10 95

10_94

10_96

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10 95 Quick Notes

10 95 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,17,12,16 X15,9,16,8 X19,7,20,6 X5,15,6,14 X7,19,8,18 X13,1,14,20 X17,13,18,12 X9,2,10,3
Gauss code	-1, 10, -2, 1, -6, 5, -7, 4, -10, 2, -3, 9, -8, 6, -4, 3, -9, 7, -5, 8
Dowker-Thistlethwaite code	4 10 14 18 2 16 20 8 12 6
Conway Notation	[.210.2.2]

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-2][-10] Hyperbolic Volume 15.0479 A-Polynomial See Data:10 95/A-polynomial

[edit Notes for 10 95'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 95's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^3-9 t^2+21 t-27+21 t^{-1} -9 t^{-2} +2 t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^6+3 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 91, 2 }
Jones polynomial	[math]\displaystyle{ -q^8+3 q^7-7 q^6+11 q^5-14 q^4+16 q^3-14 q^2+12 q-8+4 q^{-1} - q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -z^4+z^2 a^{-2} +5 z^2 a^{-4} -2 z^2 a^{-6} -z^2+3 a^{-4} -2 a^{-6} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^9 a^{-3} +2 z^9 a^{-5} +6 z^8 a^{-2} +11 z^8 a^{-4} +5 z^8 a^{-6} +7 z^7 a^{-1} +10 z^7 a^{-3} +8 z^7 a^{-5} +5 z^7 a^{-7} -6 z^6 a^{-2} -19 z^6 a^{-4} -6 z^6 a^{-6} +3 z^6 a^{-8} +4 z^6+a z^5-12 z^5 a^{-1} -25 z^5 a^{-3} -21 z^5 a^{-5} -8 z^5 a^{-7} +z^5 a^{-9} -2 z^4 a^{-2} +13 z^4 a^{-4} +4 z^4 a^{-6} -5 z^4 a^{-8} -6 z^4-a z^3+5 z^3 a^{-1} +16 z^3 a^{-3} +17 z^3 a^{-5} +5 z^3 a^{-7} -2 z^3 a^{-9} +z^2 a^{-2} -7 z^2 a^{-4} -4 z^2 a^{-6} +2 z^2 a^{-8} +2 z^2-z a^{-1} -3 z a^{-3} -5 z a^{-5} -2 z a^{-7} +z a^{-9} +3 a^{-4} +2 a^{-6} }[/math]
The A2 invariant	[math]\displaystyle{ -q^6+2 q^4-q^2-1+3 q^{-2} -3 q^{-4} +3 q^{-6} + q^{-10} +3 q^{-12} -2 q^{-14} +3 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-22} - q^{-24} }[/math]
The G2 invariant	[math]\displaystyle{ q^{32}-3 q^{30}+7 q^{28}-13 q^{26}+15 q^{24}-14 q^{22}+3 q^{20}+22 q^{18}-53 q^{16}+87 q^{14}-102 q^{12}+76 q^{10}-13 q^8-85 q^6+191 q^4-252 q^2+243-141 q^{-2} -36 q^{-4} +219 q^{-6} -340 q^{-8} +342 q^{-10} -220 q^{-12} +19 q^{-14} +179 q^{-16} -286 q^{-18} +261 q^{-20} -110 q^{-22} -88 q^{-24} +243 