10 5

10_4

10_6

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10 5 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,12,4,13 X13,1,14,20 X5,15,6,14 X7,17,8,16 X9,19,10,18 X15,7,16,6 X17,9,18,8 X19,11,20,10 X11,2,12,3
Gauss code	-1, 10, -2, 1, -4, 7, -5, 8, -6, 9, -10, 2, -3, 4, -7, 5, -8, 6, -9, 3
Dowker-Thistlethwaite code	4 12 14 16 18 2 20 6 8 10
Conway Notation	[6112]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [0][-12] Hyperbolic Volume 7.37394 A-Polynomial See Data:10 5/A-polynomial

[edit Notes for 10 5'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 5's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-3 t^3+5 t^2-5 t+5-5 t^{-1} +5 t^{-2} -3 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+5 z^6+7 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 33, 4 }
Jones polynomial	[math]\displaystyle{ -q^9+2 q^8-3 q^7+4 q^6-5 q^5+5 q^4-4 q^3+4 q^2-2 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} +17 z^4 a^{-4} -5 z^4 a^{-6} -6 z^2 a^{-2} +17 z^2 a^{-4} -7 z^2 a^{-6} - a^{-2} +5 a^{-4} -3 a^{-6} }[/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} +z^7 a^{-1} -3 z^7 a^{-3} -2 z^7 a^{-5} +2 z^7 a^{-7} -11 z^6 a^{-2} -20 z^6 a^{-4} -7 z^6 a^{-6} +2 z^6 a^{-8} -5 z^5 a^{-1} -2 z^5 a^{-3} -3 z^5 a^{-5} -4 z^5 a^{-7} +2 z^5 a^{-9} +18 z^4 a^{-2} +32 z^4 a^{-4} +10 z^4 a^{-6} -2 z^4 a^{-8} +2 z^4 a^{-10} +6 z^3 a^{-1} +7 z^3 a^{-3} +6 z^3 a^{-5} +3 z^3 a^{-7} -z^3 a^{-9} +z^3 a^{-11} -10 z^2 a^{-2} -22 z^2 a^{-4} -9 z^2 a^{-6} +z^2 a^{-8} -2 z^2 a^{-10} -z a^{-1} -2 z a^{-3} -3 z a^{-5} -z a^{-7} -z a^{-11} + a^{-2} +5 a^{-4} +3 a^{-6} }[/math]
The A2 invariant	[math]\displaystyle{ -q^2+ q^{-4} +2 q^{-6} + q^{-8} +2 q^{-10} - q^{-12} + q^{-14} - q^{-22} - q^{-26} }[/math]
The G2 invariant	[math]\displaystyle{ q^{12}-q^{10}+2 q^8-3 q^6+q^4-2 q^2-1+5 q^{-2} -8 q^{-4} +8 q^{-6} -7 q^{-8} +2 q^{-10} +3 q^{-12} -9 q^{-14} +10 q^{-16} -9 q^{-18} +5 q^{-20} +2 q^{-22} -4 q^{-24} +7 q^{-26} -3 q^{-28} +2 q^{-30} +4 q^{-32} -2 q^{-34} +2 q^{-36} +2 q^{-38} -2 q^{-40} +8 q^{-42} -4 q^{-44} +6 q^{-46} -2 q^{-48} - q^{-50} +7 q^{-52} -10 q^{-54} +9 q^{-56} -6 q^{-58} +2 q^{-60} +2 q^{-62} -6 q^{-64} +6 q^{-66} -5 q^{-68} +2 q^{-70} -3 q^{-74} + q^{-76} + q^{-78} -2 q^{-80} + q^{-82} - q^{-86} +2 q^{-88} -2 q^{-90} + q^{-92} + q^{-96} - q^{-98} - q^{-100} - q^{-104} +2 q^{-106} -4 q^{-108} +4 q^{-110} -3 q^{-112} - q^{-114} +2 q^{-116} -5 q^{-118} +5 q^{-120} -5 q^{-122} +3 q^{-124} - q^{-126} -2 q^{-128} +4 q^{-130} -4 q^{-132} +4 q^{-134} -2 q^{-136} + q^{-138} -2 q^{-142} +2 q^{-144} - q^{-146} + q^{-148} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_5.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^3+q+2 q^{-3} + q^{-7} - q^{-11} + q^{-13} - q^{-15} + q^{-17} - q^{-19} }[/math]
