10 64

10_63

10_65

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10 64 Further Notes and Views

Knot presentations

Planar diagram presentation	X8291 X10,4,11,3 X2,10,3,9 X18,12,19,11 X14,5,15,6 X4,17,5,18 X16,7,17,8 X6,15,7,16 X20,14,1,13 X12,20,13,19
Gauss code	1, -3, 2, -6, 5, -8, 7, -1, 3, -2, 4, -10, 9, -5, 8, -7, 6, -4, 10, -9
Dowker-Thistlethwaite code	8 10 14 16 2 18 20 6 4 12
Conway Notation	[31,3,3]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-9] Hyperbolic Volume 10.8681 A-Polynomial See Data:10 64/A-polynomial

[edit Notes for 10 64's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $4$ Rasmussen s-Invariant -2

[edit Notes for 10 64's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+3t^{3}-6t^{2}+10t-11+10t^{-1}-6t^{-2}+3t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-5z^{6}-8z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 51, 2 }
Jones polynomial	$q^{7}-2q^{6}+4q^{5}-7q^{4}+8q^{3}-8q^{2}+8q-6+4q^{-1}-2q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-7z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-18z^{4}a^{-2}+5z^{4}a^{-4}+5z^{4}-19z^{2}a^{-2}+8z^{2}a^{-4}+8z^{2}-6a^{-2}+3a^{-4}+4$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+5z^{8}a^{-2}+3z^{8}a^{-4}+2z^{8}+2az^{7}+2z^{7}a^{-3}+4z^{7}a^{-5}+a^{2}z^{6}-18z^{6}a^{-2}-8z^{6}a^{-4}+3z^{6}a^{-6}-6z^{6}-7az^{5}-5z^{5}a^{-1}-11z^{5}a^{-3}-11z^{5}a^{-5}+2z^{5}a^{-7}-4a^{2}z^{4}+30z^{4}a^{-2}+13z^{4}a^{-4}-5z^{4}a^{-6}+z^{4}a^{-8}+7z^{4}+6az^{3}+4z^{3}a^{-1}+16z^{3}a^{-3}+15z^{3}a^{-5}-3z^{3}a^{-7}+4a^{2}z^{2}-26z^{2}a^{-2}-8z^{2}a^{-4}+3z^{2}a^{-6}-2z^{2}a^{-8}-9z^{2}-az-3za^{-1}-6za^{-3}-4za^{-5}+6a^{-2}+3a^{-4}+4$
The A2 invariant	$q^{8}+2q^{4}+q^{-2}-2q^{-4}+2q^{-6}-2q^{-8}-q^{-12}-q^{-14}+2q^{-16}+q^{-20}$
The G2 invariant	$q^{46}-q^{44}+3q^{42}-5q^{40}+4q^{38}-3q^{36}-2q^{34}+9q^{32}-14q^{30}+20q^{28}-18q^{26}+9q^{24}+5q^{22}-22q^{20}+37q^{18}-41q^{16}+33q^{14}-10q^{12}-16q^{10}+43q^{8}-48q^{6}+42q^{4}-16q^{2}-11+32q^{-2}-38q^{-4}+24q^{-6}+4q^{-8}-26q^{-10}+42q^{-12}-30q^{-14}+4q^{-16}+21q^{-18}-49q^{-20}+58q^{-22}-52q^{-24}+18q^{-26}+13q^{-28}-48q^{-30}+68q^{-32}-63q^{-34}+33q^{-36}-5q^{-38}-28q^{-40}+46q^{-42}-49q^{-44}+26q^{-46}+3q^{-48}-21q^{-50}+34q^{-52}-22q^{-54}-3q^{-56}+28q^{-58}-36q^{-60}+34q^{-62}-18q^{-64}-5q^{-66}+29q^{-68}-39q^{-70}+43q^{-72}-27q^{-74}+9q^{-76}+8q^{-78}-21q^{-80}+23q^{-82}-23q^{-84}+18q^{-86}-8q^{-88}-q^{-90}+7q^{-92}-10q^{-94}+9q^{-96}-7q^{-98}+5q^{-100}-2q^{-102}-q^{-104}+2q^{-106}-3q^{-108}+2q^{-110}-q^{-112}+q^{-114}$

Further Quantum Invariants[hide]

Further quantum knot invariants for 10_64.

