8 5

8_4

8_6

Visit 8 5's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 5's page at Knotilus! Visit 8 5's page at the original Knot Atlas! 8 5 is also known as the pretzel knot P(3,3,2).

Symmetric alternative representation

Pretzel P(3,3,2) form Photo 01-09-2017 besalu.jpg.

Sum of 8.5 ; church of Besalu, Catalogna

Knot presentations

Planar diagram presentation	X6271 X8493 X2837 X14,10,15,9 X12,5,13,6 X4,13,5,14 X16,12,1,11 X10,16,11,15
Gauss code	1, -3, 2, -6, 5, -1, 3, -2, 4, -8, 7, -5, 6, -4, 8, -7
Dowker-Thistlethwaite code	6 8 12 2 14 16 4 10
Conway Notation	[3,3,2]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [1][-11] Hyperbolic Volume 6.99719 A-Polynomial See Data:8 5/A-polynomial

[edit Notes for 8 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{ 3 }[/math] Rasmussen s-Invariant -4

[edit Notes for 8 5's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+3 t^2-4 t+5-4 t^{-1} +3 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6-3 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 21, 4 }
Jones polynomial	[math]\displaystyle{ q^8-2 q^7+3 q^6-4 q^5+3 q^4-3 q^3+3 q^2-q+1 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6 a^{-4} +z^4 a^{-2} -5 z^4 a^{-4} +z^4 a^{-6} +4 z^2 a^{-2} -8 z^2 a^{-4} +3 z^2 a^{-6} +4 a^{-2} -5 a^{-4} +2 a^{-6} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^7 a^{-3} +z^7 a^{-5} +z^6 a^{-2} +4 z^6 a^{-4} +3 z^6 a^{-6} -3 z^5 a^{-3} +z^5 a^{-5} +4 z^5 a^{-7} -5 z^4 a^{-2} -15 z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} -10 z^3 a^{-5} -8 z^3 a^{-7} +2 z^3 a^{-9} +8 z^2 a^{-2} +15 z^2 a^{-4} +4 z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} +3 z a^{-3} +7 z a^{-5} +4 z a^{-7} -4 a^{-2} -5 a^{-4} -2 a^{-6} }[/math]
The A2 invariant	[math]\displaystyle{ 1+ q^{-2} +2 q^{-4} +2 q^{-6} -3 q^{-12} - q^{-14} - q^{-16} + q^{-20} + q^{-24} }[/math]
The G2 invariant	[math]\displaystyle{ q^{-2} +3 q^{-6} -2 q^{-8} +2 q^{-10} + q^{-12} -2 q^{-14} +7 q^{-16} -5 q^{-18} +5 q^{-20} + q^{-22} -2 q^{-24} +8 q^{-26} -5 q^{-28} +5 q^{-30} +2 q^{-32} -2 q^{-34} +3 q^{-36} -3 q^{-38} +2 q^{-42} -4 q^{-44} +2 q^{-46} -4 q^{-48} -2 q^{-50} +2 q^{-52} -9 q^{-54} +4 q^{-56} -7 q^{-58} + q^{-62} -7 q^{-64} +7 q^{-66} -7 q^{-68} +4 q^{-70} +2 q^{-72} -5 q^{-74} +5 q^{-76} -2 q^{-78} + q^{-80} +4 q^{-82} -2 q^{-84} +3 q^{-86} + q^{-88} - q^{-90} +4 q^{-92} -4 q^{-94} +3 q^{-96} -2 q^{-100} +3 q^{-102} -3 q^{-104} +3 q^{-106} - q^{-108} + q^{-110} -3 q^{-114} +2 q^{-116} -2 q^{-118} +2 q^{-120} - q^{-122} - q^{-128} + q^{-130} - q^{-132} + q^{-134} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_5.

A1 Invariants.