2	[math]\displaystyle{ q^{12}-q^{10}-2 q^8+2 q^6-3 q^2+2+3 q^{-2} -2 q^{-4} + q^{-6} +3 q^{-8} +2 q^{-14} -2 q^{-18} + q^{-22} - q^{-24} - q^{-26} + q^{-28} -2 q^{-32} + q^{-34} - q^{-38} +2 q^{-40} - q^{-42} - q^{-44} +2 q^{-46} - q^{-48} - q^{-50} + q^{-52} }[/math]
3	[math]\displaystyle{ -q^{27}+q^{25}+2 q^{23}-3 q^{19}-2 q^{17}+4 q^{15}+3 q^{13}-4 q^{11}-6 q^9+q^7+6 q^5+2 q^3-6 q-2 q^{-1} +4 q^{-3} +5 q^{-5} - q^{-7} -2 q^{-9} +3 q^{-13} +3 q^{-15} +2 q^{-17} -2 q^{-19} -2 q^{-21} +2 q^{-23} +4 q^{-25} -2 q^{-27} -5 q^{-29} +2 q^{-31} +3 q^{-33} -3 q^{-35} -6 q^{-37} +3 q^{-39} +4 q^{-41} -5 q^{-45} - q^{-47} +2 q^{-49} +3 q^{-51} + q^{-53} -5 q^{-55} -4 q^{-57} +4 q^{-59} +6 q^{-61} - q^{-63} -7 q^{-65} - q^{-67} +7 q^{-69} +3 q^{-71} -5 q^{-73} -4 q^{-75} +3 q^{-77} +5 q^{-79} -2 q^{-81} -4 q^{-83} +3 q^{-87} -2 q^{-91} + q^{-95} + q^{-97} - q^{-99} }[/math]
4	[math]\displaystyle{ q^{48}-q^{46}-2 q^{44}+q^{40}+5 q^{38}-4 q^{34}-4 q^{32}-3 q^{30}+10 q^{28}+7 q^{26}-2 q^{24}-8 q^{22}-13 q^{20}+4 q^{18}+10 q^{16}+8 q^{14}-16 q^{10}-6 q^8+3 q^6+10 q^4+9 q^2-6-5 q^{-2} -4 q^{-4} + q^{-6} +6 q^{-8} +5 q^{-12} +3 q^{-14} -2 q^{-16} -4 q^{-18} -7 q^{-20} +8 q^{-22} +15 q^{-24} +4 q^{-26} -9 q^{-28} -18 q^{-30} + q^{-32} +20 q^{-34} +11 q^{-36} -8 q^{-38} -22 q^{-40} -8 q^{-42} +16 q^{-44} +13 q^{-46} -3 q^{-48} -17 q^{-50} -8 q^{-52} +13 q^{-54} +10 q^{-56} -3 q^{-58} -12 q^{-60} -5 q^{-62} +11 q^{-64} +9 q^{-66} -2 q^{-68} -12 q^{-70} -9 q^{-72} +5 q^{-74} +12 q^{-76} +8 q^{-78} - q^{-80} -14 q^{-82} -15 q^{-84} +4 q^{-86} +19 q^{-88} +20 q^{-90} -8 q^{-92} -28 q^{-94} -13 q^{-96} +14 q^{-98} +33 q^{-100} +5 q^{-102} -27 q^{-104} -22 q^{-106} +3 q^{-108} +30 q^{-110} +10 q^{-112} -17 q^{-114} -18 q^{-116} -3 q^{-118} +22 q^{-120} +9 q^{-122} -8 q^{-124} -11 q^{-126} -6 q^{-128} +12 q^{-130} +5 q^{-132} -2 q^{-134} -5 q^{-136} -5 q^{-138} +6 q^{-140} + q^{-142} - q^{-146} -2 q^{-148} +3 q^{-150} - q^{-156} - q^{-158} + q^{-160} }[/math]
5	[math]\displaystyle{ -q^{75}+q^{73}+2 q^{71}-q^{67}-3 q^{65}-3 q^{63}+6 q^{59}+6 q^{57}+q^{55}-5 q^{53}-10 q^{51}-8 q^{49}+3 q^{47}+16 q^{45}+15 q^{43}+3 q^{41}-11 q^{39}-22 q^{37}-15 q^{35}+5 q^{33}+22 q^{31}+21 q^{29}+7 q^{27}-13 q^{25}-28 q^{23}-18 q^{21}+3 q^{19}+20 q^{17}+22 q^{15}+8 q^{13}-11 q^{11}-18 q^9-12 q^7+q^5+11 q^3+8 q- q^{-1} - q^{-3} +2 q^{-5} +6 q^{-7} +3 q^{-9} -8 q^{-11} -20 q^{-13} -11 