10 94

10_93

10_95

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10 94 Quick Notes

10 94 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X2837 X18,12,19,11 X14,5,15,6 X20,14,1,13 X8,15,9,16 X10,4,11,3 X16,9,17,10 X4,17,5,18 X12,20,13,19
Gauss code	1, -2, 7, -9, 4, -1, 2, -6, 8, -7, 3, -10, 5, -4, 6, -8, 9, -3, 10, -5
Dowker-Thistlethwaite code	6 10 14 2 16 18 20 8 4 12
Conway Notation	[.30.2.2]

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-9] Hyperbolic Volume 13.3116 A-Polynomial See Data:10 94/A-polynomial

[edit Notes for 10 94'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 1} Topological 4 genus $1$ Concordance genus $4$ Rasmussen s-Invariant -2

[edit Notes for 10 94's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+4t^{3}-9t^{2}+14t-15+14t^{-1}-9t^{-2}+4t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-4z^{6}-5z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 71, 2 }
Jones polynomial	$q^{7}-3q^{6}+6q^{5}-9q^{4}+11q^{3}-12q^{2}+11q-8+6q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-13z^{4}a^{-2}+4z^{4}a^{-4}+4z^{4}-12z^{2}a^{-2}+5z^{2}a^{-4}+5z^{2}-4a^{-2}+2a^{-4}+3$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+5z^{8}a^{-4}+4z^{8}+3az^{7}+3z^{7}a^{-3}+6z^{7}a^{-5}+a^{2}z^{6}-27z^{6}a^{-2}-9z^{6}a^{-4}+5z^{6}a^{-6}-12z^{6}-9az^{5}-11z^{5}a^{-1}-15z^{5}a^{-3}-10z^{5}a^{-5}+3z^{5}a^{-7}-3a^{2}z^{4}+31z^{4}a^{-2}+10z^{4}a^{-4}-6z^{4}a^{-6}+z^{4}a^{-8}+11z^{4}+6az^{3}+10z^{3}a^{-1}+16z^{3}a^{-3}+9z^{3}a^{-5}-3z^{3}a^{-7}+2a^{2}z^{2}-18z^{2}a^{-2}-6z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}a^{-8}-7z^{2}-az-3za^{-1}-5za^{-3}-3za^{-5}+4a^{-2}+2a^{-4}+3$
The A2 invariant	$q^{8}-q^{6}+2q^{4}+1+2q^{-2}-3q^{-4}+2q^{-6}-3q^{-8}+q^{-10}-q^{-14}+2q^{-16}-q^{-18}+q^{-20}$
The G2 invariant	$q^{46}-2q^{44}+5q^{42}-9q^{40}+10q^{38}-9q^{36}+17q^{32}-34q^{30}+51q^{28}-55q^{26}+35q^{24}+5q^{22}-60q^{20}+111q^{18}-124q^{16}+97q^{14}-28q^{12}-55q^{10}+125q^{8}-149q^{6}+116q^{4}-36q^{2}-49+109q^{-2}-110q^{-4}+59q^{-6}+24q^{-8}-87q^{-10}+112q^{-12}-89q^{-14}+15q^{-16}+69q^{-18}-141q^{-20}+167q^{-22}-135q^{-24}+54q^{-26}+49q^{-28}-143q^{-30}+185q^{-32}-173q^{-34}+99q^{-36}-95q^{-40}+144q^{-42}-127q^{-44}+62q^{-46}+25q^{-48}-86q^{-50}+94q^{-52}-53q^{-54}-19q^{-56}+84q^{-58}-110q^{-60}+97q^{-62}-39q^{-64}-29q^{-66}+84q^{-68}-109q^{-70}+99q^{-72}-63q^{-74}+18q^{-76}+23q^{-78}-53q^{-80}+65q^{-82}-58q^{-84}+44q^{-86}-20q^{-88}-2q^{-90}+17q^{-92}-28q^{-94}+26q^{-96}-19q^{-98}+11q^{-100}-2q^{-102}-3q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}$

Further Quantum Invariants

Further quantum knot invariants for 10_94.