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

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ 1+ q^{-2} +2 q^{-4} +2 q^{-6} -3 q^{-12} - q^{-14} - q^{-16} + q^{-20} + q^{-24} }[/math]
1,1	[math]\displaystyle{ q^4+6-4 q^{-2} +12 q^{-4} -10 q^{-6} +16 q^{-8} -12 q^{-10} +10 q^{-12} -8 q^{-14} -2 q^{-16} +2 q^{-18} -14 q^{-20} +12 q^{-22} -22 q^{-24} +22 q^{-26} -20 q^{-28} +22 q^{-30} -12 q^{-32} +16 q^{-34} - q^{-36} +4 q^{-40} -10 q^{-42} +5 q^{-44} -10 q^{-46} +6 q^{-48} -6 q^{-50} +5 q^{-52} -2 q^{-54} +4 q^{-56} -4 q^{-58} +3 q^{-60} -2 q^{-62} +2 q^{-64} -2 q^{-66} + q^{-68} }[/math]
2,0	[math]\displaystyle{ q^4+q^2+1+ q^{-2} +3 q^{-4} +3 q^{-6} +2 q^{-8} + q^{-12} -2 q^{-14} -4 q^{-16} -4 q^{-18} -3 q^{-20} -3 q^{-22} - q^{-24} +2 q^{-26} +3 q^{-28} +3 q^{-30} +3 q^{-32} +3 q^{-34} - q^{-36} + q^{-40} -2 q^{-44} - q^{-46} - q^{-48} - q^{-50} - q^{-52} + q^{-56} + q^{-60} }[/math]
3,0	[math]\displaystyle{ q^{12}+q^{10}+q^8+2 q^2+4+4 q^{-2} + q^{-4} - q^{-6} +3 q^{-10} +3 q^{-12} -2 q^{-14} -8 q^{-16} -6 q^{-18} -3 q^{-20} + q^{-22} -4 q^{-24} -5 q^{-26} -3 q^{-28} +5 q^{-30} +7 q^{-32} +6 q^{-34} +4 q^{-36} +6 q^{-38} +8 q^{-40} +5 q^{-42} + q^{-44} -3 q^{-46} -2 q^{-48} -6 q^{-50} -7 q^{-52} -8 q^{-54} - q^{-56} -2 q^{-58} -2 q^{-60} -3 q^{-62} +2 q^{-64} +6 q^{-66} +2 q^{-68} -2 q^{-70} -3 q^{-72} +4 q^{-74} +7 q^{-76} +3 q^{-78} -3 q^{-80} -3 q^{-82} + q^{-84} +4 q^{-86} + q^{-88} -2 q^{-90} -3 q^{-92} - q^{-96} - q^{-98} - q^{-100} + q^{-104} + q^{-108} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ 1+3 q^{-4} +3 q^{-6} +3 q^{-8} +5 q^{-10} +3 q^{-12} - q^{-14} - q^{-16} -6 q^{-18} -8 q^{-20} -4 q^{-22} -3 q^{-24} - q^{-26} +3 q^{-28} +5 q^{-30} +4 q^{-32} +2 q^{-34} + q^{-36} + q^{-38} -3 q^{-40} - q^{-42} + q^{-44} -2 q^{-46} +2 q^{-50} - q^{-52} - q^{-54} + q^{-56} }[/math]
1,0,0	[math]\displaystyle{ q^{-1} + q^{-3} +3 q^{-5} +2 q^{-7} +3 q^{-9} - q^{-13} -3 q^{-15} -3 q^{-17} -2 q^{-19} - q^{-21} + q^{-23} +2 q^{-27} + q^{-31} }[/math]
1,0,1	[math]\displaystyle{ q^2+6 q^{-2} +2 q^{-4} +11 q^{-6} +8 q^{-8} +6 q^{-10} +12 q^{-12} -7 q^{-14} +7 q^{-16} -14 q^{-18} -8 q^{-20} -11 q^{-22} -19 q^{-24} -3 q^{-26} -13 q^{-28} +7 q^{-30} -3 q^{-32} +13 q^{-34} +7 q^{-36} +11 q^{-38} +9 q^{-40} + q^{-42} +10 q^{-44} -8 q^{-46} +7 q^{-48} -8 q^{-50} -2 q^{-52} - q^{-54} -9 q^{-56} +4 q^{-58} -6 q^{-60} + q^{-62} +2 q^{-64} -2 q^{-66} +2 q^{-68} + q^{-72} +2 q^{-74} - q^{-76} - q^{-78} +2 q^{-80} -2 q^{-82} + q^{-84} + q^{-86} -2 q^{-88} + q^{-90} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ 1+3 q^{-4} - q^{-6} +3 q^{-8} - q^{-10} +3 q^{-12} - q^{-14} + q^{-16} -2 q^{-20} +2 q^{-22} -5 q^{-24} +3 q^{-26} -5 q^{-28} +3 q^{-30} -4 q^{-32} +2 q^{-34} - q^{-36} + q^{-38} + q^{-40} - q^{-42} +3 q^{-44} -2 q^{-46} +2 q^{-48} -2 q^{-50} + q^{-52} - q^{-54} + q^{-56} }[/math]
1,0	[math]\displaystyle{ q^2+3 q^{-6} +2 q^{-8} +3 q^{-14} +3 q^{-16} +2 q^{-18} -2 q^{-20} - q^{-22} + q^{-24} + q^{-26} -3 q^{-28} -5 q^{-30} -3 q^{-32} - q^{-34} - q^{-36} -3 q^{-38} - q^{-40} + q^{-42} +2 q^{-44} + q^{-48} +2 q^{-50} +4 q^{-52} + q^{-54} - q^{-56} +2 q^{-60} + q^{-62} -2 q^{-64} -2 q^{-66} +2 q^{-70} - q^{-72} -2 q^{-74} - q^{-76} + q^{-78} +2 q^{-80} - q^{-84} - q^{-86} + q^{-90} }[/math]