q^{-15} +15 q^{-17} +35 q^{-19} +27 q^{-21} -3 q^{-23} -39 q^{-25} -45 q^{-27} -7 q^{-29} +42 q^{-31} +59 q^{-33} +27 q^{-35} -31 q^{-37} -64 q^{-39} -44 q^{-41} +15 q^{-43} +62 q^{-45} +54 q^{-47} - q^{-49} -53 q^{-51} -60 q^{-53} -19 q^{-55} +40 q^{-57} +59 q^{-59} +27 q^{-61} -28 q^{-63} -55 q^{-65} -35 q^{-67} +14 q^{-69} +47 q^{-71} +37 q^{-73} -6 q^{-75} -38 q^{-77} -32 q^{-79} +2 q^{-81} +29 q^{-83} +30 q^{-85} -4 q^{-87} -29 q^{-89} -20 q^{-91} +5 q^{-93} +25 q^{-95} +21 q^{-97} -8 q^{-99} -29 q^{-101} -20 q^{-103} +6 q^{-105} +27 q^{-107} +27 q^{-109} +8 q^{-111} -19 q^{-113} -34 q^{-115} -25 q^{-117} +3 q^{-119} +35 q^{-121} +44 q^{-123} +22 q^{-125} -25 q^{-127} -59 q^{-129} -44 q^{-131} +10 q^{-133} +59 q^{-135} +59 q^{-137} +9 q^{-139} -53 q^{-141} -67 q^{-143} -21 q^{-145} +41 q^{-147} +61 q^{-149} +24 q^{-151} -30 q^{-153} -50 q^{-155} -21 q^{-157} +26 q^{-159} +40 q^{-161} +13 q^{-163} -25 q^{-165} -30 q^{-167} -4 q^{-169} +22 q^{-171} +25 q^{-173} + q^{-175} -23 q^{-177} -22 q^{-179} +2 q^{-181} +19 q^{-183} +17 q^{-185} +2 q^{-187} -13 q^{-189} -15 q^{-191} -3 q^{-193} +8 q^{-195} +9 q^{-197} +3 q^{-199} -3 q^{-201} -5 q^{-203} -3 q^{-205} + q^{-207} +3 q^{-209} + q^{-211} - q^{-217} + q^{-219} - q^{-223} - q^{-225} + q^{-231} + q^{-233} - q^{-235} }[/math]
6	[math]\displaystyle{ q^{108}-q^{106}-2 q^{104}+q^{100}+3 q^{98}+q^{96}+3 q^{94}-2 q^{92}-8 q^{90}-5 q^{88}-q^{86}+6 q^{84}+7 q^{82}+14 q^{80}+3 q^{78}-12 q^{76}-18 q^{74}-17 q^{72}-3 q^{70}+6 q^{68}+33 q^{66}+30 q^{64}+8 q^{62}-15 q^{60}-35 q^{58}-34 q^{56}-27 q^{54}+19 q^{52}+43 q^{50}+45 q^{48}+25 q^{46}-8 q^{44}-36 q^{42}-62 q^{40}-30 q^{38}+3 q^{36}+37 q^{34}+49 q^{32}+36 q^{30}+8 q^{28}-38 q^{26}-40 q^{24}-33 q^{22}-3 q^{20}+22 q^{18}+31 q^{16}+22 q^{14}-10 q^{12}-14 q^{10}-17 q^8+6 q^6+20 q^4+17 q^2-2-40 q^{-2} -43 q^{-4} -26 q^{-6} +31 q^{-8} +74 q^{-10} +75 q^{-12} +32 q^{-14} -53 q^{-16} -107 q^{-18} -109 q^{-20} -25 q^{-22} +77 q^{-24} +142 q^{-26} +136 q^{-28} +29 q^{-30} -94 q^{-32} -173 q^{-34} -136 q^{-36} -23 q^{-38} +105 q^{-40} +189 q^{-42} +143 q^{-44} +18 q^{-46} -122 q^{-48} -178 q^{-50} -139 q^{-52} -22 q^{-54} +122 q^{-56} +177 q^{-58} +131 q^{-60} +5 q^{-62} -108 q^{-64} -170 q^{-66} -137 q^{-68} -4 q^{-70} +114 q^{-72} +165 q^{-74} +113 q^{-76} +6 q^{-78} -119 q^{-80} -176 q^{-82} -100 q^{-84} +26 q^{-86} +133 q^{-88} +152 q^{-90} +85 q^{-92} -52 q^{-94} -157 q^{-96} -132 q^{-98} -31 q^{-100} +85 q^{-102} +138 q^{-104} +105 q^{-106} -10 q^{-108} -116 q^{-110} -113 q^{-112} -41 q^{-114} +49 q^{-116} +94 q^{-118} +75 q^{-120} -4 q^{-122} -76 