A1 Invariants.

Weight	Invariant
1	$q^{7}-2q^{5}+3q^{3}-2q+3q^{-1}-q^{-3}-q^{-5}+2q^{-7}-3q^{-9}+3q^{-11}-2q^{-13}+q^{-15}$
2	$q^{22}-2q^{20}-q^{18}+8q^{16}-4q^{14}-11q^{12}+15q^{10}+3q^{8}-22q^{6}+12q^{4}+15q^{2}-21+3q^{-2}+18q^{-4}-12q^{-6}-8q^{-8}+12q^{-10}+5q^{-12}-14q^{-14}-2q^{-16}+21q^{-18}-12q^{-20}-15q^{-22}+23q^{-24}-4q^{-26}-15q^{-28}+13q^{-30}-7q^{-34}+5q^{-36}-2q^{-40}+q^{-42}$
3	$q^{45}-2q^{43}-q^{41}+4q^{39}+5q^{37}-7q^{35}-16q^{33}+8q^{31}+32q^{29}+3q^{27}-47q^{25}-30q^{23}+56q^{21}+63q^{19}-43q^{17}-98q^{15}+9q^{13}+117q^{11}+36q^{9}-116q^{7}-76q^{5}+94q^{3}+110q-61q^{-1}-123q^{-3}+31q^{-5}+123q^{-7}-q^{-9}-117q^{-11}-20q^{-13}+99q^{-15}+46q^{-17}-81q^{-19}-69q^{-21}+51q^{-23}+91q^{-25}-11q^{-27}-109q^{-29}-35q^{-31}+113q^{-33}+79q^{-35}-96q^{-37}-111q^{-39}+64q^{-41}+122q^{-43}-28q^{-45}-106q^{-47}-3q^{-49}+77q^{-51}+17q^{-53}-45q^{-55}-16q^{-57}+20q^{-59}+9q^{-61}-9q^{-63}-2q^{-65}+6q^{-67}-2q^{-69}-3q^{-71}+q^{-73}+2q^{-75}-2q^{-79}+q^{-81}$
4	$q^{76}-2q^{74}-q^{72}+4q^{70}+q^{68}+2q^{66}-13q^{64}-10q^{62}+17q^{60}+22q^{58}+29q^{56}-41q^{54}-79q^{52}-18q^{50}+60q^{48}+164q^{46}+38q^{44}-156q^{42}-215q^{40}-101q^{38}+278q^{36}+340q^{34}+67q^{32}-328q^{30}-527q^{28}-29q^{26}+471q^{24}+589q^{22}+79q^{20}-698q^{18}-636q^{16}+12q^{14}+791q^{12}+756q^{10}-240q^{8}-880q^{6}-681q^{4}+394q^{2}+1041+402q^{-2}-578q^{-4}-980q^{-6}-138q^{-8}+824q^{-10}+691q^{-12}-177q^{-14}-848q^{-16}-380q^{-18}+499q^{-20}+668q^{-22}+41q^{-24}-627q^{-26}-454q^{-28}+244q^{-30}+638q^{-32}+245q^{-34}-415q^{-36}-611q^{-38}-139q^{-40}+587q^{-42}+625q^{-44}+22q^{-46}-736q^{-48}-738q^{-50}+229q^{-52}+902q^{-54}+693q^{-56}-433q^{-58}-1120q^{-60}-423q^{-62}+624q^{-64}+1076q^{-66}+207q^{-68}-836q^{-70}-766q^{-72}-10q^{-74}+770q^{-76}+534q^{-78}-215q^{-80}-501q^{-82}-324q^{-84}+222q^{-86}+341q^{-88}+90q^{-90}-104q^{-92}-211q^{-94}-20q^{-96}+81q^{-98}+61q^{-100}+34q^{-102}-55q^{-104}-19q^{-106}-3q^{-108}+2q^{-110}+23q^{-112}-8q^{-114}+q^{-116}-2q^{-118}-5q^{-120}+5q^{-122}-2q^{-124}+2q^{-126}-2q^{-130}+q^{-132}$
