10 72

10_71

10_73

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10 72 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,6,11,5 X8394 X2,9,3,10 X16,8,17,7 X18,12,19,11 X20,14,1,13 X12,20,13,19 X14,18,15,17 X6,16,7,15
Gauss code	1, -4, 3, -1, 2, -10, 5, -3, 4, -2, 6, -8, 7, -9, 10, -5, 9, -6, 8, -7
Dowker-Thistlethwaite code	4 8 10 16 2 18 20 6 14 12
Conway Notation	[211,3,2+]

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 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [1][-13] Hyperbolic Volume 12.9296 A-Polynomial See Data:10 72/A-polynomial

[edit Notes for 10 72'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 10 72's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -2 t^3+9 t^2-16 t+19-16 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	{ 73, 4 }
Jones polynomial	[math]\displaystyle{ q^{10}-4 q^9+7 q^8-10 q^7+12 q^6-12 q^5+11 q^4-8 q^3+5 q^2-2 q+1 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -3 z^4 a^{-4} -2 z^4 a^{-6} +z^4 a^{-8} +3 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} +z^2 a^{-8} +2 a^{-2} -2 a^{-4} +2 a^{-6} - a^{-8} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^9 a^{-5} +z^9 a^{-7} +2 z^8 a^{-4} +6 z^8 a^{-6} +4 z^8 a^{-8} +2 z^7 a^{-3} +4 z^7 a^{-5} +9 z^7 a^{-7} +7 z^7 a^{-9} +z^6 a^{-2} -2 z^6 a^{-4} -8 z^6 a^{-6} +2 z^6 a^{-8} +7 z^6 a^{-10} -6 z^5 a^{-3} -14 z^5 a^{-5} -19 z^5 a^{-7} -7 z^5 a^{-9} +4 z^5 a^{-11} -4 z^4 a^{-2} -6 z^4 a^{-4} -4 z^4 a^{-6} -11 z^4 a^{-8} -8 z^4 a^{-10} +z^4 a^{-12} +5 z^3 a^{-3} +11 z^3 a^{-5} +9 z^3 a^{-7} -3 z^3 a^{-11} +5 z^2 a^{-2} +8 z^2 a^{-4} +7 z^2 a^{-6} +6 z^2 a^{-8} +2 z^2 a^{-10} -z a^{-3} -3 z a^{-5} -z a^{-7} +z a^{-9} -2 a^{-2} -2 a^{-4} -2 a^{-6} - a^{-8} }[/math]
The A2 invariant	[math]\displaystyle{ 1+ q^{-4} +2 q^{-6} -2 q^{-8} +2 q^{-10} -2 q^{-12} +2 q^{-16} - q^{-18} +3 q^{-20} -2 q^{-22} -2 q^{-28} + q^{-30} }[/math]
The G2 invariant	[math]\displaystyle{ q^{-2} - q^{-4} +4 q^{-6} -5 q^{-8} +6 q^{-10} -5 q^{-12} +12 q^{-16} -22 q^{-18} +35 q^{-20} -37 q^{-22} +28 q^{-24} - q^{-26} -37 q^{-28} +79 q^{-30} -104 q^{-32} +101 q^{-34} -61 q^{-36} -13 q^{-38} +92 q^{-40} -150 q^{-42} +165 q^{-44} -120 q^{-46} +34 q^{-48} +60 q^{-50} -132 q^{-52} +142 q^{-54} -98 q^{-56} +16 q^{-58} +68 q^{-60} -110 q^{-62} +91 q^{-64} -19 q^{-66} -73 q^{-68} +150 q^{-70} -169 q^{-72} +122 q^{-74} -18 q^{-76} -106 q^{-78} +206 q^{-80} -240 q^{-82} +200 q^{-84} -92 q^{-86} -41 