10 47: Difference between revisions

Revision as of 20:06, 28 August 2005

10_46

10_48

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10 47 Quick Notes

10 47 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₄₂₅ X₃₈₄₉ X_9,17,10,16 X_5,15,6,14 X_15,7,16,6 X_11,19,12,18 X_13,1,14,20 X_17,11,18,10 X_19,13,20,12 X₇₂₈₃
Gauss code	-1, 10, -2, 1, -4, 5, -10, 2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -9, 7
Dowker-Thistlethwaite code	4 8 14 2 16 18 20 6 10 12
Conway Notation	[5,21,2]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{2,3\}}
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[0][-12]
Hyperbolic Volume	9.38519
A-Polynomial	See Data:10 47/A-polynomial

[edit Notes for 10 47's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2}
Topological 4 genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2}
Concordance genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4}
Rasmussen s-Invariant	4

[edit Notes for 10 47's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^4-3 t^3+6 t^2-7 t+7-7 t^{-1} +6 t^{-2} -3 t^{-3} + t^{-4} }
Conway polynomial	$z^{8}+5z^{6}+8z^{4}+6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 41, 4 }
Jones polynomial	$-q^{9}+2q^{8}-4q^{7}+5q^{6}-6q^{5}+7q^{4}-5q^{3}+5q^{2}-3q+2-q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-z^{6}a^{-2}+7z^{6}a^{-4}-z^{6}a^{-6}-5z^{4}a^{-2}+18z^{4}a^{-4}-5z^{4}a^{-6}-7z^{2}a^{-2}+21z^{2}a^{-4}-8z^{2}a^{-6}-3a^{-2}+9a^{-4}-5a^{-6}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+5z^{8}a^{-4}+3z^{8}a^{-6}+z^{7}a^{-1}-z^{7}a^{-3}+z^{7}a^{-5}+3z^{7}a^{-7}-10z^{6}a^{-2}-23z^{6}a^{-4}-10z^{6}a^{-6}+3z^{6}a^{-8}-5z^{5}a^{-1}-11z^{5}a^{-3}-14z^{5}a^{-5}-5z^{5}a^{-7}+3z^{5}a^{-9}+15z^{4}a^{-2}+35z^{4}a^{-4}+15z^{4}a^{-6}-3z^{4}a^{-8}+2z^{4}a^{-10}+7z^{3}a^{-1}+20z^{3}a^{-3}+19z^{3}a^{-5}+2z^{3}a^{-7}-3z^{3}a^{-9}+z^{3}a^{-11}-9z^{2}a^{-2}-26z^{2}a^{-4}-15z^{2}a^{-6}+z^{2}a^{-8}-z^{2}a^{-10}-3za^{-1}-8za^{-3}-9za^{-5}-za^{-7}+2za^{-9}-za^{-11}+3a^{-2}+9a^{-4}+5a^{-6}$
The A2 invariant	$-q^{2}-q^{-2}+q^{-6}+q^{-8}+4q^{-10}+q^{-12}+3q^{-14}-q^{-18}-q^{-20}-2q^{-22}-q^{-26}$
The G2 invariant	$q^{12}-q^{10}+3q^{8}-4q^{6}+3q^{4}-3q^{2}-2+7q^{-2}-14q^{-4}+15q^{-6}-14q^{-8}+3q^{-10}+7q^{-12}-19q^{-14}+23q^{-16}-20q^{-18}+9q^{-20}+3q^{-22}-15q^{-24}+18q^{-26}-12q^{-28}+4q^{-30}+8q^{-32}-10q^{-34}+11q^{-36}-6q^{-40}+16q^{-42}-14q^{-44}+15q^{-46}-2q^{-48}-5q^{-50}+18q^{-52}-20q^{-54}+24q^{-56}-12q^{-58}+2q^{-60}+11q^{-62}-17q^{-64}+19q^{-66}-13q^{-68}+5q^{-70}+5q^{-72}-10q^{-74}+8q^{-76}-4q^{-78}-5q^{-80}+8q^{-82}-9q^{-84}+3q^{-88}-9q^{-90}+8q^{-92}-7q^{-94}+2q^{-96}-q^{-98}-4q^{-100}+4q^{-102}-7q^{-104}+6q^{-106}-6q^{-108}+5q^{-110}-3q^{-112}-2q^{-114}+6q^{-116}-10q^{-118}+11q^{-120}-7q^{-122}+3q^{-124}+q^{-126}-5q^{-128}+6q^{-130}-6q^{-132}+5q^{-134}-2q^{-136}+q^{-140}-2q^{-142}+2q^{-144}-q^{-146}+q^{-148}$

Further Quantum Invariants

Further quantum knot invariants for 10_47.

