10 157

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10 156.gif

10_156

10 158.gif

10_158

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10 157 Quick Notes


10 157 Further Notes and Views

Knot presentations

Planar diagram presentation X1627 X10,4,11,3 X16,11,17,12 X7,15,8,14 X15,9,16,8 X13,1,14,20 X19,13,20,12 X18,6,19,5 X2,10,3,9 X4,18,5,17
Gauss code -1, -9, 2, -10, 8, 1, -4, 5, 9, -2, 3, 7, -6, 4, -5, -3, 10, -8, -7, 6
Dowker-Thistlethwaite code 6 -10 -18 14 -2 -16 20 8 -4 12
Conway Notation [-3:20:20]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 2
Maximal Thurston-Bennequin number [3][-13]
Hyperbolic Volume 12.6653
A-Polynomial See Data:10 157/A-polynomial

[edit Notes for 10 157'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 10 157's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^3+6 t^2-11 t+13-11 t^{-1} +6 t^{-2} - t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -z^6+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{7,t+1\} }[/math]
Determinant and Signature { 49, 4 }
Jones polynomial [math]\displaystyle{ q^{10}-4 q^9+6 q^8-8 q^7+9 q^6-8 q^5+7 q^4-4 q^3+2 q^2 }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^6 a^{-6} +2 z^4 a^{-4} -3 z^4 a^{-6} +z^4 a^{-8} +5 z^2 a^{-4} -2 z^2 a^{-6} +z^2 a^{-8} +2 a^{-4} - a^{-8} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^8 a^{-6} +z^8 a^{-8} +z^7 a^{-5} +5 z^7 a^{-7} +4 z^7 a^{-9} +2 z^6 a^{-6} +8 z^6 a^{-8} +6 z^6 a^{-10} +z^5 a^{-5} -3 z^5 a^{-7} +4 z^5 a^{-11} +3 z^4 a^{-4} -3 z^4 a^{-6} -15 z^4 a^{-8} -8 z^4 a^{-10} +z^4 a^{-12} -2 z^3 a^{-5} -6 z^3 a^{-7} -8 z^3 a^{-9} -4 z^3 a^{-11} -5 z^2 a^{-4} +7 z^2 a^{-8} +2 z^2 a^{-10} +4 z a^{-7} +4 z a^{-9} +2 a^{-4} - a^{-8} }[/math]
The A2 invariant [math]\displaystyle{ 2 q^{-6} - q^{-8} +2 q^{-10} +3 q^{-16} - q^{-18} +2 q^{-20} -2 q^{-22} - q^{-24} -2 q^{-28} + q^{-30} }[/math]
The G2 invariant [math]\displaystyle{ q^{-28} -2 q^{-32} +12 q^{-34} -18 q^{-36} +19 q^{-38} -9 q^{-40} -10 q^{-42} +42 q^{-44} -60 q^{-46} +66 q^{-48} -44 q^{-50} -4 q^{-52} +56 q^{-54} -96 q^{-56} +101 q^{-58} -67 q^{-60} +7 q^{-62} +49 q^{-64} -79 q^{-66} +74 q^{-68} -31 q^{-70} -19 q^{-72} +61 q^{-74} -66 q^{-76} +37 q^{-78} +18 q^{-80} -70 q^{-82} +105 q^{-84} -99 q^{-86} +61 q^{-88} +4 q^{-90} -72 q^{-92} +119 q^{-94} -134 q^{-96} +99 q^{-98} -38 q^{-100} -37 q^{-102} +85 q^{-104} -102 q^{-106} +73 q^{-108} -17 q^{-110} -38 q^{-112} +64 q^{-114} -58 q^{-116} +13 q^{-118} +43 q^{-120} -81 q^{-122} +85 q^{-124} -51 q^{-126} +52 q^{-130} -83 q^{-132} +85 q^{-134} -59 q^{-136} +20 q^{-138} +14 q^{-140} -40 q^{-142} +45 q^{-144} -34 q^{-146} +22 q^{-148} -5 q^{-150} -4 q^{-152} +8 q^{-154} -10 q^{-156} +6 q^{-158} -3 q^{-160} + q^{-162} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_157.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ 2 q^{-3} -2 q^{-5} +3 q^{-7} - q^{-9} + q^{-11} + q^{-13} -2 q^{-15} +2 q^{-17} -3 q^{-19} + q^{-21} }[/math]