q^{-26} -278 q^{-28} +160 q^{-30} +52 q^{-32} -272 q^{-34} +413 q^{-36} -401 q^{-38} +247 q^{-40} +12 q^{-42} -279 q^{-44} +460 q^{-46} -491 q^{-48} +363 q^{-50} -118 q^{-52} -143 q^{-54} +331 q^{-56} -370 q^{-58} +273 q^{-60} -72 q^{-62} -133 q^{-64} +248 q^{-66} -232 q^{-68} +85 q^{-70} +115 q^{-72} -272 q^{-74} +316 q^{-76} -223 q^{-78} +35 q^{-80} +161 q^{-82} -302 q^{-84} +329 q^{-86} -249 q^{-88} +100 q^{-90} +51 q^{-92} -161 q^{-94} +198 q^{-96} -170 q^{-98} +108 q^{-100} -33 q^{-102} -25 q^{-104} +53 q^{-106} -62 q^{-108} +50 q^{-110} -31 q^{-112} +14 q^{-114} + q^{-116} -8 q^{-118} +9 q^{-120} -8 q^{-122} +5 q^{-124} -2 q^{-126} + q^{-128} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_95.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^5+3 q^3-4 q+4 q^{-1} -2 q^{-3} +2 q^{-5} +2 q^{-7} -3 q^{-9} +4 q^{-11} -4 q^{-13} +2 q^{-15} - q^{-17} }[/math]
2	[math]\displaystyle{ q^{16}-3 q^{14}+11 q^{10}-13 q^8-12 q^6+33 q^4-10 q^2-34+41 q^{-2} +7 q^{-4} -43 q^{-6} +23 q^{-8} +21 q^{-10} -26 q^{-12} -5 q^{-14} +23 q^{-16} +5 q^{-18} -33 q^{-20} +12 q^{-22} +34 q^{-24} -41 q^{-26} -7 q^{-28} +43 q^{-30} -24 q^{-32} -17 q^{-34} +26 q^{-36} -5 q^{-38} -10 q^{-40} +7 q^{-42} -2 q^{-46} + q^{-48} }[/math]
3	[math]\displaystyle{ -q^{33}+3 q^{31}-7 q^{27}-2 q^{25}+17 q^{23}+15 q^{21}-39 q^{19}-41 q^{17}+53 q^{15}+93 q^{13}-47 q^{11}-166 q^9+13 q^7+234 q^5+60 q^3-275 q-165 q^{-1} +279 q^{-3} +262 q^{-5} -228 q^{-7} -331 q^{-9} +150 q^{-11} +358 q^{-13} -52 q^{-15} -342 q^{-17} -33 q^{-19} +288 q^{-21} +112 q^{-23} -216 q^{-25} -178 q^{-27} +131 q^{-29} +234 q^{-31} -37 q^{-33} -282 q^{-35} -59 q^{-37} +298 q^{-39} +168 q^{-41} -293 q^{-43} -259 q^{-45} +242 q^{-47} +324 q^{-49} -158 q^{-51} -345 q^{-53} +57 q^{-55} +318 q^{-57} +25 q^{-59} -244 q^{-61} -79 q^{-63} +160 q^{-65} +92 q^{-67} -86 q^{-69} -72 q^{-71} +33 q^{-73} +46 q^{-75} -10 q^{-77} -23 q^{-79} +3 q^{-81} +9 q^{-83} - q^{-85} -3 q^{-87} +2 q^{-91} - q^{-93} }[/math]