q^{-124} -67 q^{-126} -12 q^{-128} +46 q^{-130} +61 q^{-132} +30 q^{-134} -30 q^{-136} -67 q^{-138} -45 q^{-140} +9 q^{-142} +60 q^{-144} +70 q^{-146} +38 q^{-148} -22 q^{-150} -68 q^{-152} -83 q^{-154} -48 q^{-156} +16 q^{-158} +77 q^{-160} +103 q^{-162} +75 q^{-164} +7 q^{-166} -95 q^{-168} -143 q^{-170} -115 q^{-172} -11 q^{-174} +106 q^{-176} +177 q^{-178} +153 q^{-180} +8 q^{-182} -130 q^{-184} -206 q^{-186} -148 q^{-188} -10 q^{-190} +145 q^{-192} +220 q^{-194} +132 q^{-196} -9 q^{-198} -152 q^{-200} -182 q^{-202} -116 q^{-204} +25 q^{-206} +142 q^{-208} +137 q^{-210} +79 q^{-212} -31 q^{-214} -89 q^{-216} -107 q^{-218} -53 q^{-220} +22 q^{-222} +46 q^{-224} +70 q^{-226} +41 q^{-228} +17 q^{-230} -32 q^{-232} -49 q^{-234} -40 q^{-236} -34 q^{-238} +17 q^{-240} +40 q^{-242} +57 q^{-244} +22 q^{-246} -11 q^{-248} -35 q^{-250} -53 q^{-252} -18 q^{-254} +8 q^{-256} +41 q^{-258} +30 q^{-260} +14 q^{-262} -6 q^{-264} -31 q^{-266} -18 q^{-268} -12 q^{-270} +11 q^{-272} +13 q^{-274} +13 q^{-276} +7 q^{-278} -7 q^{-280} -4 q^{-282} -11 q^{-284} - q^{-286} +5 q^{-290} +5 q^{-292} +3 q^{-296} -5 q^{-298} - q^{-300} -2 q^{-302} +3 q^{-310} - q^{-312} + q^{-314} - q^{-320} - q^{-322} + q^{-324} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^2+ q^{-4} +2 q^{-6} + q^{-8} +2 q^{-10} - q^{-12} + q^{-14} - q^{-22} - q^{-26} }[/math]
1,1	[math]\displaystyle{ q^{12}-2 q^{10}+4 q^8-8 q^6+13 q^4-20 q^2+24-32 q^{-2} +30 q^{-4} -30 q^{-6} +24 q^{-8} -10 q^{-10} +6 q^{-12} +18 q^{-14} -18 q^{-16} +36 q^{-18} -36 q^{-20} +44 q^{-22} -44 q^{-24} +38 q^{-26} -35 q^{-28} +22 q^{-30} -20 q^{-32} +6 q^{-34} -4 q^{-36} -2 q^{-38} +2 q^{-40} + q^{-44} +2 q^{-48} -4 q^{-50} +5 q^{-52} -10 q^{-54} +12 q^{-56} -12 q^{-58} +13 q^{-60} -12 q^{-62} +12 q^{-64} -10 q^{-66} +7 q^{-68} -6 q^{-70} +4 q^{-72} -2 q^{-74} + q^{-76} }[/math]
2,0	[math]\displaystyle{ q^{10}-q^6-q^4-1-3 q^{-2} +2 q^{-6} + q^{-8} +4 q^{-12} +4 q^{-14} +3 q^{-16} +2 q^{-18} +3 q^{-20} - q^{-24} - q^{-30} + q^{-34} -3 q^{-36} -3 q^{-38} - q^{-40} - q^{-44} - q^{-46} + q^{-48} + q^{-50} - q^{-54} + q^{-66} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^6-q^4-3 q^{-2} +2 q^{-10} +3 q^{-12} + q^{-14} +4 q^{-16} +2 q^{-18} + q^{-20} +4 q^{-22} +2 q^{-24} - q^{-26} - q^{-30} -3 q^{-32} -3 q^{-34} - q^{-36} -2 q^{-40} +2 q^{-44} -2 q^{-46} - q^{-48} +3 q^{-50} - q^{-52} -2 q^{-54} +2 q^{-56} - q^{-60} + q^{-62} }[/math]
1,0,0	[math]\displaystyle{ -q- q^{-3} + q^{-5} +3 q^{-9} + q^{-11} +2 q^{-13} + q^{-15} + q^{-17} + q^{-19} + q^{-23} -2 q^{-25} -2 q^{-29} - q^{-33} }[/math]