2 [math]\displaystyle{ q^{-4} +3 q^{-6} -5 q^{-8} -3 q^{-10} +14 q^{-12} -4 q^{-14} -14 q^{-16} +18 q^{-18} +3 q^{-20} -18 q^{-22} +10 q^{-24} +8 q^{-26} -11 q^{-28} -2 q^{-30} +8 q^{-32} +2 q^{-34} -14 q^{-36} +4 q^{-38} +14 q^{-40} -18 q^{-42} -3 q^{-44} +19 q^{-46} -9 q^{-48} -7 q^{-50} +10 q^{-52} - q^{-54} -3 q^{-56} + q^{-58} }[/math]
3 [math]\displaystyle{ 2 q^{-5} +2 q^{-7} -12 q^{-11} -4 q^{-13} +20 q^{-15} +25 q^{-17} -19 q^{-19} -51 q^{-21} +9 q^{-23} +75 q^{-25} +19 q^{-27} -86 q^{-29} -54 q^{-31} +86 q^{-33} +82 q^{-35} -68 q^{-37} -100 q^{-39} +41 q^{-41} +105 q^{-43} -16 q^{-45} -96 q^{-47} -5 q^{-49} +79 q^{-51} +25 q^{-53} -59 q^{-55} -46 q^{-57} +34 q^{-59} +59 q^{-61} -9 q^{-63} -79 q^{-65} -19 q^{-67} +88 q^{-69} +56 q^{-71} -89 q^{-73} -81 q^{-75} +73 q^{-77} +101 q^{-79} -45 q^{-81} -105 q^{-83} +15 q^{-85} +91 q^{-87} +10 q^{-89} -62 q^{-91} -22 q^{-93} +34 q^{-95} +21 q^{-97} -19 q^{-99} -11 q^{-101} +5 q^{-103} +7 q^{-105} - q^{-107} -3 q^{-109} + q^{-111} }[/math]
5 [math]\displaystyle{ 2 q^{-3} +2 q^{-5} +4 q^{-7} -12 q^{-11} -24 q^{-13} -14 q^{-15} +6 q^{-17} +54 q^{-19} +98 q^{-21} +54 q^{-23} -66 q^{-25} -197 q^{-27} -237 q^{-29} -79 q^{-31} +273 q^{-33} +535 q^{-35} +423 q^{-37} -103 q^{-39} -775 q^{-41} -1014 q^{-43} -438 q^{-45} +748 q^{-47} +1636 q^{-49} +1350 q^{-51} -186 q^{-53} -1966 q^{-55} -2462 q^{-57} -942 q^{-59} +1726 q^{-61} +3388 q^{-63} +2420 q^{-65} -764 q^{-67} -3756 q^{-69} -3916 q^{-71} -748 q^{-73} +3407 q^{-75} +4987 q^{-77} +2429 q^{-79} -2355 q^{-81} -5384 q^{-83} -3920 q^{-85} +933 q^{-87} +5085 q^{-89} +4872 q^{-91} +496 q^{-93} -4226 q^{-95} -5202 q^{-97} -1663 q^{-99} +3129 q^{-101} +4966 q^{-103} +2386 q^{-105} -2053 q^{-107} -4345 q^{-109} -2691 q^{-111} +1141 q^{-113} +3596 q^{-115} +2688 q^{-117} -467 q^{-119} -2879 q^{-121} -2543 q^{-123} -35 q^{-125} +2286 q^{-127} +2442 q^{-129} +473 q^{-131} -1827 q^{-133} -2490 q^{-135} -974 q^{-137} +1424 q^{-139} +2684 q^{-141} +1676 q^{-143} -928 q^{-145} -2999 q^{-147} -2587 q^{-149} +235 q^{-151} +3207 q^{-153} +3657 q^{-155} +825 q^{-157} -3155 q^{-159} -4695 q^{-161} -2126 q^{-163} +2590 q^{-165} +5395 q^{-167} +3579 q^{-169} -1510 q^{-171} -5506 q^{-173} -4812 q^{-175} +32 q^{-177} +4847 q^{-179} +5499 q^{-181} +1509 q^{-183} -3539 q^{-185} -5406 q^{-187} -2730 q^{-189} +1925 q^{-191} +4553 q^{-193} +3288 q^{-195} -396 q^{-197} -3228 q^{-199} -3148 