A1 Invariants.

Weight	Invariant
1	$q^{7}-q^{5}+2q^{3}-2q+2q^{-1}+q^{-7}-3q^{-9}+2q^{-11}-q^{-13}+q^{-15}$
2	$q^{22}-q^{20}-q^{18}+4q^{16}-q^{14}-5q^{12}+7q^{10}+q^{8}-11q^{6}+6q^{4}+6q^{2}-11+2q^{-2}+10q^{-4}-5q^{-6}-3q^{-8}+7q^{-10}+2q^{-12}-6q^{-14}-2q^{-16}+9q^{-18}-7q^{-20}-7q^{-22}+13q^{-24}-3q^{-26}-7q^{-28}+7q^{-30}-4q^{-34}+2q^{-36}-q^{-40}+q^{-42}$
3	$q^{45}-q^{43}-q^{41}+q^{39}+3q^{37}-q^{35}-5q^{33}+9q^{29}+2q^{27}-13q^{25}-9q^{23}+16q^{21}+18q^{19}-14q^{17}-29q^{15}+6q^{13}+37q^{11}+6q^{9}-37q^{7}-20q^{5}+33q^{3}+33q-25q^{-1}-39q^{-3}+15q^{-5}+43q^{-7}-3q^{-9}-41q^{-11}-3q^{-13}+38q^{-15}+11q^{-17}-29q^{-19}-19q^{-21}+16q^{-23}+26q^{-25}-7q^{-27}-33q^{-29}-11q^{-31}+37q^{-33}+22q^{-35}-33q^{-37}-33q^{-39}+30q^{-41}+40q^{-43}-19q^{-45}-34q^{-47}+5q^{-49}+29q^{-51}-18q^{-55}-5q^{-57}+9q^{-59}+4q^{-61}-4q^{-63}-2q^{-65}+2q^{-67}+q^{-69}-q^{-71}-q^{-79}+q^{-81}$
4	$q^{76}-q^{74}-q^{72}+q^{70}+3q^{66}-3q^{64}-3q^{62}+3q^{60}+9q^{56}-6q^{54}-13q^{52}-q^{50}+4q^{48}+31q^{46}+3q^{44}-29q^{42}-32q^{40}-20q^{38}+60q^{36}+57q^{34}+q^{32}-66q^{30}-103q^{28}+23q^{26}+103q^{24}+105q^{22}-7q^{20}-166q^{18}-102q^{16}+32q^{14}+175q^{12}+141q^{10}-96q^{8}-186q^{6}-121q^{4}+117q^{2}+233+46q^{-2}-150q^{-4}-214q^{-6}+2q^{-8}+212q^{-10}+137q^{-12}-70q^{-14}-205q^{-16}-58q^{-18}+144q^{-20}+143q^{-22}-19q^{-24}-154q^{-26}-79q^{-28}+76q^{-30}+134q^{-32}+26q^{-34}-102q^{-36}-114q^{-38}-8q^{-40}+128q^{-42}+109q^{-44}-12q^{-46}-157q^{-48}-140q^{-50}+80q^{-52}+191q^{-54}+129q^{-56}-127q^{-58}-252q^{-60}-39q^{-62}+168q^{-64}+231q^{-66}-9q^{-68}-227q^{-70}-133q^{-72}+38q^{-74}+192q^{-76}+90q^{-78}-91q^{-80}-108q^{-82}-54q^{-84}+73q^{-86}+76q^{-88}+4q^{-90}-25q^{-92}-50q^{-94}+2q^{-96}+22q^{-98}+12q^{-100}+8q^{-102}-19q^{-104}-3q^{-106}+2q^{-108}+q^{-110}+7q^{-112}-6q^{-114}+q^{-118}-q^{-120}+3q^{-122}-2q^{-124}-q^{-130}+q^{-132}$
5	$q^{115}-q^{113}-q^{111}+q^{109}+q^{103}-q^{101}-2q^{99}+3q^{97}+4q^{95}-2q^{93}-3q^{91}-6q^{89}-7q^{87}+4q^{85}+18q^{83}+17q^{81}+2q^{79}-22q^{77}-41q^{75}-29q^{73}+17q^{71}+66q^{69}+76q^{67}+21q^{65}-71q^{63}-134q^{61}-106q^{59}+25q^{57}+178q^{55}+222q^{53}+89q^{51}-146q^{49}-322q^{47}-278q^{45}+10q^{43}+349q^{41}+467q^{39}+237q^{37}-224q^{35}-581q^{33}-539q^{31}-56q^{29}+540q^{27}+782q^{25}+440q^{23}-293q^{21}-881q^{19}-831q^{17}-103q^{15}+775q^{13}+1098q^{11}+560q^{9}-473q^{7}-1188q^{5}-962q^{3}+74q+1082q^{-1}+1208q^{-3}+334q^{-5}-834q^{-7}-1285q^{-9}-641q^{-11}+539q^{-13}+1207q^{-15}+801q^{-17}-258q^{-19}-1024q^{-21}-836q^{-23}+74q^{-25}+821q^{-27}+764q^{-29}+29q^{-31}-642q^{-33}-652q^{-35}-69q^{-37}+519q^{-39}+561q^{-41}+75q^{-43}-444q^{-45}-521q^{-47}-121q^{-49}+382q^{-51}+540q^{-53}+235q^{-55}-289q^{-57}-610q^{-59}-438q^{-61}+130q^{-63}+650q^{-65}+706q^{-67}+157q^{-69}-629q^{-71}-983q^{-73}-511q^{-75}+469q^{-77}+1167q^{-79}+931q^{-81}-167q^{-83}-1211q^{-85}-1259q^{-87}-229q^{-89}+1034q^{-91}+1442q^{-93}+631q^{-95