q^{-88} +152 q^{-90} -206 q^{-92} +185 q^{-94} -103 q^{-96} -5 q^{-98} +89 q^{-100} -120 q^{-102} +85 q^{-104} -9 q^{-106} -71 q^{-108} +117 q^{-110} -105 q^{-112} +40 q^{-114} +45 q^{-116} -124 q^{-118} +161 q^{-120} -142 q^{-122} +80 q^{-124} +3 q^{-126} -77 q^{-128} +118 q^{-130} -121 q^{-132} +93 q^{-134} -45 q^{-136} -2 q^{-138} +34 q^{-140} -53 q^{-142} +50 q^{-144} -34 q^{-146} +19 q^{-148} -2 q^{-150} -6 q^{-152} +9 q^{-154} -10 q^{-156} +6 q^{-158} -3 q^{-160} + q^{-162} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_72.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q- q^{-1} +3 q^{-3} -3 q^{-5} +3 q^{-7} - q^{-9} +2 q^{-13} -3 q^{-15} +3 q^{-17} -3 q^{-19} + q^{-21} }[/math]
2	[math]\displaystyle{ q^6-q^4-q^2+5-3 q^{-2} -7 q^{-4} +14 q^{-6} - q^{-8} -19 q^{-10} +20 q^{-12} +9 q^{-14} -28 q^{-16} +13 q^{-18} +17 q^{-20} -21 q^{-22} - q^{-24} +15 q^{-26} -3 q^{-28} -14 q^{-30} +4 q^{-32} +18 q^{-34} -19 q^{-36} -9 q^{-38} +29 q^{-40} -14 q^{-42} -16 q^{-44} +23 q^{-46} -4 q^{-48} -11 q^{-50} +9 q^{-52} -3 q^{-56} + q^{-58} }[/math]
3	[math]\displaystyle{ q^{15}-q^{13}-q^{11}+q^9+4 q^7-3 q^5-8 q^3+3 q+18 q^{-1} - q^{-3} -30 q^{-5} -9 q^{-7} +48 q^{-9} +28 q^{-11} -58 q^{-13} -59 q^{-15} +61 q^{-17} +98 q^{-19} -48 q^{-21} -129 q^{-23} +19 q^{-25} +153 q^{-27} +20 q^{-29} -159 q^{-31} -57 q^{-33} +145 q^{-35} +88 q^{-37} -119 q^{-39} -111 q^{-41} +85 q^{-43} +117 q^{-45} -45 q^{-47} -114 q^{-49} + q^{-51} +104 q^{-53} +44 q^{-55} -85 q^{-57} -88 q^{-59} +54 q^{-61} +127 q^{-63} -21 q^{-65} -152 q^{-67} -19 q^{-69} +163 q^{-71} +53 q^{-73} -147 q^{-75} -79 q^{-77} +120 q^{-79} +89 q^{-81} -85 q^{-83} -82 q^{-85} +51 q^{-87} +63 q^{-89} -25 q^{-91} -43 q^{-93} +10 q^{-95} +27 q^{-97} -7 q^{-99} -11 q^{-101} + q^{-103} +6 q^{-105} -3 q^{-109} + q^{-111} }[/math]
4	[math]\displaystyle{ q^{28}-q^{26}-q^{24}+q^{22}+4 q^{18}-5 q^{16}-6 q^{14}+5 q^{12}+5 q^{10}+18 q^8-15 q^6-33 q^4-3 q^2+23+78 q^{-2} -5 q^{-4} -98 q^{-6} -84 q^{-8} +3 q^{-10} +215 q^{-12} +128 q^{-14} -118 q^{-16} -279 q^{-18} -209 q^{-20} +297 q^{-22} +428 q^{-24} +119 q^{-26} -401 q^{-28} -644 q^{-30} +61 q^{-32} +639 q^{-34} +620 q^{-36} -156 q^{-38} -974 q^{-40} -476 q^{-42} +449 q^{-44} +1022 q^{-46} +387 q^{-48} -883 q^{-50} -917 q^{-52} -38 q^{-54} +1021 q^{-56} +828 q^{-58} -472 q^{-60} -999 q^{-62} -461 q^{-64} +719 q^{-66} +942 q^{-68} -44 q^{-70} -804 q^{-72} -656 q^{-74} +331 q^{-76} +821 q^{-78} +317 