q^{-201} -663 q^{-203} +1861 q^{-205} +2472 q^{-207} +1112 q^{-209} -761 q^{-211} -1605 q^{-213} -1085 q^{-215} +103 q^{-217} +886 q^{-219} +774 q^{-221} +145 q^{-223} -371 q^{-225} -453 q^{-227} -193 q^{-229} +135 q^{-231} +231 q^{-233} +109 q^{-235} -27 q^{-237} -92 q^{-239} -64 q^{-241} -2 q^{-243} +42 q^{-245} +30 q^{-247} -2 q^{-249} -13 q^{-251} -9 q^{-253} -2 q^{-255} +2 q^{-257} +7 q^{-259} - q^{-261} -3 q^{-263} + q^{-265} }[/math]
6 [math]\displaystyle{ 1+3 q^{-2} +3 q^{-4} +3 q^{-6} -3 q^{-8} -11 q^{-10} -27 q^{-12} -33 q^{-14} -9 q^{-16} +43 q^{-18} +99 q^{-20} +129 q^{-22} +113 q^{-24} -45 q^{-26} -267 q^{-28} -445 q^{-30} -381 q^{-32} -53 q^{-34} +477 q^{-36} +1073 q^{-38} +1130 q^{-40} +478 q^{-42} -828 q^{-44} -2046 q^{-46} -2515 q^{-48} -1544 q^{-50} +1000 q^{-52} +3654 q^{-54} +4885 q^{-56} +3239 q^{-58} -749 q^{-60} -5699 q^{-62} -8412 q^{-64} -6194 q^{-66} +354 q^{-68} +8364 q^{-70} +12685 q^{-72} +10360 q^{-74} +548 q^{-76} -11562 q^{-78} -18333 q^{-80} -14889 q^{-82} -1383 q^{-84} +14847 q^{-86} +24778 q^{-88} +19812 q^{-90} +1786 q^{-92} -19340 q^{-94} -30753 q^{-96} -23892 q^{-98} -1614 q^{-100} +24502 q^{-102} +36410 q^{-104} +26299 q^{-106} -1001 q^{-108} -29247 q^{-110} -40339 q^{-112} -26570 q^{-114} +5472 q^{-116} +34040 q^{-118} +41714 q^{-120} +22840 q^{-122} -10634 q^{-124} -37368 q^{-126} -40159 q^{-128} -16268 q^{-130} +16700 q^{-132} +38262 q^{-134} +34261 q^{-136} +8499 q^{-138} -21677 q^{-140} -36303 q^{-142} -25761 q^{-144} +101 q^{-146} +24307 q^{-148} +30561 q^{-150} +16556 q^{-152} -7303 q^{-154} -24199 q^{-156} -23017 q^{-158} -7265 q^{-160} +12008 q^{-162} +21086 q^{-164} +15416 q^{-166} -312 q^{-168} -14278 q^{-170} -17026 q^{-172} -8260 q^{-174} +5758 q^{-176} +14712 q^{-178} +13350 q^{-180} +2510 q^{-182} -9742 q^{-184} -15244 q^{-186} -10312 q^{-188} +2144 q^{-190} +13543 q^{-192} +16584 q^{-194} +8064 q^{-196} -6636 q^{-198} -18576 q^{-200} -18720 q^{-202} -5820 q^{-204} +12496 q^{-206} +24642 q^{-208} +20997 q^{-210} +2705 q^{-212} -20211 q^{-214} -31317 q^{-216} -22258 q^{-218} +2772 q^{-220} +28476 q^{-222} +37046 q^{-224} +21702 q^{-226} -10072 q^{-228} -36442 q^{-230} -40322 q^{-232} -17887 q^{-234} +17147 q^{-236} +42003 q^{-238} +40479 q^{-240} +12046 q^{-242} -23262 q^{-244} -43806 q^{-246} -36344 q^{-248} -6373 q^{-250} +26589 q^{-252} +41873 q^{-254} +29729 q^{-256} +1035 q^{-258} -26503 q^{-260} -35741 q^{-262} -22871 q^{-264} +2567 q^{-266} +23853 q^{-268} +27849 q^{-270} +15756 q^{-272} -4144 q^{-274} -18656 q^{-276} -20412 q^{-278} -9881 q^{-280} +4539 q^{-282} +13189 q^{-284} +13297 q^{-286} +5691 q^{-288} -3444 q^{-290} -8886 q^{-292} -7864 q^{-294} -2695 