5	$q^{115}-2q^{113}-q^{111}+4q^{109}+q^{107}-2q^{105}-4q^{103}-7q^{101}-2q^{99}+20q^{97}+29q^{95}+3q^{93}-35q^{91}-70q^{89}-57q^{87}+29q^{85}+153q^{83}+182q^{81}+47q^{79}-189q^{77}-377q^{75}-304q^{73}+77q^{71}+555q^{69}+716q^{67}+315q^{65}-473q^{63}-1126q^{61}-1045q^{59}-73q^{57}+1238q^{55}+1851q^{53}+1128q^{51}-637q^{49}-2308q^{47}-2492q^{45}-736q^{43}+1933q^{41}+3540q^{39}+2665q^{37}-437q^{35}-3709q^{33}-4540q^{31}-1916q^{29}+2582q^{27}+5605q^{25}+4549q^{23}-253q^{21}-5375q^{19}-6677q^{17}-2714q^{15}+3787q^{13}+7675q^{11}+5543q^{9}-1226q^{7}-7321q^{5}-7597q^{3}-1582q+5857q^{-1}+8459q^{-3}+3990q^{-5}-3761q^{-7}-8193q^{-9}-5571q^{-11}+1674q^{-13}+7126q^{-15}+6149q^{-17}-10q^{-19}-5670q^{-21}-5959q^{-23}-1021q^{-25}+4337q^{-27}+5278q^{-29}+1421q^{-31}-3286q^{-33}-4535q^{-35}-1499q^{-37}+2679q^{-39}+4036q^{-41}+1513q^{-43}-2314q^{-45}-3909q^{-47}-1884q^{-49}+1935q^{-51}+4143q^{-53}+2769q^{-55}-1183q^{-57}-4446q^{-59}-4162q^{-61}-249q^{-63}+4388q^{-65}+5821q^{-67}+2395q^{-69}-3569q^{-71}-7183q^{-73}-4998q^{-75}+1740q^{-77}+7677q^{-79}+7530q^{-81}+910q^{-83}-6921q^{-85}-9239q^{-87}-3829q^{-89}+4883q^{-91}+9579q^{-93}+6295q^{-95}-2085q^{-97}-8438q^{-99}-7586q^{-101}-702q^{-103}+6134q^{-105}+7465q^{-107}+2787q^{-109}-3470q^{-111}-6159q^{-113}-3707q^{-115}+1141q^{-117}+4233q^{-119}+3573q^{-121}+401q^{-123}-2380q^{-125}-2774q^{-127}-1047q^{-129}+987q^{-131}+1771q^{-133}+1086q^{-135}-177q^{-137}-952q^{-139}-819q^{-141}-142q^{-143}+408q^{-145}+490q^{-147}+214q^{-149}-121q^{-151}-261q^{-153}-163q^{-155}+16q^{-157}+108q^{-159}+92q^{-161}+23q^{-163}-38q^{-165}-49q^{-167}-17q^{-169}+16q^{-171}+14q^{-173}+6q^{-175}+2q^{-177}-7q^{-179}-5q^{-181}+3q^{-183}+2q^{-185}-2q^{-187}+2q^{-189}-2q^{-193}+q^{-195}$
6	$q^{162}-2q^{160}-q^{158}+4q^{156}+q^{154}-2q^{152}-8q^{150}+2q^{148}+q^{146}+q^{144}+26q^{142}+16q^{140}-10q^{138}-57q^{136}-51q^{134}-36q^{132}+11q^{130}+143q^{128}+195q^{126}+137q^{124}-99q^{122}-287q^{120}-459q^{118}-410q^{116}+68q^{114}+640q^{112}+1060q^{110}+830q^{108}+173q^{106}-1009q^{104}-2023q^{102}-1941q^{100}-678q^{98}+1463q^{96}+3152q^{94}+3780q^{92}+2018q^{90}-1481q^{88}-4878q^{86}-6262q^{84}-4220q^{82}+439q^{80}+6485q^{78}+9698q^{76}+7976q^{74}+1368q^{72}-7280q^{70}-13406q^{68}-13474q^{66}-5098q^{64}+7203q^{62}+17621q^{60}+19672q^{58}+11084q^{56}-5