A1 Invariants.

Weight	Invariant
1	$-q^{3}+q-q^{-1}+2q^{-3}+2q^{-7}+q^{-9}-q^{-11}+q^{-13}-2q^{-15}+q^{-17}-q^{-19}$
2	$q^{12}-q^{10}-2q^{8}+3q^{6}-5q^{2}+3+4q^{-2}-5q^{-4}+5q^{-8}-2q^{-10}-q^{-12}+5q^{-14}+q^{-16}-2q^{-18}+2q^{-20}+3q^{-22}-3q^{-24}-2q^{-26}+3q^{-28}-5q^{-32}+2q^{-34}+q^{-36}-4q^{-38}+2q^{-40}-q^{-42}-2q^{-44}+3q^{-46}-q^{-50}+q^{-52}$
3	$-q^{27}+q^{25}+2q^{23}-4q^{19}-2q^{17}+6q^{15}+5q^{13}-6q^{11}-10q^{9}+2q^{7}+13q^{5}+3q^{3}-13q-8q^{-1}+8q^{-3}+14q^{-5}-3q^{-7}-12q^{-9}-4q^{-11}+10q^{-13}+9q^{-15}-3q^{-17}-11q^{-19}+q^{-21}+11q^{-23}+7q^{-25}-9q^{-27}-6q^{-29}+10q^{-31}+9q^{-33}-10q^{-35}-10q^{-37}+9q^{-39}+9q^{-41}-5q^{-43}-12q^{-45}+q^{-47}+8q^{-49}+6q^{-51}-7q^{-53}-11q^{-55}-2q^{-57}+14q^{-59}+4q^{-61}-11q^{-63}-12q^{-65}+6q^{-67}+11q^{-69}-7q^{-73}-4q^{-75}+6q^{-77}+6q^{-79}-q^{-81}-5q^{-83}-q^{-85}+4q^{-87}+2q^{-89}-2q^{-91}-q^{-93}+q^{-97}-q^{-99}$
4	$q^{48}-q^{46}-2q^{44}+q^{40}+6q^{38}-6q^{34}-6q^{32}-4q^{30}+15q^{28}+12q^{26}-2q^{24}-15q^{22}-25q^{20}+7q^{18}+24q^{16}+23q^{14}+q^{12}-39q^{10}-23q^{8}+2q^{6}+32q^{4}+36q^{2}-10-27q^{-2}-33q^{-4}-3q^{-6}+34q^{-8}+25q^{-10}+14q^{-12}-23q^{-14}-38q^{-16}-12q^{-18}+15q^{-20}+48q^{-22}+25q^{-24}-26q^{-26}-48q^{-28}-29q^{-30}+40q^{-32}+62q^{-34}+11q^{-36}-48q^{-38}-57q^{-40}+12q^{-42}+62q^{-44}+31q^{-46}-30q^{-48}-55q^{-50}-3q^{-52}+49q^{-54}+27q^{-56}-26q^{-58}-44q^{-60}-5q^{-62}+44q^{-64}+29q^{-66}-21q^{-68}-43q^{-70}-23q^{-72}+29q^{-74}+46q^{-76}+14q^{-78}-24q^{-80}-57q^{-82}-27q^{-84}+38q^{-86}+65q^{-88}+32q^{-90}-58q^{-92}-83q^{-94}-13q^{-96}+70q^{-98}+87q^{-100}-12q^{-102}-86q^{-104}-54q^{-106}+29q^{-108}+86q^{-110}+23q^{-112}-46q^{-114}-51q^{-116}-7q^{-118}+52q^{-120}+25q^{-122}-14q^{-124}-28q^{-126}-16q^{-128}+23q^{-130}+14q^{-132}+q^{-134}-11q^{-136}-13q^{-138}+7q^{-140}+4q^{-142}+3q^{-144}-2q^{-146}-5q^{-148}+2q^{-150}+q^{-154}-q^{-158}+q^{-160}$