q^{-296} +2329 q^{-298} +5135 q^{-300} +4255 q^{-302} +1325 q^{-304} -1704 q^{-306} -2686 q^{-308} -1904 q^{-310} -528 q^{-312} +898 q^{-314} +1300 q^{-316} +892 q^{-318} -2 q^{-320} -452 q^{-322} -478 q^{-324} -334 q^{-326} +26 q^{-328} +217 q^{-330} +227 q^{-332} +39 q^{-334} -39 q^{-336} -64 q^{-338} -74 q^{-340} -12 q^{-342} +26 q^{-344} +44 q^{-346} -2 q^{-348} -4 q^{-350} - q^{-352} -12 q^{-354} -2 q^{-356} +2 q^{-358} +7 q^{-360} - q^{-362} -3 q^{-364} + q^{-366} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ 2 q^{-6} - q^{-8} +2 q^{-10} +3 q^{-16} - q^{-18} +2 q^{-20} -2 q^{-22} - q^{-24} -2 q^{-28} + q^{-30} }[/math]
1,1 [math]\displaystyle{ 2 q^{-10} -4 q^{-14} +28 q^{-16} -60 q^{-18} +118 q^{-20} -190 q^{-22} +272 q^{-24} -340 q^{-26} +385 q^{-28} -374 q^{-30} +316 q^{-32} -196 q^{-34} +37 q^{-36} +150 q^{-38} -346 q^{-40} +502 q^{-42} -635 q^{-44} +696 q^{-46} -698 q^{-48} +630 q^{-50} -493 q^{-52} +324 q^{-54} -130 q^{-56} -56 q^{-58} +206 q^{-60} -314 q^{-62} +358 q^{-64} -354 q^{-66} +314 q^{-68} -248 q^{-70} +184 q^{-72} -120 q^{-74} +69 q^{-76} -38 q^{-78} +18 q^{-80} -6 q^{-82} + q^{-84} }[/math]
2,0 [math]\displaystyle{ q^{-10} +4 q^{-12} -2 q^{-14} -4 q^{-16} +5 q^{-18} +4 q^{-20} -4 q^{-22} -2 q^{-24} +7 q^{-26} +6 q^{-28} -6 q^{-30} +2 q^{-32} +8 q^{-34} -4 q^{-36} -3 q^{-38} +4 q^{-40} - q^{-42} -6 q^{-44} +2 q^{-48} -7 q^{-50} -5 q^{-52} +5 q^{-54} +2 q^{-56} -9 q^{-58} +2 q^{-60} +7 q^{-62} - q^{-66} +4 q^{-70} -2 q^{-72} -2 q^{-74} + q^{-76} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ 3 q^{-12} -3 q^{-14} +2 q^{-16} +9 q^{-18} -9 q^{-20} +4 q^{-22} +13 q^{-24} -13 q^{-26} +4 q^{-28} +11 q^{-30} -9 q^{-32} -2 q^{-34} +5 q^{-36} -3 q^{-38} -5 q^{-40} -4 q^{-42} +6 q^{-44} -2 q^{-46} -11 q^{-48} +13 q^{-50} -13 q^{-54} +12 q^{-56} + q^{-58} -8 q^{-60} +7 q^{-62} -3 q^{-66} + q^{-68} }[/math]
1,0,0 [math]\displaystyle{ 2 q^{-9} - q^{-11} +3 q^{-13} - q^{-15} +2 q^{-17} +2 q^{-21} + q^{-23} + q^{-27} -2 q^{-29} -3 q^{-33} + q^{-35} -2 q^{-37} + q^{-39} }[/math]
1,0,1 [math]\displaystyle{ 2 q^{-16} + q^{-18} -5 q^{-20} +20 q^{-22} -10 q^{-24} -12 q^{-26} +80 q^{-28} -122 q^{-30} +123 q^{-32} -23 q^{-34} -142 q^{-36} +301 q^{-38} -363 q^{-40} +274 q^{-42} -35 q^{-44} -229 q^{-46} +427 q^{-48} -447 q^{-50} +281 q^{-52} -60 q^{-54} -153 q^{-56} +180 q^{-58} -114 q^{-60} -4 q^{-62} +38 q^{-64} +73 q^{-66} -226 q^{-68} +349 q^{-70} -294 q^{-72} +85 q^{-74} +185 q^{-76} -396 q^{-78} +423 q^{-80} -286 q^{-82} +48 q^{-84} +170 q^{-86} -267 q^{-88} +235 q^{-90} -108 q^{-92} -23 q^{-94} +94 q^{-96} -91 q^{-98} +46 q^{-100} -3 q^{-102} -17 q^{-104} +15 q^{-106} -6 q^{-108} + q^{-110} }[/math]