4	[math]\displaystyle{ q^{56}-3 q^{54}+7 q^{50}-2 q^{48}-2 q^{46}-20 q^{44}+2 q^{42}+49 q^{40}+17 q^{38}-15 q^{36}-129 q^{34}-69 q^{32}+170 q^{30}+217 q^{28}+112 q^{26}-389 q^{24}-505 q^{22}+75 q^{20}+660 q^{18}+855 q^{16}-312 q^{14}-1340 q^{12}-897 q^{10}+628 q^8+2133 q^6+913 q^4-1538 q^2-2513-775 q^{-2} +2617 q^{-4} +2803 q^{-6} -195 q^{-8} -3208 q^{-10} -2770 q^{-12} +1435 q^{-14} +3634 q^{-16} +1750 q^{-18} -2218 q^{-20} -3640 q^{-22} -396 q^{-24} +2807 q^{-26} +2725 q^{-28} -555 q^{-30} -3004 q^{-32} -1583 q^{-34} +1303 q^{-36} +2581 q^{-38} +751 q^{-40} -1791 q^{-42} -2111 q^{-44} -71 q^{-46} +2095 q^{-48} +1781 q^{-50} -554 q^{-52} -2525 q^{-54} -1450 q^{-56} +1476 q^{-58} +2807 q^{-60} +950 q^{-62} -2587 q^{-64} -2922 q^{-66} +204 q^{-68} +3264 q^{-70} +2706 q^{-72} -1554 q^{-74} -3645 q^{-76} -1619 q^{-78} +2276 q^{-80} +3619 q^{-82} +351 q^{-84} -2691 q^{-86} -2685 q^{-88} +301 q^{-90} +2772 q^{-92} +1572 q^{-94} -772 q^{-96} -2079 q^{-98} -946 q^{-100} +1047 q^{-102} +1269 q^{-104} +393 q^{-106} -769 q^{-108} -814 q^{-110} +30 q^{-112} +420 q^{-114} +403 q^{-116} -65 q^{-118} -276 q^{-120} -96 q^{-122} +26 q^{-124} +126 q^{-126} +27 q^{-128} -44 q^{-130} -16 q^{-132} -14 q^{-134} +20 q^{-136} +5 q^{-138} -7 q^{-140} +2 q^{-142} -3 q^{-144} +3 q^{-146} -2 q^{-150} + q^{-152} }[/math]
5	[math]\displaystyle{ -q^{85}+3 q^{83}-7 q^{79}+2 q^{77}+6 q^{75}+5 q^{73}+3 q^{71}-12 q^{69}-36 q^{67}-14 q^{65}+57 q^{63}+93 q^{61}+51 q^{59}-90 q^{57}-240 q^{55}-234 q^{53}+86 q^{51}+541 q^{49}+638 q^{47}+129 q^{45}-824 q^{43}-1462 q^{41}-928 q^{39}+882 q^{37}+2657 q^{35}+2557 q^{33}-56 q^{31}-3726 q^{29}-5263 q^{27}-2350 q^{25}+3879 q^{23}+8535 q^{21}+6693 q^{19}-1858 q^{17}-11172 q^{15}-12772 q^{13}-3169 q^{11}+11624 q^9+19211 q^7+11109 q^5-8402 q^3-23950 q-20838 q^{-1} +1133 q^{-3} +25148 q^{-5} +30006 q^{-7} +9156 q^{-9} -21625 q^{-11} -36241 q^{-13} -20461 q^{-15} +13964 q^{-17} +37948 q^{-19} +30068 q^{-21} -3791 q^{-23} -34785 q^{-25} -36085 q^{-27} -6534 q^{-29} +27940 q^{-31} +37600 q^{-33} +14968 q^{-35} -19241 q^{-37} -35145 q^{-39} -20297 q^{-41} +10569 q^{-43} +30063 q^{-45} +22581 q^{-47} -3290 q^{-49} -24034 q^{-51} -22572 q^{-53} -2186 q^{-55} +18287 q^{-57} +21612 q^{-59} +6230 q^{-61} -13611 q^{-63} -20783 q^{-65} -9643 q^{-67} +9842 q^{-69} +20739 q^{-71} +13429 q^{-73} -6337 q^{-75} -21582 q^{-77} -18138 q^{-79} +2248 q^{-81} +22434 q^{-83} +23889 q^{-85} +3420 q^{-87} -22344 