q^{-80} -491 q^{-82} -723 q^{-84} -102 q^{-86} +580 q^{-88} +670 q^{-90} -56 q^{-92} -698 q^{-94} -617 q^{-96} +175 q^{-98} +946 q^{-100} +498 q^{-102} -459 q^{-104} -1030 q^{-106} -387 q^{-108} +899 q^{-110} +937 q^{-112} +26 q^{-114} -1038 q^{-116} -828 q^{-118} +488 q^{-120} +934 q^{-122} +449 q^{-124} -625 q^{-126} -836 q^{-128} +46 q^{-130} +542 q^{-132} +507 q^{-134} -184 q^{-136} -502 q^{-138} -106 q^{-140} +160 q^{-142} +296 q^{-144} +4 q^{-146} -191 q^{-148} -59 q^{-150} +9 q^{-152} +107 q^{-154} +14 q^{-156} -54 q^{-158} -7 q^{-160} -8 q^{-162} +27 q^{-164} +3 q^{-166} -14 q^{-168} + q^{-170} -2 q^{-172} +6 q^{-174} -3 q^{-178} + q^{-180} }[/math]
5	[math]\displaystyle{ q^{45}-q^{43}-q^{41}+q^{39}+2 q^{33}-3 q^{31}-5 q^{29}+5 q^{27}+8 q^{25}+3 q^{23}-17 q^{19}-26 q^{17}+2 q^{15}+41 q^{13}+51 q^{11}+21 q^9-57 q^7-124 q^5-82 q^3+67 q+218 q^{-1} +222 q^{-3} -3 q^{-5} -328 q^{-7} -457 q^{-9} -197 q^{-11} +358 q^{-13} +790 q^{-15} +605 q^{-17} -209 q^{-19} -1082 q^{-21} -1234 q^{-23} -299 q^{-25} +1194 q^{-27} +1998 q^{-29} +1189 q^{-31} -865 q^{-33} -2625 q^{-35} -2445 q^{-37} -56 q^{-39} +2858 q^{-41} +3794 q^{-43} +1550 q^{-45} -2392 q^{-47} -4857 q^{-49} -3454 q^{-51} +1144 q^{-53} +5326 q^{-55} +5348 q^{-57} +728 q^{-59} -4937 q^{-61} -6824 q^{-63} -2941 q^{-65} +3758 q^{-67} +7597 q^{-69} +5017 q^{-71} -2030 q^{-73} -7521 q^{-75} -6611 q^{-77} +82 q^{-79} +6752 q^{-81} +7517 q^{-83} +1680 q^{-85} -5546 q^{-87} -7698 q^{-89} -3024 q^{-91} +4151 q^{-93} +7346 q^{-95} +3877 q^{-97} -2855 q^{-99} -6657 q^{-101} -4284 q^{-103} +1736 q^{-105} +5822 q^{-107} +4459 q^{-109} -785 q^{-111} -5006 q^{-113} -4552 q^{-115} -104 q^{-117} +4201 q^{-119} +4693 q^{-121} +1118 q^{-123} -3345 q^{-125} -4922 q^{-127} -2337 q^{-129} +2293 q^{-131} +5139 q^{-133} +3769 q^{-135} -889 q^{-137} -5149 q^{-139} -5329 q^{-141} -879 q^{-143} +4753 q^{-145} +6710 q^{-147} +2931 q^{-149} -3746 q^{-151} -7655 q^{-153} -5034 q^{-155} +2215 q^{-157} +7831 q^{-159} +6774 q^{-161} -248 q^{-163} -7158 q^{-165} -7847 q^{-167} -1729 q^{-169} +5726 q^{-171} +7985 q^{-173} +3349 q^{-175} -3837 q^{-177} -7234 q^{-179} -4287 q^{-181} +1912 q^{-183} +5836 q^{-185} +4447 q^{-187} -369 q^{-189} -4139 q^{-191} -3938 q^{-193} -624 q^{-195} +2551 q^{-197} +3065 q^{-199} +1021 q^{-201} -1342 q^{-203} -2076 q^{-205} -1000 q^{-207} +555 q^{-209} +1241 q^{-211} +786 q^{-213} -163 q^{-215} -675 q^{-217} -480 q^{-219} +4 q^{-221} +300 