1,0,1	[math]\displaystyle{ q^{12}-2 q^{10}+3 q^8-3 q^6+q^4+3 q^2-6+9 q^{-2} -12 q^{-4} +3 q^{-6} -6 q^{-8} -7 q^{-10} +3 q^{-12} -6 q^{-14} +10 q^{-16} - q^{-18} +21 q^{-20} - q^{-22} +26 q^{-24} +2 q^{-26} +11 q^{-28} +14 q^{-30} -16 q^{-32} +17 q^{-34} -23 q^{-36} +7 q^{-38} -20 q^{-40} -10 q^{-44} -10 q^{-46} +7 q^{-48} -19 q^{-50} +17 q^{-52} -12 q^{-54} +9 q^{-56} +3 q^{-58} -2 q^{-60} +6 q^{-62} -2 q^{-64} +2 q^{-66} -5 q^{-68} +6 q^{-70} -8 q^{-72} +4 q^{-74} +2 q^{-76} -7 q^{-78} +11 q^{-80} -7 q^{-82} +2 q^{-84} +4 q^{-86} -8 q^{-88} +6 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- q^{-4} -3 q^{-6} -2 q^{-8} - q^{-10} - q^{-12} + q^{-16} +4 q^{-18} +3 q^{-20} +4 q^{-22} +4 q^{-24} +7 q^{-26} +6 q^{-28} +6 q^{-30} +3 q^{-32} +4 q^{-34} + q^{-36} -4 q^{-38} -4 q^{-40} -4 q^{-42} -6 q^{-44} -5 q^{-46} - q^{-48} -3 q^{-50} -2 q^{-52} + q^{-54} + q^{-56} -2 q^{-58} - q^{-60} +2 q^{-62} -2 q^{-66} + q^{-68} +2 q^{-70} + q^{-76} }[/math]
1,0,0,0	[math]\displaystyle{ -1- q^{-4} + q^{-10} +2 q^{-12} + q^{-14} +3 q^{-16} + q^{-18} +3 q^{-20} + q^{-22} +2 q^{-24} - q^{-30} -2 q^{-32} - q^{-34} -2 q^{-36} - q^{-40} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^6+q^4-2 q^2+2-3 q^{-2} +4 q^{-4} -4 q^{-6} +4 q^{-8} -2 q^{-10} +3 q^{-12} + q^{-14} +4 q^{-18} -3 q^{-20} +6 q^{-22} -6 q^{-24} +7 q^{-26} -6 q^{-28} +5 q^{-30} -5 q^{-32} +3 q^{-34} -3 q^{-36} -2 q^{-42} +2 q^{-44} -2 q^{-46} +3 q^{-48} -3 q^{-50} +3 q^{-52} -2 q^{-54} +2 q^{-56} -2 q^{-58} + q^{-60} - q^{-62} }[/math]
1,0	[math]\displaystyle{ q^{12}-q^8-q^6+q^4+q^2-2-3 q^{-2} +3 q^{-6} -3 q^{-10} - q^{-12} +4 q^{-14} +4 q^{-16} + q^{-18} -3 q^{-20} + q^{-22} +4 q^{-24} +4 q^{-26} - q^{-28} -2 q^{-30} +3 q^{-34} + q^{-36} - q^{-38} +2 q^{-42} -2 q^{-46} - q^{-48} + q^{-50} + q^{-52} -2 q^{-54} -3 q^{-56} - q^{-58} +2 q^{-60} -2 q^{-64} -2 q^{-66} + q^{-68} +2 q^{-70} -2 q^{-74} -2 q^{-76} + q^{-78} +3 q^{-80} + q^{-82} -2 q^{-84} -2 q^{-86} +2 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+q^2-2+2 q^{-2} -4 q^{-4} +2 q^{-6} -4 q^{-8} +3 q^{-10} -3 q^{-12} +2 q^{-14} - q^{-16} +3 q^{-18} +2 q^{-20} + q^{-22} +5 q^{-24} +2 q^{-26} +8 q^{-28} +8 q^{-32} -3 q^{-34} +6 q^{-36} -6 q^{-38} +2 q^{-40} -7 q^{-42} -6 q^{-46} -3 q^{-50} - q^{-56} + q^{-58} - q^{-60} +2 q^{-62} -2 q^{-64} + q^{-66} -2 q^{-68} +3 q^{-70} -2 q^{-72} + q^{-74} -2 q^{-76} +2 q^{-78} - q^{-80} + q^{-82} - q^{-84} + q^{-86} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{12}-q^{10}+2 q^8-3 q^6+q^4-2 q^2-1+5 q^{-2} -8 q^{-4} +8 q^{-6} -7 q^{-8} +2 q^{-10} +3 q^{-12} -9 q^{-14} +10 q^{-16} -9 