5	$-q^{75}+q^{73}+2q^{71}-q^{67}-3q^{65}-4q^{63}+8q^{59}+8q^{57}+2q^{55}-7q^{53}-16q^{51}-14q^{49}+4q^{47}+26q^{45}+28q^{43}+10q^{41}-21q^{39}-47q^{37}-38q^{35}+5q^{33}+53q^{31}+64q^{29}+32q^{27}-31q^{25}-81q^{23}-74q^{21}-12q^{19}+64q^{17}+95q^{15}+65q^{13}-14q^{11}-83q^{9}-97q^{7}-48q^{5}+34q^{3}+84q+86q^{-1}+41q^{-3}-30q^{-5}-84q^{-7}-91q^{-9}-49q^{-11}+19q^{-13}+100q^{-15}+127q^{-17}+69q^{-19}-48q^{-21}-152q^{-23}-164q^{-25}-48q^{-27}+135q^{-29}+230q^{-31}+153q^{-33}-56q^{-35}-248q^{-37}-243q^{-39}-39q^{-41}+214q^{-43}+300q^{-45}+136q^{-47}-148q^{-49}-307q^{-51}-207q^{-53}+65q^{-55}+284q^{-57}+256q^{-59}+6q^{-61}-235q^{-63}-261q^{-65}-64q^{-67}+176q^{-69}+248q^{-71}+90q^{-73}-131q^{-75}-211q^{-77}-93q^{-79}+96q^{-81}+171q^{-83}+77q^{-85}-90q^{-87}-150q^{-89}-51q^{-91}+95q^{-93}+138q^{-95}+45q^{-97}-104q^{-99}-157q^{-101}-63q^{-103}+94q^{-105}+172q^{-107}+115q^{-109}-40q^{-111}-179q^{-113}-186q^{-115}-53q^{-117}+134q^{-119}+250q^{-121}+173q^{-123}-47q^{-125}-260q^{-127}-294q^{-129}-81q^{-131}+231q^{-133}+365q^{-135}+203q^{-137}-131q^{-139}-380q^{-141}-303q^{-143}+38q^{-145}+333q^{-147}+334q^{-149}+52q^{-151}-250q^{-153}-315q^{-155}-99q^{-157}+175q^{-159}+257q^{-161}+105q^{-163}-115q^{-165}-188q^{-167}-89q^{-169}+74q^{-171}+139q^{-173}+62q^{-175}-54q^{-177}-97q^{-179}-43q^{-181}+38q^{-183}+67q^{-185}+32q^{-187}-27q^{-189}-49q^{-191}-26q^{-193}+17q^{-195}+31q^{-197}+18q^{-199}-5q^{-201}-19q^{-203}-14q^{-205}+q^{-207}+12q^{-209}+6q^{-211}+q^{-213}-3q^{-215}-5q^{-217}+3q^{-221}+q^{-223}-q^{-229}+q^{-233}-q^{-235}$
6	$q^{108}-q^{106}-2q^{104}+q^{100}+3q^{98}+q^{96}+4q^{94}-2q^{92}-10q^{90}-7q^{88}-2q^{86}+8q^{84}+10q^{82}+21q^{80}+8q^{78}-16q^{76}-30q^{74}-32q^{72}-11q^{70}+8q^{68}+59q^{66}+64q^{64}+30q^{62}-23q^{60}-78q^{58}-93q^{56}-82q^{54}+23q^{52}+108q^{50}+147q^{48}+110q^{46}+11q^{44}-106q^{42}-213q^{40}-164q^{38}-53q^{36}+109q^{34}+216q^{32}+229q^{30}+127q^{28}-85q^{26}-209q^{24}-263q^{22}-171q^{20}-13q^{18}+167q^{16}+265q^{14}+206q^{12}+103q^{10