q^{-223} +273 q^{-225} +43 q^{-227} -150 q^{-229} -123 q^{-231} -12 q^{-233} +47 q^{-235} +51 q^{-237} +11 q^{-239} -21 q^{-241} -25 q^{-243} +2 q^{-245} +15 q^{-247} +3 q^{-249} -4 q^{-251} -2 q^{-253} -2 q^{-255} -2 q^{-257} +6 q^{-259} -3 q^{-263} + q^{-265} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ 1+ q^{-4} +2 q^{-6} -2 q^{-8} +2 q^{-10} -2 q^{-12} +2 q^{-16} - q^{-18} +3 q^{-20} -2 q^{-22} -2 q^{-28} + q^{-30} }[/math]
1,1	[math]\displaystyle{ q^4-2 q^2+8-16 q^{-2} +37 q^{-4} -68 q^{-6} +120 q^{-8} -194 q^{-10} +293 q^{-12} -410 q^{-14} +534 q^{-16} -642 q^{-18} +719 q^{-20} -726 q^{-22} +654 q^{-24} -488 q^{-26} +243 q^{-28} +78 q^{-30} -436 q^{-32} +784 q^{-34} -1095 q^{-36} +1324 q^{-38} -1450 q^{-40} +1446 q^{-42} -1320 q^{-44} +1088 q^{-46} -774 q^{-48} +418 q^{-50} -66 q^{-52} -248 q^{-54} +490 q^{-56} -644 q^{-58} +709 q^{-60} -698 q^{-62} +632 q^{-64} -528 q^{-66} +409 q^{-68} -298 q^{-70} +206 q^{-72} -128 q^{-74} +71 q^{-76} -38 q^{-78} +18 q^{-80} -6 q^{-82} + q^{-84} }[/math]
2,0	[math]\displaystyle{ q^4-1+ q^{-2} +4 q^{-4} -5 q^{-8} + q^{-10} +9 q^{-12} -3 q^{-14} -10 q^{-16} +6 q^{-18} +9 q^{-20} -7 q^{-22} -6 q^{-24} +10 q^{-26} +6 q^{-28} -7 q^{-30} +3 q^{-32} +9 q^{-34} -8 q^{-36} -6 q^{-38} +7 q^{-40} -5 q^{-42} -11 q^{-44} +4 q^{-46} +10 q^{-48} -7 q^{-50} -6 q^{-52} +10 q^{-54} +5 q^{-56} -10 q^{-58} -3 q^{-60} +7 q^{-62} -2 q^{-66} +3 q^{-70} - q^{-72} -2 q^{-74} + q^{-76} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ 1- q^{-2} +2 q^{-4} +3 q^{-6} -4 q^{-8} +5 q^{-10} +6 q^{-12} -13 q^{-14} +9 q^{-16} +9 q^{-18} -21 q^{-20} +11 q^{-22} +12 q^{-24} -21 q^{-26} +6 q^{-28} +12 q^{-30} -10 q^{-32} -3 q^{-34} +6 q^{-36} +5 q^{-38} -9 q^{-40} -6 q^{-42} +18 q^{-44} -10 q^{-46} -14 q^{-48} +23 q^{-50} -7 q^{-52} -15 q^{-54} +17 q^{-56} -2 q^{-58} -9 q^{-60} +8 q^{-62} -3 q^{-66} + q^{-68} }[/math]
1,0,0	[math]\displaystyle{ q^{-1} +2 q^{-5} +3 q^{-9} -2 q^{-11} +2 q^{-13} -3 q^{-15} - q^{-19} + q^{-21} +2 q^{-23} +3 q^{-27} -2 q^{-29} + q^{-31} -3 q^{-33} + q^{-35} -2 q^{-37} + q^{-39} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ 1- q^{-2} +4 q^{-4} -5 q^{-6} +10 q^{-8} -13 q^{-10} +18 q^{-12} -21 q^{-14} +23 q^{-16} -23 q^{-18} +19 q^{-20} -13 q^{-22} +2 q^{-24} +9 q^{-26} -20 q^{-28} +32 q^{-30} -40 q^{-32} +47 q^{-34} -46 q^{-36} +43 q^{-38} -35 q^{-40} +24 q^{-42} -12 q^{-44} +10 q^{-48} -19 q^{-50} +23 q^{-52} -25 q^{-54} +23 q^{-56} -20 q^{-58} +15 q^{-60} -10 q^{-62} +6 q^{-64} -3 q^{-66} + q^{-68} }[/math]