q^{-89} -29999 q^{-91} -10776 q^{-93} +19913 q^{-95} +35029 q^{-97} +19587 q^{-99} -14435 q^{-101} -37440 q^{-103} -28345 q^{-105} +5937 q^{-107} +35806 q^{-109} +35172 q^{-111} +4428 q^{-113} -29726 q^{-115} -38195 q^{-117} -14591 q^{-119} +20089 q^{-121} +36389 q^{-123} +22115 q^{-125} -8823 q^{-127} -30021 q^{-129} -25468 q^{-131} -1438 q^{-133} +20855 q^{-135} +24133 q^{-137} +8550 q^{-139} -11133 q^{-141} -19273 q^{-143} -11657 q^{-145} +3213 q^{-147} +12844 q^{-149} +11108 q^{-151} +1722 q^{-153} -6768 q^{-155} -8422 q^{-157} -3705 q^{-159} +2438 q^{-161} +5199 q^{-163} +3534 q^{-165} -73 q^{-167} -2533 q^{-169} -2481 q^{-171} -748 q^{-173} +921 q^{-175} +1365 q^{-177} +714 q^{-179} -169 q^{-181} -590 q^{-183} -443 q^{-185} -65 q^{-187} +206 q^{-189} +212 q^{-191} +64 q^{-193} -52 q^{-195} -70 q^{-197} -38 q^{-199} + q^{-201} +30 q^{-203} +14 q^{-205} -7 q^{-207} -2 q^{-209} - q^{-211} -4 q^{-213} +2 q^{-215} +3 q^{-217} -3 q^{-219} +2 q^{-223} - q^{-225} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^6+2 q^4-q^2-1+3 q^{-2} -3 q^{-4} +3 q^{-6} + q^{-10} +3 q^{-12} -2 q^{-14} +3 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-22} - q^{-24} }[/math]
1,1	[math]\displaystyle{ q^{20}-6 q^{18}+20 q^{16}-50 q^{14}+109 q^{12}-214 q^{10}+376 q^8-600 q^6+876 q^4-1184 q^2+1472-1672 q^{-2} +1720 q^{-4} -1564 q^{-6} +1184 q^{-8} -572 q^{-10} -200 q^{-12} +1064 q^{-14} -1902 q^{-16} +2622 q^{-18} -3128 q^{-20} +3366 q^{-22} -3296 q^{-24} +2936 q^{-26} -2322 q^{-28} +1538 q^{-30} -692 q^{-32} -134 q^{-34} +827 q^{-36} -1328 q^{-38} +1598 q^{-40} -1654 q^{-42} +1540 q^{-44} -1310 q^{-46} +1030 q^{-48} -756 q^{-50} +518 q^{-52} -332 q^{-54} +200 q^{-56} -114 q^{-58} +60 q^{-60} -28 q^{-62} +12 q^{-64} -4 q^{-66} + q^{-68} }[/math]
2,0	[math]\displaystyle{ q^{18}-2 q^{16}-2 q^{14}+6 q^{12}+3 q^{10}-9 q^8-8 q^6+12 q^4+10 q^2-18-7 q^{-2} +21 q^{-4} +6 q^{-6} -18 q^{-8} + q^{-10} +17 q^{-12} -4 q^{-14} -8 q^{-16} +8 q^{-18} +5 q^{-20} -12 q^{-22} +8 q^{-24} +10 q^{-26} -15 q^{-28} -4 q^{-30} +18 q^{-32} +3 q^{-34} -21 q^{-36} - q^{-38} +17 q^{-40} -3 q^{-42} -17 q^{-44} + q^{-46} +11 q^{-48} - q^{-50} -5 q^{-52} +3 q^{-56} - q^{-60} + q^{-62} }[/math]