q^{-18} +5 q^{-20} +2 q^{-22} -4 q^{-24} +7 q^{-26} -3 q^{-28} +2 q^{-30} +4 q^{-32} -2 q^{-34} +2 q^{-36} +2 q^{-38} -2 q^{-40} +8 q^{-42} -4 q^{-44} +6 q^{-46} -2 q^{-48} - q^{-50} +7 q^{-52} -10 q^{-54} +9 q^{-56} -6 q^{-58} +2 q^{-60} +2 q^{-62} -6 q^{-64} +6 q^{-66} -5 q^{-68} +2 q^{-70} -3 q^{-74} + q^{-76} + q^{-78} -2 q^{-80} + q^{-82} - q^{-86} +2 q^{-88} -2 q^{-90} + q^{-92} + q^{-96} - q^{-98} - q^{-100} - q^{-104} +2 q^{-106} -4 q^{-108} +4 q^{-110} -3 q^{-112} - q^{-114} +2 q^{-116} -5 q^{-118} +5 q^{-120} -5 q^{-122} +3 q^{-124} - q^{-126} -2 q^{-128} +4 q^{-130} -4 q^{-132} +4 q^{-134} -2 q^{-136} + q^{-138} -2 q^{-142} +2 q^{-144} - q^{-146} + q^{-148} }[/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 5"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-3 t^3+5 t^2-5 t+5-5 t^{-1} +5 t^{-2} -3 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+5 z^6+7 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]=

{ 33, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^9+2 q^8-3 q^7+4 q^6-5 q^5+5 q^4-4 q^3+4 q^2-2 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} +17 z^4 a^{-4} -5 z^4 a^{-6} -6 z^2 a^{-2} +17 z^2 a^{-4} -7 z^2 a^{-6} - a^{-2} +5 a^{-4} -3 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} +4 z^8 a^{-4} +2 z^8 a^{-6} +z^7 a^{-1} -3 z^7 a^{-3} -2 z^7 a^{-5} +2 z^7 a^{-7} -11 z^6 a^{-2} -20 z^6 a^{-4} -7 z^6 a^{-6} +2 z^6 a^{-8} -5 z^5 a^{-1} -2 z^5 a^{-3} -3 z^5 a^{-5} -4 z^5 a^{-7} +2 z^5 a^{-9} +18 z^4 a^{-2} +32 z^4 a^{-4} +10 z^4 a^{-6} -2 z^4 a^{-8} +2 z^4 a^{-10} +6 z^3 a^{-1} +7 z^3 a^{-3} +6 z^3 a^{-5} +3 z^3 a^{-7} -z^3 a^{-9} +z^3 a^{-11} -10 z^2 a^{-2} -22 z^2 a^{-4} -9 z^2 a^{-6} +z^2 a^{-8} -2 z^2 a^{-10} -z a^{-1} -2 z a^{-3} -3 z a^{-5} -z a^{-7} -z a^{-11} + a^{-2} +5 a^{-4} +3 a^{-6} }[/math]

Vassiliev invariants

V2 and V3:

(4, 7)

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{ 56 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{728}{3} }[/math] [math]\displaystyle{ \frac{16}{3} }[/math] [math]\displaystyle{ 896 }[/math] [math]\displaystyle{ \frac{3824}{3} }[/math] [math]\displaystyle{ \frac{608}{3} }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 1568 }[/math] [math]\displaystyle{ \frac{11648}{3} }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ \frac{104222}{15} }[/math] [math]\displaystyle{ \frac{18512}{15} }[/math] [math]\displaystyle{ \frac{38288}{45} }[/math] [math]\displaystyle{ \frac{130}{9} }[/math] [math]\displaystyle{ -\frac{658}{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]4 is the signature of 10 5. 