1,0	[math]\displaystyle{ q^2- q^{-2} - q^{-4} +3 q^{-6} +4 q^{-8} - q^{-10} -7 q^{-12} -2 q^{-14} +10 q^{-16} +11 q^{-18} -7 q^{-20} -18 q^{-22} -2 q^{-24} +22 q^{-26} +15 q^{-28} -16 q^{-30} -24 q^{-32} +4 q^{-34} +27 q^{-36} +8 q^{-38} -21 q^{-40} -15 q^{-42} +12 q^{-44} +17 q^{-46} -6 q^{-48} -16 q^{-50} +2 q^{-52} +16 q^{-54} -16 q^{-58} -4 q^{-60} +16 q^{-62} +9 q^{-64} -15 q^{-66} -15 q^{-68} +12 q^{-70} +20 q^{-72} -6 q^{-74} -25 q^{-76} -5 q^{-78} +24 q^{-80} +16 q^{-82} -15 q^{-84} -23 q^{-86} +2 q^{-88} +20 q^{-90} +9 q^{-92} -11 q^{-94} -12 q^{-96} +2 q^{-98} +9 q^{-100} +3 q^{-102} -3 q^{-104} -3 q^{-106} + q^{-110} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{-2} - q^{-4} +3 q^{-6} -2 q^{-8} +7 q^{-10} -6 q^{-12} +11 q^{-14} -10 q^{-16} +16 q^{-18} -17 q^{-20} +17 q^{-22} -18 q^{-24} +18 q^{-26} -17 q^{-28} +10 q^{-30} -7 q^{-32} +3 q^{-34} +5 q^{-36} -14 q^{-38} +19 q^{-40} -21 q^{-42} +31 q^{-44} -35 q^{-46} +35 q^{-48} -34 q^{-50} +36 q^{-52} -32 q^{-54} +23 q^{-56} -22 q^{-58} +16 q^{-60} -6 q^{-62} -2 q^{-64} +3 q^{-66} -10 q^{-68} +18 q^{-70} -18 q^{-72} +16 q^{-74} -20 q^{-76} +20 q^{-78} -14 q^{-80} +12 q^{-82} -12 q^{-84} +9 q^{-86} -4 q^{-88} +3 q^{-90} -3 q^{-92} + q^{-94} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{-2} - q^{-4} +4 q^{-6} -5 q^{-8} +6 q^{-10} -5 q^{-12} +12 q^{-16} -22 q^{-18} +35 q^{-20} -37 q^{-22} +28 q^{-24} - q^{-26} -37 q^{-28} +79 q^{-30} -104 q^{-32} +101 q^{-34} -61 q^{-36} -13 q^{-38} +92 q^{-40} -150 q^{-42} +165 q^{-44} -120 q^{-46} +34 q^{-48} +60 q^{-50} -132 q^{-52} +142 q^{-54} -98 q^{-56} +16 q^{-58} +68 q^{-60} -110 q^{-62} +91 q^{-64} -19 q^{-66} -73 q^{-68} +150 q^{-70} -169 q^{-72} +122 q^{-74} -18 q^{-76} -106 q^{-78} +206 q^{-80} -240 q^{-82} +200 q^{-84} -92 q^{-86} -41 q^{-88} +152 q^{-90} -206 q^{-92} +185 q^{-94} -103 q^{-96} -5 q^{-98} +89 q^{-100} -120 q^{-102} +85 q^{-104} -9 q^{-106} -71 q^{-108} +117 q^{-110} -105 q^{-112} +40 q^{-114} +45 q^{-116} -124 q^{-118} +161 q^{-120} -142 q^{-122} +80 q^{-124} +3 q^{-126} -77 q^{-128} +118 q^{-130} -121 q^{-132} +93 q^{-134} -45 q^{-136} -2 q^{-138} +34 q^{-140} -53 q^{-142} +50 q^{-144} -34 q^{-146} +19 q^{-148} -2 q^{-150} -6 q^{-152} +9 q^{-154} -10 q^{-156} +6 q^{-158} -3 q^{-160} + q^{-162} }[/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 72"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -2 t^3+9 t^2-16 t+19-16 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]=