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-28} -2 q^{-32} +12 q^{-34} -18 q^{-36} +19 q^{-38} -9 q^{-40} -10 q^{-42} +42 q^{-44} -60 q^{-46} +66 q^{-48} -44 q^{-50} -4 q^{-52} +56 q^{-54} -96 q^{-56} +101 q^{-58} -67 q^{-60} +7 q^{-62} +49 q^{-64} -79 q^{-66} +74 q^{-68} -31 q^{-70} -19 q^{-72} +61 q^{-74} -66 q^{-76} +37 q^{-78} +18 q^{-80} -70 q^{-82} +105 q^{-84} -99 q^{-86} +61 q^{-88} +4 q^{-90} -72 q^{-92} +119 q^{-94} -134 q^{-96} +99 q^{-98} -38 q^{-100} -37 q^{-102} +85 q^{-104} -102 q^{-106} +73 q^{-108} -17 q^{-110} -38 q^{-112} +64 q^{-114} -58 q^{-116} +13 q^{-118} +43 q^{-120} -81 q^{-122} +85 q^{-124} -51 q^{-126} +52 q^{-130} -83 q^{-132} +85 q^{-134} -59 q^{-136} +20 q^{-138} +14 q^{-140} -40 q^{-142} +45 q^{-144} -34 q^{-146} +22 q^{-148} -5 q^{-150} -4 q^{-152} +8 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["10 157"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^3+6 t^2-11 t+13-11 t^{-1} +6 t^{-2} - t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^6+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{ \{7,t+1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 49, 4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^{10}-4 q^9+6 q^8-8 q^7+9 q^6-8 q^5+7 q^4-4 q^3+2 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^{-6} +2 z^4 a^{-4} -3 z^4 a^{-6} +z^4 a^{-8} +5 z^2 a^{-4} -2 z^2 a^{-6} +z^2 a^{-8} +2 a^{-4} - a^{-8} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^8 a^{-6} +z^8 a^{-8} +z^7 a^{-5} +5 z^7 a^{-7} +4 z^7 a^{-9} +2 z^6 a^{-6} +8 z^6 a^{-8} +6 z^6 a^{-10} +z^5 a^{-5} -3 z^5 a^{-7} +4 z^5 a^{-11} +3 z^4 a^{-4} -3 z^4 a^{-6} -15 z^4 a^{-8} -8 z^4 a^{-10} +z^4 a^{-12} -2 z^3 a^{-5} -6 z^3 a^{-7} -8 z^3 a^{-9} -4 z^3 a^{-11} -5 z^2 a^{-4} +7 z^2 a^{-8} +2 z^2 a^{-10} +4 z a^{-7} +4 z a^{-9} +2 a^{-4} - a^{-8} }[/math]