}-67q^{8}-176q^{6}-243q^{4}-193q^{2}-75+42q^{-2}+200q^{-4}+278q^{-6}+284q^{-8}+122q^{-10}-113q^{-12}-346q^{-14}-484q^{-16}-361q^{-18}-17q^{-20}+434q^{-22}+678q^{-24}+597q^{-26}+161q^{-28}-465q^{-30}-888q^{-32}-837q^{-34}-251q^{-36}+504q^{-38}+1045q^{-40}+1027q^{-42}+359q^{-44}-566q^{-46}-1199q^{-48}-1118q^{-50}-396q^{-52}+602q^{-54}+1296q^{-56}+1197q^{-58}+367q^{-60}-691q^{-62}-1316q^{-64}-1180q^{-66}-317q^{-68}+776q^{-70}+1356q^{-72}+1090q^{-74}+161q^{-76}-834q^{-78}-1323q^{-80}-971q^{-82}+26q^{-84}+949q^{-86}+1228q^{-88}+723q^{-90}-212q^{-92}-997q^{-94}-1099q^{-96}-439q^{-98}+443q^{-100}+965q^{-102}+820q^{-104}+153q^{-106}-578q^{-108}-861q^{-110}-491q^{-112}+154q^{-114}+598q^{-116}+578q^{-118}+153q^{-120}-336q^{-122}-521q^{-124}-253q^{-126}+185q^{-128}+419q^{-130}+298q^{-132}-74q^{-134}-403q^{-136}-425q^{-138}-90q^{-140}+350q^{-142}+558q^{-144}+391q^{-146}-60q^{-148}-510q^{-150}-679q^{-152}-444q^{-154}+100q^{-156}+627q^{-158}+836q^{-160}+576q^{-162}-44q^{-164}-716q^{-166}-1066q^{-168}-791q^{-170}-10q^{-172}+862q^{-174}+1295q^{-176}+997q^{-178}+52q^{-180}-1039q^{-182}-1516q^{-184}-1090q^{-186}+34q^{-188}+1155q^{-190}+1607q^{-192}+1067q^{-194}-184q^{-196}-1246q^{-198}-1514q^{-200}-831q^{-202}+302q^{-204}+1199q^{-206}+1278q^{-208}+515q^{-210}-424q^{-212}-997q^{-214}-864q^{-216}-262q^{-218}+432q^{-220}+722q^{-222}+468q^{-224}+42q^{-226}-324q^{-228}-374q^{-230}-217q^{-232}+52q^{-234}+186q^{-236}+114q^{-238}+37q^{-240}-39q^{-242}-21q^{-244}-9q^{-246}+32q^{-248}+6q^{-250}-67q^{-252}-54q^{-254}-25q^{-256}+49q^{-258}+61q^{-260}+66q^{-262}+11q^{-264}-61q^{-266}-58q^{-268}-40q^{-270}+15q^{-272}+30q^{-274}+47q^{-276}+22q^{-278}-16q^{-280}-20q^{-282}-24q^{-284}-3q^{-286}+2q^{-288}+17q^{-290}+10q^{-292}-2q^{-294}-2q^{-296}-8q^{-298}-q^{-300}-2q^{-302}+4q^{-304}+2q^{-306}-q^{-308}+q^{-310}-3q^{-312}+q^{-318}-q^{-322}+q^{-324}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{2}-q^{-2}+q^{-6}+q^{-8}+4q^{-10}+q^{-12}+3q^{-14}-q^{-18}-q^{-20}-2q^{-22}-q^{-26}$