A3 Invariants.

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

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{16}-2 q^{14}-q^{12}+5 q^{10}-2 q^8-8 q^6+7 q^4+10 q^2-12-9 q^{-2} +19 q^{-4} +7 q^{-6} -27 q^{-8} +29 q^{-12} -11 q^{-14} -26 q^{-16} +21 q^{-18} +15 q^{-20} -22 q^{-22} +3 q^{-24} +31 q^{-26} -3 q^{-28} -13 q^{-30} +27 q^{-32} +13 q^{-34} -30 q^{-36} -3 q^{-38} +22 q^{-40} -18 q^{-42} -28 q^{-44} +11 q^{-46} +12 q^{-48} -16 q^{-50} -8 q^{-52} +14 q^{-54} +5 q^{-56} -8 q^{-58} + q^{-60} +5 q^{-62} - q^{-64} - q^{-66} + q^{-68} }[/math]
1,0,0,0	[math]\displaystyle{ -q^8+2 q^6-2 q^4+q^2+1-2 q^{-2} +3 q^{-4} -3 q^{-6} +2 q^{-8} - q^{-10} + q^{-12} + q^{-14} +2 q^{-16} +2 q^{-18} +2 q^{-20} +4 q^{-22} -2 q^{-24} +3 q^{-26} -3 q^{-28} - q^{-30} - q^{-32} -3 q^{-34} + q^{-36} - q^{-38} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{14}+3 q^{12}-7 q^{10}+13 q^8-21 q^6+30 q^4-36 q^2+39-39 q^{-2} +34 q^{-4} -22 q^{-6} +6 q^{-8} +14 q^{-10} -33 q^{-12} +53 q^{-14} -66 q^{-16} +76 q^{-18} -75 q^{-20} +70 q^{-22} -55 q^{-24} +38 q^{-26} -18 q^{-28} -2 q^{-30} +18 q^{-32} -31 q^{-34} +37 q^{-36} -39 q^{-38} +36 q^{-40} -30 q^{-42} +22 q^{-44} -15 q^{-46} +9 q^{-48} -5 q^{-50} +2 q^{-52} - q^{-54} }[/math]
1,0	[math]\displaystyle{ q^{24}-3 q^{20}-3 q^{18}+4 q^{16}+10 q^{14}-17 q^{10}-12 q^8+18 q^6+27 q^4-7 q^2-37-13 q^{-2} +36 q^{-4} +32 q^{-6} -22 q^{-8} -41 q^{-10} +2 q^{-12} +39 q^{-14} +13 q^{-16} -29 q^{-18} -20 q^{-20} +22 q^{-22} +24 q^{-24} -12 q^{-26} -24 q^{-28} +9 q^{-30} +28 q^{-32} -28 q^{-36} -4 q^{-38} +30 q^{-40} +14 q^{-42} -30 q^{-44} -27 q^{-46} +22 q^{-48} +37 q^{-50} -9 q^{-52} -44 q^{-54} -13 q^{-56} +34 q^{-58} +28 q^{-60} -17 q^{-62} -32 q^{-64} -2 q^{-66} +23 q^{-68} +12 q^{-70} -9 q^{-72} -12 q^{-74} +7 q^{-78} +3 q^{-80} -2 q^{-82} -2 q^{-84} + q^{-88} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{18}-3 q^{16}+4 q^{14}-6 q^{12}+11 q^{10}-17 q^8+20 q^6-23 q^4+30 q^2-32+30 q^{-2} -29 q^{-4} +29 q^{-6} -21 q^{-8} +7 q^{-10} -2 q^{-12} -6 q^{-14} +22 q^{-16} -35 q^{-18} +41 q^{-20} -45 q^{-22} +62 q^{-24} -55 q^{-26} +59 q^{-28} -52 q^{-30} +56 q^{-32} -37 q^{-34} +28 q^{-36} -25 q^{-38} +7 q^{-40} +3 q^{-42} -16 q^{-44} +14 q^{-46} -29 q^{-48} +31 q^{-50} -29 q^{-52} +28 q^{-54} -29 q^{-56} +25 q^{-58} -17 q^{-60} +14 q^{-62} -12 q^{-64} +8 q^{-66} -4 q^{-68} +3 q^{-70} -2 q^{-72} + q^{-74} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{32}-3 q^{30}+7 q^{28}-13 q^{26}+15 q^{24}-14 q^{22}+3 q^{20}+22 q^{18}-53 q^{16}+87 q^{14}-102 q^{12}+76 q^{10}-13 q^8-85 q^6+191 q^4-252 q^2+243-141 q^{-2} -36 q^{-4} +219 q^{-6} -340 q^{-8} +342 q^{-10} -220 q^{-12} +19 q^{-14} +179 q^{-16} -286 q^{-18} +261 q^{-20} -110 q^{-22} -88 q^{-24} +243 q^{-26} -278 q^{-28} +160 q^{-30} +52 q^{-32} -272 q^{-34} +413 q^{-36} -401 q^{-38} +247 q^{-40} +12 q^{-42} -279 q^{-44} +460 q^{-46} -491 q^{-48} +363 q^{-50} -118 q^{-52} -143 q^{-54} +331 q^{-56} -370 q^{-58} +273 q^{-60} -72 q^{-62} -133 q^{-64} +248 q^{-66} -232 q^{-68} +85 q^{-70} +115 q^{-72} -272 q^{-74} +316 q^{-76} -223 q^{-78} +35 q^{-80} +161 q^{-82} -302 q^{-84} +329 q^{-86} -249 q^{-88} +100 q^{-90} +51 q^{-92} -161 q^{-94} +198 q^{-96} -170 q^{-98} +108 q^{-100} -33 q^{-102} -25 q^{-104} +53 q^{-106} -62 q^{-108} +50 q^{-110} -31 q^{-112} +14 q^{-114} + q^{-116} -8 q^{-118} +9 q^{-120} -8 q^{-122} +5 q^{-124} -2 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 95"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^3-9 t^2+21 t-27+21 t^{-1} -9 t^{-2} +2 t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^6+3 z^4+3 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]=