}-697q^{-97}-1415q^{-99}-919q^{-101}+278q^{-103}+1148q^{-105}+1039q^{-107}+119q^{-109}-785q^{-111}-943q^{-113}-361q^{-115}+379q^{-117}+711q^{-119}+456q^{-121}-75q^{-123}-430q^{-125}-400q^{-127}-102q^{-129}+187q^{-131}+274q^{-133}+155q^{-135}-33q^{-137}-148q^{-139}-134q^{-141}-31q^{-143}+58q^{-145}+82q^{-147}+46q^{-149}-11q^{-151}-42q^{-153}-34q^{-155}-2q^{-157}+18q^{-159}+16q^{-161}+5q^{-163}-4q^{-165}-11q^{-167}-3q^{-169}+6q^{-171}+q^{-173}-q^{-175}+q^{-177}-2q^{-179}-q^{-181}+3q^{-183}+q^{-185}-2q^{-187}-q^{-193}+q^{-195}$
6	$q^{162}-q^{160}-q^{158}+q^{156}-2q^{150}+3q^{148}-2q^{144}+5q^{142}+2q^{140}-2q^{138}-12q^{136}-q^{134}-2q^{132}-2q^{130}+18q^{128}+21q^{126}+13q^{124}-22q^{122}-20q^{120}-36q^{118}-39q^{116}+13q^{114}+61q^{112}+90q^{110}+40q^{108}+10q^{106}-85q^{104}-168q^{102}-136q^{100}-16q^{98}+156q^{96}+230q^{94}+283q^{92}+105q^{90}-197q^{88}-435q^{86}-465q^{84}-213q^{82}+153q^{80}+662q^{78}+796q^{76}+484q^{74}-180q^{72}-855q^{70}-1166q^{68}-956q^{66}+49q^{64}+1097q^{62}+1710q^{60}+1413q^{58}+301q^{56}-1199q^{54}-2367q^{52}-2123q^{50}-747q^{48}+1304q^{46}+2843q^{44}+3052q^{42}+1454q^{40}-1330q^{38}-3457q^{36}-3970q^{34}-2171q^{32}+1038q^{30}+4145q^{28}+5036q^{26}+2908q^{24}-936q^{22}-4704q^{20}-5934q^{18}-3789q^{16}+1007q^{14}+5409q^{12}+6709q^{10}+4177q^{8}-1149q^{6}-5988q^{4}-7443q^{2}-4118+1718q^{-2}+6545q^{-4}+7487q^{-6}+3689q^{-8}-2416q^{-10}-7082q^{-12}-6985q^{-14}-2608q^{-16}+3284q^{-18}+6952q^{-20}+6005q^{-22}+1267q^{-24}-4129q^{-26}-6309q^{-28}-4384q^{-30}+240q^{-32}+4315q^{-34}+5192q^{-36}+2570q^{-38}-1545q^{-40}-3988q^{-42}-3573q^{-44}-753q^{-46}+2156q^{-48}+3214q^{-50}+1870q^{-52}-679q^{-54}-2296q^{-56}-2125q^{-58}-319q^{-60}+1503q^{-62}+2130q^{-64}+1094q^{-66}-793q^{-68}-2071q^{-70}-1920q^{-72}-298q^{-74}+1552q^{-76}+2589q^{-78}+1922q^{-80}-92q^{-82}-2336q^{-84}-3328q^{-86}-2201q^{-88}+394q^{-90}+3259q^{-92}+4354q^{-94}+2810q^{-96}-938q^{-98}-4531q^{-100}-5553q^{-102}-3206q^{-104}+1687q^{-106}+5972q^{-108}+6856q^{-110}+3130q^{-112}-2840q^{-114}-7325q^{-116}-7515q^{-118}-2727q^{-120}+3991q^{-122}+8464q^{-124}+7393q^{-126}+1768q^{-128}-4898q^{-130}-8594q^{-132}-6745q^{-134}-793q^{-136}+5505q^{-138}+7863q^{-140}+5436q^{-142}-17q^{-144}-5161q^{-146}-6676q^{-148}-4087q^{-150}+690q^{-152}+4276q^{-154}+5025q^{-156}+2857q^{-158}-717q^{-160}-3277q^{-162}-3556q^{-164}-1749q^{-166}+523q^{-168}+2151q^{-170}+2339q^{-172}+1156q^{-174}-367q^{-176}-1342q^{-178}-1342q^{-180}-762q^{-182}+130q^{-184}+785q^{-186}+817q^{-188}+432q^{-190}-55q^{-192}-360q^{-194}-483q^{-196}-291q^{-198}+33q^{-200}+213q^{-202}+236q^{-204}+147q^{-206}+26q^{-208}-129q^{-210}-147q^{-212}-57q^{-214}+8q^{-216}+51q^{-218}+61q^{-220}+49q^{-222}-18q^{-224}-42q^{-226}-17q^{-228}-7q^{-230}+4q^{-232}+10q^{-234}+19q^{-236}-3q^{-238}-11q^{-240}+q^{-242}-q^{-244}-2q^{-248}+5q^{-250}-q^{-252}-4q^{-254}+3q^{-256}+q^{-258}+q^{-260}-2q^{-262}-q^{-268}+q^{-270}$