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{-2} +3 q^{-6} -2 q^{-8} +2 q^{-10} + q^{-12} -2 q^{-14} +7 q^{-16} -5 q^{-18} +5 q^{-20} + q^{-22} -2 q^{-24} +8 q^{-26} -5 q^{-28} +5 q^{-30} +2 q^{-32} -2 q^{-34} +3 q^{-36} -3 q^{-38} +2 q^{-42} -4 q^{-44} +2 q^{-46} -4 q^{-48} -2 q^{-50} +2 q^{-52} -9 q^{-54} +4 q^{-56} -7 q^{-58} + q^{-62} -7 q^{-64} +7 q^{-66} -7 q^{-68} +4 q^{-70} +2 q^{-72} -5 q^{-74} +5 q^{-76} -2 q^{-78} + q^{-80} +4 q^{-82} -2 q^{-84} +3 q^{-86} + q^{-88} - q^{-90} +4 q^{-92} -4 q^{-94} +3 q^{-96} -2 q^{-100} +3 q^{-102} -3 q^{-104} +3 q^{-106} - q^{-108} + q^{-110} -3 q^{-114} +2 q^{-116} -2 q^{-118} +2 q^{-120} - q^{-122} - q^{-128} + q^{-130} - q^{-132} + q^{-134} }[/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["8 5"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6-3 z^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]=

{ 21, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^7 a^{-3} +z^7 a^{-5} +z^6 a^{-2} +4 z^6 a^{-4} +3 z^6 a^{-6} -3 z^5 a^{-3} +z^5 a^{-5} +4 z^5 a^{-7} -5 z^4 a^{-2} -15 z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} -10 z^3 a^{-5} -8 z^3 a^{-7} +2 z^3 a^{-9} +8 z^2 a^{-2} +15 z^2 a^{-4} +4 z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} +3 z a^{-3} +7 z a^{-5} +4 z a^{-7} -4 a^{-2} -5 a^{-4} -2 a^{-6} }[/math]