1,1	$q^{12}-2q^{10}+6q^{8}-12q^{6}+21q^{4}-34q^{2}+48-62q^{-2}+65q^{-4}-72q^{-6}+58q^{-8}-42q^{-10}+14q^{-12}+16q^{-14}-38q^{-16}+78q^{-18}-80q^{-20}+108q^{-22}-94q^{-24}+102q^{-26}-84q^{-28}+60q^{-30}-48q^{-32}+16q^{-34}-9q^{-36}-14q^{-38}+14q^{-40}-20q^{-42}+17q^{-44}-12q^{-46}+12q^{-48}-14q^{-50}+19q^{-52}-20q^{-54}+22q^{-56}-28q^{-58}+26q^{-60}-22q^{-62}+18q^{-64}-14q^{-66}+11q^{-68}-6q^{-70}+4q^{-72}-2q^{-74}+q^{-76}$
2,0	$q^{10}-q^{6}+q^{2}-1-3q^{-2}+2q^{-6}-2q^{-8}-3q^{-10}+q^{-12}+q^{-14}-q^{-16}+q^{-18}+5q^{-20}+3q^{-22}+5q^{-24}+5q^{-26}+5q^{-28}+q^{-30}+4q^{-32}+4q^{-34}-2q^{-36}-3q^{-38}-2q^{-40}-4q^{-42}-8q^{-44}-6q^{-46}-3q^{-48}-q^{-50}-q^{-52}+2q^{-56}+2q^{-58}+2q^{-60}+q^{-62}+q^{-66}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{6}-q^{4}+q^{2}+1-3q^{-2}-3q^{-6}-6q^{-8}-q^{-10}-q^{-12}+9q^{-16}+8q^{-18}+7q^{-20}+11q^{-22}+6q^{-24}-2q^{-26}-q^{-28}-4q^{-30}-5q^{-32}-7q^{-34}-2q^{-36}-5q^{-40}+3q^{-44}-3q^{-46}-q^{-48}+5q^{-50}-2q^{-52}-2q^{-54}+3q^{-56}-q^{-60}+q^{-62}$
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q-2 q^{-3} -2 q^{-7} +2 q^{-9} + q^{-11} +4 q^{-13} +4 q^{-15} +4 q^{-17} +3 q^{-19} -4 q^{-25} - q^{-27} -3 q^{-29} - q^{-33} }
1,0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{12}-2 q^{10}+5 q^8-5 q^6+4 q^4+2 q^2-10+20 q^{-2} -22 q^{-4} +18 q^{-6} -12 q^{-8} -11 q^{-10} +7 q^{-12} -35 q^{-14} +17 q^{-16} -39 q^{-18} +23 q^{-20} -18 q^{-22} +31 q^{-24} +22 q^{-26} +22 q^{-28} +61 q^{-30} -9 q^{-32} +66 q^{-34} -34 q^{-36} +24 q^{-38} -38 q^{-40} -20 q^{-42} -19 q^{-44} -44 q^{-46} +18 q^{-48} -51 q^{-50} +34 q^{-52} -20 q^{-54} +7 q^{-56} +14 q^{-58} -6 q^{-60} +14 q^{-62} -5 q^{-64} +8 q^{-66} -6 q^{-68} +10 q^{-70} -12 q^{-72} +8 q^{-74} -3 q^{-76} -10 q^{-78} +17 q^{-80} -15 q^{-82} +5 q^{-84} +5 q^{-86} -9 q^{-88} +9 q^{-90} -3 q^{-92} - q^{-94} +3 q^{-96} -2 q^{-98} + q^{-100} }