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 19                     1 -1 17                   1   1 15                 2 1   -1 13               2 1     1 11             3 2       -1 9           2 2         0 7         2 3           1 5       2 2             0 3     1 3               2 1   1 1                 0 -1   1                   1 -3 1                     -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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, 5]]
Out[2]=	10
In[3]:=	PD[Knot[10, 5]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[13, 1, 14, 20], X[5, 15, 6, 14], X[7, 17, 8, 16], X[9, 19, 10, 18], X[15, 7, 16, 6], X[17, 9, 18, 8], X[19, 11, 20, 10], X[11, 2, 12, 3]]
In[4]:=	GaussCode[Knot[10, 5]]
Out[4]=	GaussCode[-1, 10, -2, 1, -4, 7, -5, 8, -6, 9, -10, 2, -3, 4, -7, 5, -8, 6, -9, 3]
In[5]:=	BR[Knot[10, 5]]
Out[5]=	BR[3, {1, 1, 1, 1, 1, 1, -2, 1, -2, -2}]
In[6]:=	alex = Alexander[Knot[10, 5]][t]
Out[6]=	-4 3 5 5 2 3 4 5 + t - -- + -- - - - 5 t + 5 t - 3 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 5]][z]
Out[7]=	2 4 6 8 1 + 4 z + 7 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 5]}
In[9]:=	{KnotDet[Knot[10, 5]], KnotSignature[Knot[10, 5]]}
Out[9]=	{33, 4}
In[10]:=	J=Jones[Knot[10, 5]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 2 - - - 2 q + 4 q - 4 q + 5 q - 5 q + 4 q - 3 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 5]}
In[12]:=	A2Invariant[Knot[10, 5]][q]
Out[12]=	-2 4 6 8 10 12 14 22 26 -q + q + 2 q + q + 2 q - q + q - q - q
In[13]:=	Kauffman[Knot[10, 5]][a, z]
Out[13]=	2 2 2 2 3 5 -2 z z 3 z 2 z z 2 z z 9 z 22 z -- + -- + a - --- - -- - --- - --- - - - ---- + -- - ---- - ----- - 6 4 11 7 5 3 a 10 8 6 4 a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 10 z z z 3 z 6 z 7 z 6 z 2 z 2 z 10 z ----- + --- - -- + ---- + ---- + ---- + ---- + ---- - ---- + ----- + 2 11 9 7 5 3 a 10 8 6 a a a a a a a a a 4 4 5 5 5 5 5 6 6 32 z 18 z 2 z 4 z 3 z 2 z 5 z 2 z 7 z ----- + ----- + ---- - ---- - ---- - ---- - ---- + ---- - ---- - 4 2 9 7 5 3 a 8 6 a a a a a a a a 6 6 7 7 7 7 8 8 8 9 9 20 z 11 z 2 z 2 z 3 z z 2 z 4 z 2 z z z ----- - ----- + ---- - ---- - ---- + -- + ---- + ---- + ---- + -- + -- 4 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a a
In[14]:=	{Vassiliev[2][Knot[10, 5]], Vassiliev[3][Knot[10, 5]]}
Out[14]=	{0, 7}
In[15]:=	Kh[Knot[10, 5]][q, t]
Out[15]=	3 3 5 1 1 q q q 5 7 7 2 3 q + 2 q + ----- + ---- + -- + - + -- + 2 q t + 2 q t + 3 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 2 q t + 2 q t + 3 q t + 2 q t + 2 q t + q t + 15 5 15 6 17 6 19 7 2 q t + q t + q t + q t