A2 Invariants.

Weight	Invariant
1,0	$q^{8}+2q^{4}+q^{-2}-2q^{-4}+2q^{-6}-2q^{-8}-q^{-12}-q^{-14}+2q^{-16}+q^{-20}$
1,1	$q^{28}-2q^{26}+6q^{24}-14q^{22}+25q^{20}-36q^{18}+62q^{16}-84q^{14}+106q^{12}-130q^{10}+148q^{8}-154q^{6}+130q^{4}-100q^{2}+58+16q^{-2}-90q^{-4}+158q^{-6}-214q^{-8}+258q^{-10}-288q^{-12}+282q^{-14}-256q^{-16}+216q^{-18}-152q^{-20}+92q^{-22}-22q^{-24}-32q^{-26}+85q^{-28}-116q^{-30}+116q^{-32}-124q^{-34}+117q^{-36}-102q^{-38}+76q^{-40}-60q^{-42}+54q^{-44}-36q^{-46}+24q^{-48}-18q^{-50}+13q^{-52}-8q^{-54}+4q^{-56}-2q^{-58}+q^{-60}$
2,0	$q^{24}+2q^{18}+2q^{16}-q^{14}+q^{12}+3q^{10}-5q^{6}-2q^{4}+q^{2}-6-5q^{-2}+4q^{-4}+2q^{-6}+5q^{-10}+7q^{-12}+q^{-14}+4q^{-18}-q^{-20}-7q^{-22}-q^{-24}+3q^{-26}-4q^{-28}+4q^{-32}+q^{-34}-3q^{-36}-2q^{-38}+q^{-40}-q^{-42}-q^{-44}+q^{-46}+q^{-48}+q^{-52}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}-q^{18}+q^{16}+q^{14}-4q^{12}+2q^{10}+3q^{8}-4q^{6}+8q^{4}+6q^{2}-6+7q^{-2}+3q^{-4}-11q^{-6}-q^{-8}-6q^{-12}-4q^{-14}+q^{-16}+4q^{-18}-2q^{-20}+q^{-22}+11q^{-24}-3q^{-26}-4q^{-28}+10q^{-30}-5q^{-32}-6q^{-34}+6q^{-36}-q^{-38}-3q^{-40}+2q^{-42}-q^{-46}+q^{-48}$
1,0,0	$q^{9}+3q^{5}+3q-q^{-1}+q^{-3}-2q^{-5}-q^{-7}-q^{-9}-2q^{-11}-2q^{-15}+2q^{-17}-q^{-19}+3q^{-21}+q^{-25}$
1,0,1	$q^{34}-2q^{32}+5q^{30}-7q^{28}+4q^{26}+3q^{24}-12q^{22}+31q^{20}-23q^{18}+16q^{16}+14q^{14}-53q^{12}+72q^{10}-72q^{8}+33q^{6}+28q^{4}-85q^{2}+125-105q^{-2}+59q^{-4}-3q^{-6}-67q^{-8}+69q^{-10}-72q^{-12}+34q^{-14}-8q^{-16}+22q^{-18}-30q^{-20}+77q^{-22}-47q^{-24}+12q^{-26}+63q^{-28}-114q^{-30}+112q^{-32}-82q^{-34}+8q^{-36}+51q^{-38}-93q^{-40}+80q^{-42}-36q^{-44}-19q^{-46}+52q^{-48}-51q^{-50}+18q^{-52}+11q^{-54}-21q^{-56}+26q^{-58}-10q^{-60}-2q^{-62}+8q^{-64}-10q^{-66}+7q^{-68}-q^{-70}-3q^{-72}+3q^{-74