 5 z 4 z 4 z 4 z -1 - -- + -- + -- - 3 z - ---- + ---- + ---- - z - -- + ---- + ---- + 6 4 2 6 4 2 6 4 2 a a a a a a a a a 6 6 z z -- + -- 4 2 a a
In[18]:=	Kauffman[Knot[10, 12]][a, z]
Out[18]=	2 2 2 2 2 2 z 3 z z z 2 2 z 8 z -1 + -- + -- - -- + --- - --- - -- + - + a z + 4 z + ---- - ---- - 6 4 2 9 5 3 a 8 6 a a a a a a a a 2 2 3 3 3 3 4 12 z 2 z 3 z z 4 z 5 z 3 4 5 z ----- + ---- - ---- - -- + ---- + ---- - 3 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 6 8 z 23 z 4 z z 3 z z 4 z 5 6 2 z ---- + ----- + ---- + -- - ---- - -- - ---- + a z + 2 z + ---- - 6 4 2 9 7 3 a 8 a a a a a a a 6 6 6 7 7 7 7 8 8 8 5 z 14 z 5 z 2 z z z 2 z 2 z 4 z 2 z ---- - ----- - ---- + ---- - -- - -- + ---- + ---- + ---- + ---- + 6 4 2 7 5 3 a 6 4 2 a a a a a a a a a 9 9 z z -- + -- 5 3 a a
In[19]:=	{Vassiliev[2][Knot[10, 12]], Vassiliev[3][Knot[10, 12]]}
Out[19]=	{4, 6}
In[20]:=	Kh[Knot[10, 12]][q, t]
Out[20]=	3 1 1 1 2 q 3 5 4 q + 3 q + ----- + ----- + ---- + --- + - + 4 q t + 3 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 4 q t + 4 q t + 3 q t + 4 q t + 3 q t + 3 q t + q t + 13 5 13 6 15 6 17 7 3 q t + q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 12], 2][q]
Out[21]=	-7 2 4 7 -2 12 2 3 4 -15 + q - -- + -- - -- + q + -- - 2 q + 27 q - 25 q - 11 q + 6 4 3 q q q q 5 6 7 8 9 10 11 12 44 q - 30 q - 20 q + 52 q - 28 q - 24 q + 46 q - 18 q - 13 14 15 16 17 18 19 20 23 q + 31 q - 7 q - 16 q + 15 q - q - 7 q + 5 q - 22 23 2 q + q

See/edit the Rolfsen_Splice_Template.