{ 73, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{10}-4 q^9+7 q^8-10 q^7+12 q^6-12 q^5+11 q^4-8 q^3+5 q^2-2 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^6 a^{-6} +z^4 a^{-2} -3 z^4 a^{-4} -2 z^4 a^{-6} +z^4 a^{-8} +3 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} +z^2 a^{-8} +2 a^{-2} -2 a^{-4} +2 a^{-6} - a^{-8} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^9 a^{-5} +z^9 a^{-7} +2 z^8 a^{-4} +6 z^8 a^{-6} +4 z^8 a^{-8} +2 z^7 a^{-3} +4 z^7 a^{-5} +9 z^7 a^{-7} +7 z^7 a^{-9} +z^6 a^{-2} -2 z^6 a^{-4} -8 z^6 a^{-6} +2 z^6 a^{-8} +7 z^6 a^{-10} -6 z^5 a^{-3} -14 z^5 a^{-5} -19 z^5 a^{-7} -7 z^5 a^{-9} +4 z^5 a^{-11} -4 z^4 a^{-2} -6 z^4 a^{-4} -4 z^4 a^{-6} -11 z^4 a^{-8} -8 z^4 a^{-10} +z^4 a^{-12} +5 z^3 a^{-3} +11 z^3 a^{-5} +9 z^3 a^{-7} -3 z^3 a^{-11} +5 z^2 a^{-2} +8 z^2 a^{-4} +7 z^2 a^{-6} +6 z^2 a^{-8} +2 z^2 a^{-10} -z a^{-3} -3 z a^{-5} -z a^{-7} +z a^{-9} -2 a^{-2} -2 a^{-4} -2 a^{-6} - a^{-8} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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

Vassiliev invariants

V2 and V3:

(2, 4)

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{ 32 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{460}{3} }[/math] [math]\displaystyle{ \frac{116}{3} }[/math] [math]\displaystyle{ 256 }[/math] [math]\displaystyle{ \frac{2144}{3} }[/math] [math]\displaystyle{ \frac{512}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ \frac{3680}{3} }[/math] [math]\displaystyle{ \frac{928}{3} }[/math] [math]\displaystyle{ \frac{49111}{15} }[/math] [math]\displaystyle{ \frac{1076}{15} }[/math] [math]\displaystyle{ \frac{71164}{45} }[/math] [math]\displaystyle{ \frac{281}{9} }[/math] [math]\displaystyle{ \frac{3271}{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 72. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 21                     1 1 19                   3   -3 17                 4 1   3 15               6 3     -3 13             6 4       2 11           6 6         0 9         5 6           -1 7       3 6             3 5     2 5               -3 3   1 4                 3 1   1                   -1 -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/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^{28}-4 q^{27}+3 q^{26}+10 q^{25}-24 q^{24}+10 q^{23}+37 q^{22}-63 q^{21}+12 q^{20}+80 q^{19}-101 q^{18}+2 q^{17}+117 q^{16}-115 q^{15}-16 q^{14}+128 q^{13}-97 q^{12}-32 q^{11}+108 q^{10}-59 q^9-36 q^8+67 q^7-22 q^6-25 q^5+28 q^4-4 q^3-10 q^2+7 q-2 q^{-1} + q^{-2} }[/math]