Vassiliev invariants

V2 and V3:

(-1, -3)

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{ -4 }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{62}{3} }[/math] [math]\displaystyle{ \frac{86}{3} }[/math] [math]\displaystyle{ 96 }[/math] [math]\displaystyle{ 208 }[/math] [math]\displaystyle{ 96 }[/math] [math]\displaystyle{ 104 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ \frac{248}{3} }[/math] [math]\displaystyle{ -\frac{344}{3} }[/math] [math]\displaystyle{ \frac{31409}{30} }[/math] [math]\displaystyle{ \frac{834}{5} }[/math] [math]\displaystyle{ \frac{16618}{45} }[/math] [math]\displaystyle{ \frac{2095}{18} }[/math] [math]\displaystyle{ -\frac{1231}{30} }[/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 8 5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 χ 17                 1 1 15               1   -1 13             2 1   1 11           2 1     -1 9         1 2       -1 7       2 2         0 5     1 1           0 3   1 3             2 1                   0 -1 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=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/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[8, 5]]
Out[2]=	8
In[3]:=	PD[Knot[8, 5]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[2, 8, 3, 7], X[14, 10, 15, 9], X[12, 5, 13, 6], X[4, 13, 5, 14], X[16, 12, 1, 11], X[10, 16, 11, 15]]
In[4]:=	GaussCode[Knot[8, 5]]
Out[4]=	GaussCode[1, -3, 2, -6, 5, -1, 3, -2, 4, -8, 7, -5, 6, -4, 8, -7]
In[5]:=	BR[Knot[8, 5]]
Out[5]=	BR[3, {1, 1, 1, -2, 1, 1, 1, -2}]
In[6]:=	alex = Alexander[Knot[8, 5]][t]
Out[6]=	-3 3 4 2 3 5 - t + -- - - - 4 t + 3 t - t 2 t t
In[7]:=	Conway[Knot[8, 5]][z]
Out[7]=	2 4 6 1 - z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 5], Knot[10, 141]}
In[9]:=	{KnotDet[Knot[8, 5]], KnotSignature[Knot[8, 5]]}
Out[9]=	{21, 4}
In[10]:=	J=Jones[Knot[8, 5]][q]
Out[10]=	2 3 4 5 6 7 8 1 - q + 3 q - 3 q + 3 q - 4 q + 3 q - 2 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 5]}
In[12]:=	A2Invariant[Knot[8, 5]][q]
Out[12]=	2 4 6 12 14 16 20 24 1 + q + 2 q + 2 q - 3 q - q - q + q + q
In[13]:=	Kauffman[Knot[8, 5]][a, z]
Out[13]=	2 2 2 2 2 -2 5 4 4 z 7 z 3 z z 2 z 4 z 15 z 8 z -- - -- - -- + --- + --- + --- + --- - ---- + ---- + ----- + ---- + 6 4 2 7 5 3 10 8 6 4 2 a a a a a a a a a a a 3 3 3 4 4 4 4 5 5 5 2 z 8 z 10 z 3 z 7 z 15 z 5 z 4 z z 3 z ---- - ---- - ----- + ---- - ---- - ----- - ---- + ---- + -- - ---- + 9 7 5 8 6 4 2 7 5 3 a a a a a a a a a a 6 6 6 7 7 3 z 4 z z z z ---- + ---- + -- + -- + -- 6 4 2 5 3 a a a a a
In[14]:=	{Vassiliev[2][Knot[8, 5]], Vassiliev[3][Knot[8, 5]]}
Out[14]=	{0, -3}
In[15]:=	Kh[Knot[8, 5]][q, t]
Out[15]=	3 3 5 1 q 5 7 7 2 9 2 9 3 3 q + q + ---- + -- + q t + 2 q t + 2 q t + q t + 2 q t + 2 t q t 11 3 11 4 13 4 13 5 15 5 17 6 2 q t + q t + 2 q t + q t + q t + q t