}-2q^{-76}+q^{-78}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{22}+q^{18}+2q^{16}-q^{14}-q^{12}+2q^{10}-q^{8}-q^{6}+8q^{4}+5q^{2}+2+7q^{-2}+8q^{-4}-4q^{-6}-11q^{-8}-2q^{-10}-8q^{-12}-17q^{-14}-7q^{-16}+6q^{-18}-6q^{-20}+2q^{-22}+15q^{-24}+6q^{-26}+q^{-28}+8q^{-30}+7q^{-32}-6q^{-34}-3q^{-36}+3q^{-38}-2q^{-40}-8q^{-42}+q^{-44}+3q^{-46}-3q^{-48}+2q^{-52}+q^{-58}$
1,0,0,0	$q^{10}+3q^{6}+q^{4}+3q^{2}+2+q^{-4}-3q^{-6}-q^{-8}-4q^{-10}-q^{-12}-3q^{-14}-q^{-18}+q^{-20}+2q^{-22}+3q^{-26}+q^{-30}$

B2 Invariants.

Weight	Invariant
0,1	$q^{20}-q^{18}+3q^{16}-5q^{14}+6q^{12}-8q^{10}+11q^{8}-10q^{6}+12q^{4}-8q^{2}+6-q^{-2}-3q^{-4}+9q^{-6}-15q^{-8}+18q^{-10}-22q^{-12}+20q^{-14}-21q^{-16}+16q^{-18}-12q^{-20}+7q^{-22}-q^{-24}-3q^{-26}+8q^{-28}-10q^{-30}+11q^{-32}-10q^{-34}+10q^{-36}-7q^{-38}+5q^{-40}-4q^{-42}+2q^{-44}-q^{-46}+q^{-48}$
1,0	$q^{34}-q^{30}-q^{28}+2q^{26}+3q^{24}-2q^{22}-5q^{20}-q^{18}+6q^{16}+6q^{14}-5q^{12}-8q^{10}+2q^{8}+13q^{6}+6q^{4}-9q^{2}-9+5q^{-2}+11q^{-4}+q^{-6}-11q^{-8}-5q^{-10}+6q^{-12}+4q^{-14}-6q^{-16}-6q^{-18}+4q^{-20}+5q^{-22}-3q^{-24}-8q^{-26}+q^{-28}+8q^{-30}+2q^{-32}-8q^{-34}-4q^{-36}+9q^{-38}+9q^{-40}-4q^{-42}-10q^{-44}+q^{-46}+12q^{-48}+5q^{-50}-8q^{-52}-9q^{-54}+q^{-56}+8q^{-58}+2q^{-60}-4q^{-62}-4q^{-64}+3q^{-68}+q^{-70}-q^{-72}-q^{-74}+q^{-78}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{26}-q^{24}+2q^{22}-3q^{20}+4q^{18}-6q^{16}+5q^{14}-7q^{12}+10q^{10}-7q^{8}+11q^{6}-4q^{4}+12q^{2}+5q^{-2}+2q^{-4}-3q^{-6}+3q^{-8}-15q^{-10}+7q^{-12}-20q^{-14}+10q^{-16}-22q^{-18}+13q^{-20}-15q^{-22}+16q^{-24}-9q^{-26}+12q^{-28}-2q^{-30}+8q^{-32}+4q^{-34}-2q^{-36}+4q^{-38}-7q^{-40}+9q^{-42}-9q^{-44}+6q^{-46}-9q^{-48}+8q^{-50}-5q^{-52}+4q^{-54}-4q^{-56}+3q^{-58}-2q^{-60}+q^{-62}-q^{-64}+q^{-66}$