A4 Invariants.

Weight

Invariant

0,1,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4+1+2 q^{-2} + q^{-4} - q^{-6} - q^{-8} -5 q^{-10} -8 q^{-12} -10 q^{-14} -11 q^{-16} -7 q^{-18} - q^{-20} +6 q^{-22} +11 q^{-24} +21 q^{-26} +24 q^{-28} +22 q^{-30} +13 q^{-32} +14 q^{-34} +4 q^{-36} -8 q^{-38} -10 q^{-40} -10 q^{-42} -16 q^{-44} -14 q^{-46} -6 q^{-48} -8 q^{-50} -6 q^{-52} +3 q^{-56} -2 q^{-58} +5 q^{-62} +2 q^{-64} -2 q^{-66} +2 q^{-68} +3 q^{-70} + q^{-76} }

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1-2 q^{-4} - q^{-6} -2 q^{-8} - q^{-10} + q^{-12} + q^{-14} +5 q^{-16} +4 q^{-18} +7 q^{-20} +4 q^{-22} +4 q^{-24} - q^{-28} -3 q^{-30} -4 q^{-32} -2 q^{-34} -3 q^{-36} - q^{-40} }

B2 Invariants.

Weight

Invariant

0,1

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^6+q^4-3 q^2+3-5 q^{-2} +6 q^{-4} -7 q^{-6} +6 q^{-8} -5 q^{-10} +5 q^{-12} + q^{-16} +6 q^{-18} -5 q^{-20} +11 q^{-22} -10 q^{-24} +12 q^{-26} -11 q^{-28} +10 q^{-30} -9 q^{-32} +5 q^{-34} -4 q^{-36} + q^{-40} -4 q^{-42} +5 q^{-44} -5 q^{-46} +5 q^{-48} -5 q^{-50} +4 q^{-52} -4 q^{-54} +3 q^{-56} -2 q^{-58} + q^{-60} - q^{-62} }

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{12}-q^8-q^6+2 q^4+2 q^2-2-4 q^{-2} +4 q^{-6} -7 q^{-10} -5 q^{-12} +3 q^{-14} +5 q^{-16} - q^{-18} -7 q^{-20} +7 q^{-24} +8 q^{-26} - q^{-30} +3 q^{-32} +8 q^{-34} +4 q^{-36} +4 q^{-42} + q^{-44} -4 q^{-46} -4 q^{-48} + q^{-50} +2 q^{-52} -4 q^{-54} -7 q^{-56} -2 q^{-58} +4 q^{-60} -5 q^{-64} -4 q^{-66} +2 q^{-68} +4 q^{-70} -4 q^{-74} -3 q^{-76} +2 q^{-78} +5 q^{-80} + q^{-82} -3 q^{-84} -3 q^{-86} + q^{-88} +3 q^{-90} + q^{-92} - q^{-94} - q^{-96} + q^{-100} }

D4 Invariants.

Weight

Invariant

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^6-q^4+2 q^2-2+4 q^{-2} -5 q^{-4} +3 q^{-6} -8 q^{-8} +2 q^{-10} -10 q^{-12} - q^{-14} -7 q^{-16} +2 q^{-18} + q^{-20} +4 q^{-22} +11 q^{-24} +8 q^{-26} +18 q^{-28} +4 q^{-30} +16 q^{-32} -4 q^{-34} +10 q^{-36} -12 q^{-38} +3 q^{-40} -14 q^{-42} -11 q^{-46} -5 q^{-50} - q^{-52} -3 q^{-56} +2 q^{-58} -3 q^{-60} +4 q^{-62} -4 q^{-64} +3 q^{-66} -3 q^{-68} +5 q^{-70} -3 q^{-72} +2 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} + q^{-82} - q^{-84} + q^{-86} }

G2 Invariants.

Weight

Invariant

1,0

q^{12}-q^{10}+3q^{8}-4q^{6}+3q^{4}-3q^{2}-2+7q^{-2}-14q^{-4}+15q^{-6}-14q^{-8}+3q^{-10}+7q^{-12}-19q^{-14}+23q^{-16}-20q^{-18}+9q^{-20}+3q^{-22}-15q^{-24}+18q^{-26}-12q^{-28}+4q^{-30}+8q^{-32}-10q^{-34}+11q^{-36}-6q^{-40}+16q^{-42}-14q^{-44}+15q^{-46}-2q^{-48}-5q^{-50}+18q^{-52}-20q^{-54}+24q^{-56}-12q^{-58}+2q^{-60}+11q^{-62}-17q^{-64}+19q^{-66}-13q^{-68}+5q^{-70}+5q^{-72}-10q^{-74}+8q^{-76}-4q^{-78}-5q^{-80}+8q^{-82}-9q^{-84}+3q^{-88}-9q^{-90}+8q^{-92}-7q^{-94}+2q^{-96}-q^{-98}-4q^{-100}+4q^{-102}-7q^{-104}+6q^{-106}-6q^{-108}+5q^{-110}-3q^{-112}-2q^{-114}+6q^{-116}-10q^{-118}+11q^{-120}-7q^{-122}+3q^{-124}+q^{-126}-5q^{-128}+6q^{-130}-6q^{-132}+5q^{-134}-2q^{-136}+q^{-140}-2q^{-142}+2q^{-144}-q^{-146}+q^{-148}

.