3	[math]\displaystyle{ q^{54}-4 q^{53}+3 q^{52}+6 q^{51}-4 q^{50}-16 q^{49}+7 q^{48}+40 q^{47}-21 q^{46}-69 q^{45}+25 q^{44}+128 q^{43}-33 q^{42}-202 q^{41}+22 q^{40}+302 q^{39}-2 q^{38}-401 q^{37}-46 q^{36}+502 q^{35}+108 q^{34}-583 q^{33}-179 q^{32}+633 q^{31}+256 q^{30}-656 q^{29}-321 q^{28}+636 q^{27}+385 q^{26}-596 q^{25}-424 q^{24}+521 q^{23}+454 q^{22}-434 q^{21}-456 q^{20}+325 q^{19}+446 q^{18}-227 q^{17}-399 q^{16}+123 q^{15}+344 q^{14}-48 q^{13}-266 q^{12}-11 q^{11}+196 q^{10}+33 q^9-120 q^8-48 q^7+76 q^6+34 q^5-34 q^4-28 q^3+19 q^2+13 q-5-9 q^{-1} +4 q^{-2} +2 q^{-3} -2 q^{-5} + q^{-6} }[/math]
4	[math]\displaystyle{ q^{88}-4 q^{87}+3 q^{86}+6 q^{85}-8 q^{84}+4 q^{83}-19 q^{82}+20 q^{81}+30 q^{80}-43 q^{79}+5 q^{78}-66 q^{77}+88 q^{76}+123 q^{75}-141 q^{74}-63 q^{73}-198 q^{72}+283 q^{71}+415 q^{70}-277 q^{69}-329 q^{68}-594 q^{67}+601 q^{66}+1106 q^{65}-242 q^{64}-825 q^{63}-1476 q^{62}+812 q^{61}+2180 q^{60}+243 q^{59}-1271 q^{58}-2792 q^{57}+602 q^{56}+3244 q^{55}+1154 q^{54}-1309 q^{53}-4078 q^{52}-41 q^{51}+3815 q^{50}+2111 q^{49}-861 q^{48}-4849 q^{47}-833 q^{46}+3734 q^{45}+2753 q^{44}-135 q^{43}-4939 q^{42}-1515 q^{41}+3113 q^{40}+2985 q^{39}+673 q^{38}-4435 q^{37}-2005 q^{36}+2126 q^{35}+2837 q^{34}+1433 q^{33}-3449 q^{32}-2228 q^{31}+946 q^{30}+2299 q^{29}+1960 q^{28}-2149 q^{27}-2035 q^{26}-113 q^{25}+1420 q^{24}+1994 q^{23}-879 q^{22}-1400 q^{21}-686 q^{20}+495 q^{19}+1496 q^{18}-61 q^{17}-624 q^{16}-667 q^{15}-83 q^{14}+791 q^{13}+182 q^{12}-104 q^{11}-358 q^{10}-214 q^9+285 q^8+112 q^7+57 q^6-112 q^5-127 q^4+73 q^3+25 q^2+43 q-19-44 q^{-1} +18 q^{-2} - q^{-3} +13 q^{-4} - q^{-5} -11 q^{-6} +5 q^{-7} - q^{-8} +2 q^{-9} -2 q^{-11} + q^{-12} }[/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, 72]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], X[16, 8, 17, 7], X[18, 12, 19, 11], X[20, 14, 1, 13], X[12, 20, 13, 19], X[14, 18, 15, 17], X[6, 16, 7, 15]]
In[3]:=	GaussCode[Knot[10, 72]]
Out[3]=	GaussCode[1, -4, 3, -1, 2, -10, 5, -3, 4, -2, 6, -8, 7, -9, 10, -5, 9, -6, 8, -7]
In[4]:=	DTCode[Knot[10, 72]]
Out[4]=	DTCode[4, 8, 10, 16, 2, 18, 20, 6, 14, 12]
In[5]:=	br = BR[Knot[10, 72]]