{ 91, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^8+3 q^7-7 q^6+11 q^5-14 q^4+16 q^3-14 q^2+12 q-8+4 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} +2 z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -z^4+z^2 a^{-2} +5 z^2 a^{-4} -2 z^2 a^{-6} -z^2+3 a^{-4} -2 a^{-6} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 z^9 a^{-3} +2 z^9 a^{-5} +6 z^8 a^{-2} +11 z^8 a^{-4} +5 z^8 a^{-6} +7 z^7 a^{-1} +10 z^7 a^{-3} +8 z^7 a^{-5} +5 z^7 a^{-7} -6 z^6 a^{-2} -19 z^6 a^{-4} -6 z^6 a^{-6} +3 z^6 a^{-8} +4 z^6+a z^5-12 z^5 a^{-1} -25 z^5 a^{-3} -21 z^5 a^{-5} -8 z^5 a^{-7} +z^5 a^{-9} -2 z^4 a^{-2} +13 z^4 a^{-4} +4 z^4 a^{-6} -5 z^4 a^{-8} -6 z^4-a z^3+5 z^3 a^{-1} +16 z^3 a^{-3} +17 z^3 a^{-5} +5 z^3 a^{-7} -2 z^3 a^{-9} +z^2 a^{-2} -7 z^2 a^{-4} -4 z^2 a^{-6} +2 z^2 a^{-8} +2 z^2-z a^{-1} -3 z a^{-3} -5 z a^{-5} -2 z a^{-7} +z a^{-9} +3 a^{-4} +2 a^{-6} }[/math]

Vassiliev invariants

V2 and V3:

(3, 5)

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{ 12 }[/math] [math]\displaystyle{ 40 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 158 }[/math] [math]\displaystyle{ 10 }[/math] [math]\displaystyle{ 480 }[/math] [math]\displaystyle{ \frac{2320}{3} }[/math] [math]\displaystyle{ \frac{352}{3} }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 800 }[/math] [math]\displaystyle{ 1896 }[/math] [math]\displaystyle{ 120 }[/math] [math]\displaystyle{ \frac{37871}{10} }[/math] [math]\displaystyle{ \frac{1258}{5} }[/math] [math]\displaystyle{ \frac{16502}{15} }[/math] [math]\displaystyle{ \frac{209}{6} }[/math] [math]\displaystyle{ \frac{911}{10} }[/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 95. 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										2		2
13									5	1		-4
11								6	2			4
9							8	5				-3
7						8	6					2
5					6	8						2
3				6	8							-2
1			3	7								4
-1		1	5									-4
-3		3										3
-5	1											-1