Vassiliev invariants

V2 and V3: (4, 8)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
[math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{1016}{3} }[/math] [math]\displaystyle{ \frac{184}{3} }[/math] [math]\displaystyle{ 1024 }[/math] [math]\displaystyle{ \frac{5824}{3} }[/math] [math]\displaystyle{ \frac{1120}{3} }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 2048 }[/math] [math]\displaystyle{ \frac{16256}{3} }[/math] [math]\displaystyle{ \frac{2944}{3} }[/math] [math]\displaystyle{ \frac{168062}{15} }[/math] [math]\displaystyle{ \frac{584}{5} }[/math] [math]\displaystyle{ \frac{218888}{45} }[/math] [math]\displaystyle{ \frac{610}{9} }[/math] [math]\displaystyle{ \frac{10142}{15} }[/math]

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]4 is the signature of 10 157. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 012345678χ
21        11
19       3 -3
17      31 2
15     53  -2
13    43   1
11   45    1
9  34     -1
7 14      3
513       -2
32        2

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, 157]]
Out[2]=  
10
In[3]:=
PD[Knot[10, 157]]
Out[3]=  
PD[X[1, 6, 2, 7], X[10, 4, 11, 3], X[16, 11, 17, 12], X[7, 15, 8, 14], 

 X[15, 9, 16, 8], X[13, 1, 14, 20], X[19, 13, 20, 12], 



  X[18, 6, 19, 5], X[2, 10, 3, 9], X[4, 18, 5, 17]]
In[4]:=
GaussCode[Knot[10, 157]]
Out[4]=  
GaussCode[-1, -9, 2, -10, 8, 1, -4, 5, 9, -2, 3, 7, -6, 4, -5, -3, 10, 
  -8, -7, 6]
In[5]:=
BR[Knot[10, 157]]
Out[5]=  
BR[3, {1, 1, 1, 2, 2, -1, 2, -1, 2, 2}]
In[6]:=
alex = Alexander[Knot[10, 157]][t]
Out[6]=  
      -3   6    11             2    3

13 - t   + -- - -- - 11 t + 6 t  - t



           2   t


           t
In[7]:=
Conway[Knot[10, 157]][z]
Out[7]=  
       2    6
1 + 4 z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[10, 157]}
In[9]:=
{KnotDet[Knot[10, 157]], KnotSignature[Knot[10, 157]]}
Out[9]=  
{49, 4}
In[10]:=
J=Jones[Knot[10, 157]][q]
Out[10]=  
   2      3      4      5      6      7      8      9    10
2 q  - 4 q  + 7 q  - 8 q  + 9 q  - 8 q  + 6 q  - 4 q  + q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[10, 157]}
In[12]:=
A2Invariant[Knot[10, 157]][q]
Out[12]=  
   6    8      10      16    18      20      22    24      28    30
2 q  - q  + 2 q   + 3 q   - q   + 2 q   - 2 q   - q   - 2 q   + q
In[13]:=
Kauffman[Knot[10, 157]][a, z]
Out[13]=  
                           2      2      2      3      3      3

 -8   2    4 z   4 z   2 z    7 z    5 z    4 z    8 z    6 z



-a   + -- + --- + --- + ---- + ---- - ---- - ---- - ---- - ---- - 



       4    9     7     10      8      4     11      9      7
      a    a     a     a       a      a     a       a      a

    3    4       4       4      4      4      5      5    5      6
 2 z    z     8 z    15 z    3 z    3 z    4 z    3 z    z    6 z
 ---- + --- - ---- - ----- - ---- + ---- + ---- - ---- + -- + ---- + 
   5     12    10      8       6      4     11      7     5    10
  a     a     a       a       a      a     a       a     a    a

    6      6      7      7    7    8    8
 8 z    2 z    4 z    5 z    z    z    z
 ---- + ---- + ---- + ---- + -- + -- + --
   8      6      9      7     5    8    6


   a      a      a      a     a    a    a
In[14]:=
{Vassiliev[2][Knot[10, 157]], Vassiliev[3][Knot[10, 157]]}
Out[14]=  
{0, 8}
In[15]:=
Kh[Knot[10, 157]][q, t]
Out[15]=  
   3    5      5      7        7  2      9  2      9  3      11  3

2 q  + q  + 3 q  t + q  t + 4 q  t  + 3 q  t  + 4 q  t  + 4 q   t  + 



    11  4      13  4      13  5      15  5      15  6      17  6
 5 q   t  + 4 q   t  + 3 q   t  + 5 q   t  + 3 q   t  + 3 q   t  + 

  17  7      19  7    21  8


  q   t  + 3 q   t  + q   t
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