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.

In[3]:= K = Knot["10 47"];

In[4]:= Alexander[K][t]

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

Out[4]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^4-3 t^3+6 t^2-7 t+7-7 t^{-1} +6 t^{-2} -3 t^{-3} + t^{-4} }

In[5]:= Conway[K][z]

Out[5]= $z^{8}+5z^{6}+8z^{4}+6z^{2}+1$

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]= $\{1\}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 41, 4 }

In[8]:= Jones[K][q]

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

Out[8]= $-q^{9}+2q^{8}-4q^{7}+5q^{6}-6q^{5}+7q^{4}-5q^{3}+5q^{2}-3q+2-q^{-1}$

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= $z^{8}a^{-4}-z^{6}a^{-2}+7z^{6}a^{-4}-z^{6}a^{-6}-5z^{4}a^{-2}+18z^{4}a^{-4}-5z^{4}a^{-6}-7z^{2}a^{-2}+21z^{2}a^{-4}-8z^{2}a^{-6}-3a^{-2}+9a^{-4}-5a^{-6}$

In[10]:= Kauffman[K][a, z]

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

Out[10]= $z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+5z^{8}a^{-4}+3z^{8}a^{-6}+z^{7}a^{-1}-z^{7}a^{-3}+z^{7}a^{-5}+3z^{7}a^{-7}-10z^{6}a^{-2}-23z^{6}a^{-4}-10z^{6}a^{-6}+3z^{6}a^{-8}-5z^{5}a^{-1}-11z^{5}a^{-3}-14z^{5}a^{-5}-5z^{5}a^{-7}+3z^{5}a^{-9}+15z^{4}a^{-2}+35z^{4}a^{-4}+15z^{4}a^{-6}-3z^{4}a^{-8}+2z^{4}a^{-10}+7z^{3}a^{-1}+20z^{3}a^{-3}+19z^{3}a^{-5}+2z^{3}a^{-7}-3z^{3}a^{-9}+z^{3}a^{-11}-9z^{2}a^{-2}-26z^{2}a^{-4}-15z^{2}a^{-6}+z^{2}a^{-8}-z^{2}a^{-10}-3za^{-1}-8za^{-3}-9za^{-5}-za^{-7}+2za^{-9}-za^{-11}+3a^{-2}+9a^{-4}+5a^{-6}$

Vassiliev invariants

V₂ and V₃:

(6, 11)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 24}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 88}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 288}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 540}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 76}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2112}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{10096}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1696}{3}}

440

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2304}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3872}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 12960}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1824}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{110351}{5}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{11108}{15}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{40988}{5}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 155}

{\frac {5151}{5}}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 4 is the signature of 10 47. 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

15

3

1

-2

13

2

1

11

4

3

-1

9

3

2

1

7

2

4

2

5

3

0

3

1

3

2

1

2

-1

1

-3

1

-1

Integral Khovanov Homology

(db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=5}
$r=-3$	${\mathbb {Z} }$
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=5}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=6}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=7}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{Include}(\textrm{ColouredJonesM.mhtml})}