G2 Invariants.

Weight

Invariant

1,0

$q^{46}-q^{44}+3q^{42}-5q^{40}+4q^{38}-3q^{36}-2q^{34}+9q^{32}-14q^{30}+20q^{28}-18q^{26}+9q^{24}+5q^{22}-22q^{20}+37q^{18}-41q^{16}+33q^{14}-10q^{12}-16q^{10}+43q^{8}-48q^{6}+42q^{4}-16q^{2}-11+32q^{-2}-38q^{-4}+24q^{-6}+4q^{-8}-26q^{-10}+42q^{-12}-30q^{-14}+4q^{-16}+21q^{-18}-49q^{-20}+58q^{-22}-52q^{-24}+18q^{-26}+13q^{-28}-48q^{-30}+68q^{-32}-63q^{-34}+33q^{-36}-5q^{-38}-28q^{-40}+46q^{-42}-49q^{-44}+26q^{-46}+3q^{-48}-21q^{-50}+34q^{-52}-22q^{-54}-3q^{-56}+28q^{-58}-36q^{-60}+34q^{-62}-18q^{-64}-5q^{-66}+29q^{-68}-39q^{-70}+43q^{-72}-27q^{-74}+9q^{-76}+8q^{-78}-21q^{-80}+23q^{-82}-23q^{-84}+18q^{-86}-8q^{-88}-q^{-90}+7q^{-92}-10q^{-94}+9q^{-96}-7q^{-98}+5q^{-100}-2q^{-102}-q^{-104}+2q^{-106}-3q^{-108}+2q^{-110}-q^{-112}+q^{-114}$

.

Computer Talk[hide]

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 64"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+3t^{3}-6t^{2}+10t-11+10t^{-1}-6t^{-2}+3t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-5z^{6}-8z^{4}-3z^{2}+1$

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]=

{ 51, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-2q^{6}+4q^{5}-7q^{4}+8q^{3}-8q^{2}+8q-6+4q^{-1}-2q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-7z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-18z^{4}a^{-2}+5z^{4}a^{-4}+5z^{4}-19z^{2}a^{-2}+8z^{2}a^{-4}+8z^{2}-6a^{-2}+3a^{-4}+4$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+5z^{8}a^{-2}+3z^{8}a^{-4}+2z^{8}+2az^{7}+2z^{7}a^{-3}+4z^{7}a^{-5}+a^{2}z^{6}-18z^{6}a^{-2}-8z^{6}a^{-4}+3z^{6}a^{-6}-6z^{6}-7az^{5}-5z^{5}a^{-1}-11z^{5}a^{-3}-11z^{5}a^{-5}+2z^{5}a^{-7}-4a^{2}z^{4}+30z^{4}a^{-2}+13z^{4}a^{-4}-5z^{4}a^{-6}+z^{4}a^{-8}+7z^{4}+6az^{3}+4z^{3}a^{-1}+16z^{3}a^{-3}+15z^{3}a^{-5}-3z^{3}a^{-7}+4a^{2}z^{2}-26z^{2}a^{-2}-8z^{2}a^{-4}+3z^{2}a^{-6}-2z^{2}a^{-8}-9z^{2}-az-3za^{-1}-6za^{-3}-4za^{-5}+6a^{-2}+3a^{-4}+4$