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, 95]]
Out[2]=	10
In[3]:=	PD[Knot[10, 95]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 17, 12, 16], X[15, 9, 16, 8], X[19, 7, 20, 6], X[5, 15, 6, 14], X[7, 19, 8, 18], X[13, 1, 14, 20], X[17, 13, 18, 12], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[10, 95]]
Out[4]=	GaussCode[-1, 10, -2, 1, -6, 5, -7, 4, -10, 2, -3, 9, -8, 6, -4, 3, -9, 7, -5, 8]
In[5]:=	BR[Knot[10, 95]]
Out[5]=	BR[4, {-1, -1, 2, 2, -3, 2, -1, 2, 3, 3, 2}]
In[6]:=	alex = Alexander[Knot[10, 95]][t]
Out[6]=	2 9 21 2 3 -27 + -- - -- + -- + 21 t - 9 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 95]][z]
Out[7]=	2 4 6 1 + 3 z + 3 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 95]}
In[9]:=	{KnotDet[Knot[10, 95]], KnotSignature[Knot[10, 95]]}
Out[9]=	{91, 2}
In[10]:=	J=Jones[Knot[10, 95]][q]
Out[10]=	-2 4 2 3 4 5 6 7 8 -8 - q + - + 12 q - 14 q + 16 q - 14 q + 11 q - 7 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 95]}
In[12]:=	A2Invariant[Knot[10, 95]][q]
Out[12]=	-6 2 -2 2 4 6 10 12 14 -1 - q + -- - q + 3 q - 3 q + 3 q + q + 3 q - 2 q + 4 q 16 18 20 22 24 3 q - 2 q - 2 q + q - q
In[13]:=	Kauffman[Knot[10, 95]][a, z]
Out[13]=	2 2 2 2 2 3 z 2 z 5 z 3 z z 2 2 z 4 z 7 z z -- + -- + -- - --- - --- - --- - - + 2 z + ---- - ---- - ---- + -- - 6 4 9 7 5 3 a 8 6 4 2 a a a a a a a a a a 3 3 3 3 3 4 4 2 z 5 z 17 z 16 z 5 z 3 4 5 z 4 z ---- + ---- + ----- + ----- + ---- - a z - 6 z - ---- + ---- + 9 7 5 3 a 8 6 a a a a a a 4 4 5 5 5 5 5 13 z 2 z z 8 z 21 z 25 z 12 z 5 6 ----- - ---- + -- - ---- - ----- - ----- - ----- + a z + 4 z + 4 2 9 7 5 3 a a a a a a a 6 6 6 6 7 7 7 7 8 3 z 6 z 19 z 6 z 5 z 8 z 10 z 7 z 5 z ---- - ---- - ----- - ---- + ---- + ---- + ----- + ---- + ---- + 8 6 4 2 7 5 3 a 6 a a a a a a a a 8 8 9 9 11 z 6 z 2 z 2 z ----- + ---- + ---- + ---- 4 2 5 3 a a a a
In[14]:=	{Vassiliev[2][Knot[10, 95]], Vassiliev[3][Knot[10, 95]]}
Out[14]=	{0, 5}
In[15]:=	Kh[Knot[10, 95]][q, t]
Out[15]=	3 1 3 1 5 3 q 3 5 7 q + 6 q + ----- + ----- + ---- + --- + --- + 8 q t + 6 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 8 q t + 8 q t + 6 q t + 8 q t + 5 q t + 6 q t + 11 5 13 5 13 6 15 6 17 7 2 q t + 5 q t + q t + 2 q t + q t