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 47]]
Out[2]=	10
In[3]:=	PD[Knot[10, 47]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[9, 17, 10, 16], X[5, 15, 6, 14], X[15, 7, 16, 6], X[11, 19, 12, 18], X[13, 1, 14, 20], X[17, 11, 18, 10], X[19, 13, 20, 12], X[7, 2, 8, 3]]
In[4]:=	GaussCode[Knot[10, 47]]
Out[4]=	GaussCode[-1, 10, -2, 1, -4, 5, -10, 2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -9, 7]
In[5]:=	BR[Knot[10, 47]]
Out[5]=	BR[3, {1, 1, 1, 1, 1, -2, 1, 1, -2, -2}]
In[6]:=	alex = Alexander[Knot[10, 47]][t]
Out[6]=	-4 3 6 7 2 3 4 7 + t - -- + -- - - - 7 t + 6 t - 3 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 47]][z]
Out[7]=	2 4 6 8 1 + 6 z + 8 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 47]}
In[9]:=	{KnotDet[Knot[10, 47]], KnotSignature[Knot[10, 47]]}
Out[9]=	{41, 4}
In[10]:=	J=Jones[Knot[10, 47]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 2 - - - 3 q + 5 q - 5 q + 7 q - 6 q + 5 q - 4 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 47]}
In[12]:=	A2Invariant[Knot[10, 47]][q]
Out[12]=	-2 2 6 8 10 12 14 18 20 22 26 -q - q + q + q + 4 q + q + 3 q - q - q - 2 q - q
In[13]:=	Kauffman[Knot[10, 47]][a, z]
Out[13]=	2 2 2 5 9 3 z 2 z z 9 z 8 z 3 z z z 15 z -- + -- + -- - --- + --- - -- - --- - --- - --- - --- + -- - ----- - 6 4 2 11 9 7 5 3 a 10 8 6 a a a a a a a a a a a 2 2 3 3 3 3 3 3 4 26 z 9 z z 3 z 2 z 19 z 20 z 7 z 2 z ----- - ---- + --- - ---- + ---- + ----- + ----- + ---- + ---- - 4 2 11 9 7 5 3 a 10 a a a a a a a a 4 4 4 4 5 5 5 5 5 3 z 15 z 35 z 15 z 3 z 5 z 14 z 11 z 5 z ---- + ----- + ----- + ----- + ---- - ---- - ----- - ----- - ---- + 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 7 7 7 7 8 8 3 z 10 z 23 z 10 z 3 z z z z 3 z 5 z ---- - ----- - ----- - ----- + ---- + -- - -- + -- + ---- + ---- + 8 6 4 2 7 5 3 a 6 4 a a a a a a a a a 8 9 9 2 z z z ---- + -- + -- 2 5 3 a a a
In[14]:=	{Vassiliev[2][Knot[10, 47]], Vassiliev[3][Knot[10, 47]]}
Out[14]=	{0, 11}
In[15]:=	Kh[Knot[10, 47]][q, t]
Out[15]=	3 3 5 1 1 q 2 q q 5 7 3 q + 3 q + ----- + ---- + -- + --- + -- + 3 q t + 2 q t + 3 3 2 2 t t q t q t t 7 2 9 2 9 3 11 3 11 4 13 4 4 q t + 3 q t + 2 q t + 4 q t + 3 q t + 2 q t + 13 5 15 5 15 6 17 6 19 7 q t + 3 q t + q t + q t + q t

@@ Line 1: / Line 1: @@
 <!--  -->
+<!-- -->
+<!--  -->
+<!-- -->
 <!-- provide an anchor so we can return to the top of the page -->
 <span id="top"></span>
+<!-- -->
 <!-- this relies on transclusion for next and previous links -->
 {{Knot Navigation Links|ext=gif}}
+{{Rolfsen Knot Page Header|n=10|k=47|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,10,-2,1,-4,5,-10,2,-3,8,-6,9,-7,4,-5,3,-8,6,-9,7/goTop.html}}
-{| align=left
-|- valign=top
-|[[Image:{{PAGENAME}}.gif]]
-|{{Rolfsen Knot Site Links|n=10|k=47|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,10,-2,1,-4,5,-10,2,-3,8,-6,9,-7,4,-5,3,-8,6,-9,7/goTop.html}}
-|{{:{{PAGENAME}} Quick Notes}}
-|}
 <br style="clear:both" />
@@ Line 24: / Line 21: @@
 {{Vassiliev Invariants}}
-===[[Khovanov Homology]]===
+{{Khovanov Homology|table=<table border=1>
-The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
-<center><table border=1>
 <tr align=center>
 <td width=13.3333%><table cellpadding=0 cellspacing=0>
@@ Line 48: / Line 41: @@
 <tr align=center><td>-1</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
 <tr align=center><td>-3</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
-</table></center>
+</table>}}
 {{Computer Talk Header}}
@@ Line 139: / Line 131: @@
   q   t  + 3 q   t  + q   t  + q   t  + q   t</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

10 47: Difference between revisions

Revision as of 20:06, 28 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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