Vassiliev invariants

V2 and V3:

(-3, -3)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-12$ $-24$ $72$ $146$ $102$ $288$ $624$ $192$ $168$ $-288$ $288$ $-1752$ $-1224$ $-{\frac {7551}{10}}$ ${\frac {20866}{15}}$ $-{\frac {39862}{15}}$ ${\frac {2879}{6}}$ $-{\frac {8031}{10}}$

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$2 is the signature of 10 64. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                     1 1 13                   1   -1 11                 3 1   2 9               4 1     -3 7             4 3       1 5           4 4         0 3         4 4           0 1       3 5             2 -1     1 3               -2 -3   1 3                 2 -5   1                   -1 -7 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

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

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 64]]
Out[2]=	10
In[3]:=	PD[Knot[10, 64]]
Out[3]=	PD[X[8, 2, 9, 1], X[10, 4, 11, 3], X[2, 10, 3, 9], X[18, 12, 19, 11], X[14, 5, 15, 6], X[4, 17, 5, 18], X[16, 7, 17, 8], X[6, 15, 7, 16], X[20, 14, 1, 13], X[12, 20, 13, 19]]
In[4]:=	GaussCode[Knot[10, 64]]
Out[4]=	GaussCode[1, -3, 2, -6, 5, -8, 7, -1, 3, -2, 4, -10, 9, -5, 8, -7, 6, -4, 10, -9]
In[5]:=	BR[Knot[10, 64]]
Out[5]=	BR[3, {1, 1, 1, -2, 1, 1, 1, -2, -2, -2}]
In[6]:=	alex = Alexander[Knot[10, 64]][t]
Out[6]=	-4 3 6 10 2 3 4 -11 - t + -- - -- + -- + 10 t - 6 t + 3 t - t 3 2 t t t
In[7]:=	Conway[Knot[10, 64]][z]
Out[7]=	2 4 6 8 1 - 3 z - 8 z - 5 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 64]}
In[9]:=	{KnotDet[Knot[10, 64]], KnotSignature[Knot[10, 64]]}
Out[9]=	{51, 2}
In[10]:=	J=Jones[Knot[10, 64]][q]
Out[10]=	-3 2 4 2 3 4 5 6 7 -6 + q - -- + - + 8 q - 8 q + 8 q - 7 q + 4 q - 2 q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 64]}
In[12]:=	A2Invariant[Knot[10, 64]][q]
Out[12]=	-8 2 2 4 6 8 12 14 16 20 q + -- + q - 2 q + 2 q - 2 q - q - q + 2 q + q 4 q
In[13]:=	Kauffman[Knot[10, 64]][a, z]
Out[13]=	2 2 2 3 6 4 z 6 z 3 z 2 2 z 3 z 8 z 4 + -- + -- - --- - --- - --- - a z - 9 z - ---- + ---- - ---- - 4 2 5 3 a 8 6 4 a a a a a a a 2 3 3 3 3 4 26 z 2 2 3 z 15 z 16 z 4 z 3 4 z ----- + 4 a z - ---- + ----- + ----- + ---- + 6 a z + 7 z + -- - 2 7 5 3 a 8 a a a a a 4 4 4 5 5 5 5 5 z 13 z 30 z 2 4 2 z 11 z 11 z 5 z ---- + ----- + ----- - 4 a z + ---- - ----- - ----- - ---- - 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 5 6 3 z 8 z 18 z 2 6 4 z 2 z 7 7 a z - 6 z + ---- - ---- - ----- + a z + ---- + ---- + 2 a z + 6 4 2 5 3 a a a a a 8 8 9 9 8 3 z 5 z z z 2 z + ---- + ---- + -- + -- 4 2 3 a a a a
In[14]:=	{Vassiliev[2][Knot[10, 64]], Vassiliev[3][Knot[10, 64]]}
Out[14]=	{0, -3}
In[15]:=	Kh[Knot[10, 64]][q, t]
Out[15]=	3 1 1 1 3 1 3 3 q 5 q + 4 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 4 q t + 4 q t + 4 q t + 4 q t + 3 q t + 4 q t + q t + 11 4 11 5 13 5 15 6 3 q t + q t + q t + q t