Out[5]=	BR[4, {1, 1, 1, 1, 2, 2, -1, 2, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 72]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 72]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 72]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 72]][t]
Out[10]=	2 9 16 2 3 19 - -- + -- - -- - 16 t + 9 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 72]][z]
Out[11]=	2 4 6 1 + 2 z - 3 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 72]}
In[13]:=	{KnotDet[Knot[10, 72]], KnotSignature[Knot[10, 72]]}
Out[13]=	{73, 4}
In[14]:=	Jones[Knot[10, 72]][q]
Out[14]=	2 3 4 5 6 7 8 9 1 - 2 q + 5 q - 8 q + 11 q - 12 q + 12 q - 10 q + 7 q - 4 q + 10 q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 72]}
In[16]:=	A2Invariant[Knot[10, 72]][q]
Out[16]=	4 6 8 10 12 16 18 20 22 1 + q + 2 q - 2 q + 2 q - 2 q + 2 q - q + 3 q - 2 q - 28 30 2 q + q
In[17]:=	HOMFLYPT[Knot[10, 72]][a, z]
Out[17]=	2 2 2 2 4 4 4 4 -8 2 2 2 z z 3 z 3 z z 2 z 3 z z -a + -- - -- + -- + -- + -- - ---- + ---- + -- - ---- - ---- + -- - 6 4 2 8 6 4 2 8 6 4 2 a a a a a a a a a a a 6 6 z z -- - -- 6 4 a a
In[18]:=	Kauffman[Knot[10, 72]][a, z]
Out[18]=	2 2 2 2 -8 2 2 2 z z 3 z z 2 z 6 z 7 z 8 z -a - -- - -- - -- + -- - -- - --- - -- + ---- + ---- + ---- + ---- + 6 4 2 9 7 5 3 10 8 6 4 a a a a a a a a a a a 2 3 3 3 3 4 4 4 4 5 z 3 z 9 z 11 z 5 z z 8 z 11 z 4 z ---- - ---- + ---- + ----- + ---- + --- - ---- - ----- - ---- - 2 11 7 5 3 12 10 8 6 a a a a a a a a a 4 4 5 5 5 5 5 6 6 6 z 4 z 4 z 7 z 19 z 14 z 6 z 7 z 2 z ---- - ---- + ---- - ---- - ----- - ----- - ---- + ---- + ---- - 4 2 11 9 7 5 3 10 8 a a a a a a a a a 6 6 6 7 7 7 7 8 8 8 8 z 2 z z 7 z 9 z 4 z 2 z 4 z 6 z 2 z ---- - ---- + -- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + 6 4 2 9 7 5 3 8 6 4 a a a a a a a a a a 9 9 z z -- + -- 7 5 a a
In[19]:=	{Vassiliev[2][Knot[10, 72]], Vassiliev[3][Knot[10, 72]]}
Out[19]=	{2, 4}
In[20]:=	Kh[Knot[10, 72]][q, t]
Out[20]=	3 3 5 1 q q 5 7 7 2 9 2 4 q + 2 q + ---- + - + -- + 5 q t + 3 q t + 6 q t + 5 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 6 q t + 6 q t + 6 q t + 6 q t + 4 q t + 6 q t + 15 6 17 6 17 7 19 7 21 8 3 q t + 4 q t + q t + 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 72], 2][q]
Out[21]=	-2 2 2 3 4 5 6 7 8 q - - + 7 q - 10 q - 4 q + 28 q - 25 q - 22 q + 67 q - 36 q - q 9 10 11 12 13 14 15 59 q + 108 q - 32 q - 97 q + 128 q - 16 q - 115 q + 16 17 18 19 20 21 22 117 q + 2 q - 101 q + 80 q + 12 q - 63 q + 37 q + 23 24 25 26 27 28 10 q - 24 q + 10 q + 3 q - 4 q + q

See/edit the Rolfsen_Splice_Template.