308q^{54}-21775q^{52}-27401q^{50}-18674q^{48}+2279q^{46}+24627q^{44}+36206q^{42}+28037q^{40}+1933q^{38}-27631q^{36}-44995q^{34}-37534q^{32}-7434q^{30}+30828q^{28}+54032q^{26}+46625q^{24}+11390q^{22}-34009q^{20}-61963q^{18}-54770q^{16}-13037q^{14}+38452q^{12}+68727q^{10}+58848q^{8}+12048q^{6}-43438q^{4}-73984q^{2}-58407-6957q^{-2}+49025q^{-4}+74855q^{-6}+53464q^{-8}-1068q^{-10}-54382q^{-12}-71205q^{-14}-43129q^{-16}+11182q^{-18}+56201q^{-20}+63073q^{-22}+29354q^{-24}-21399q^{-26}-54030q^{-28}-49904q^{-30}-13807q^{-32}+27979q^{-34}+47554q^{-36}+33963q^{-38}-831q^{-40}-30454q^{-42}-36637q^{-44}-17135q^{-46}+11459q^{-48}+28743q^{-50}+23642q^{-52}+2020q^{-54}-18399q^{-56}-23685q^{-58}-10475q^{-60}+9765q^{-62}+22408q^{-64}+17744q^{-66}-709q^{-68}-19502q^{-70}-25310q^{-72}-12673q^{-74}+9875q^{-76}+28391q^{-78}+29061q^{-80}+9920q^{-82}-18642q^{-84}-38325q^{-86}-34615q^{-88}-8043q^{-90}+27888q^{-92}+49828q^{-94}+42101q^{-96}+5305q^{-98}-38941q^{-100}-62377q^{-102}-48353q^{-104}-1070q^{-106}+50855q^{-108}+74553q^{-110}+51242q^{-112}-5894q^{-114}-62108q^{-116}-81915q^{-118}-50060q^{-120}+14050q^{-122}+70810q^{-124}+82726q^{-126}+43634q^{-128}-21859q^{-130}-72909q^{-132}-77196q^{-134}-34176q^{-136}+27864q^{-138}+68391q^{-140}+65402q^{-142}+23586q^{-144}-29019q^{-146}-59027q^{-148}-50902q^{-150}-13460q^{-152}+26397q^{-154}+45962q^{-156}+36219q^{-158}+6720q^{-160}-21791q^{-162}-32919q^{-164}-23055q^{-166}-2551q^{-168}+15735q^{-170}+21601q^{-172}+13820q^{-174}+60q^{-176}-10532q^{-178}-12604q^{-180}-7602q^{-182}+590q^{-184}+6519q^{-186}+7067q^{-188}+3581q^{-190}-861q^{-192}-3460q^{-194}-3688q^{-196}-1669q^{-198}+802q^{-200}+1930q^{-202}+1655q^{-204}+571q^{-206}-408q^{-208}-1035q^{-210}-799q^{-212}-100q^{-214}+329q^{-216}+448q^{-218}+281q^{-220}+68q^{-222}-216q^{-224}-242q^{-226}-70q^{-228}+29q^{-230}+84q^{-232}+70q^{-234}+54q^{-236}-36q^{-238}-53q^{-240}-11q^{-242}+10q^{-246}+4q^{-248}+16q^{-250}-6q^{-252}-10q^{-254}+3q^{-256}+2q^{-260}-2q^{-262}+2q^{-264}-2q^{-268}+q^{-270}$

A2 Invariants.

Weight	Invariant
1,0	$q^{8}-q^{6}+2q^{4}+1+2q^{-2}-3q^{-4}+2q^{-6}-3q^{-8}+q^{-10}-q^{-14}+2q^{-16}-q^{-18}+q^{-20}$
1,1	$q^{28}-4q^{26}+12q^{24}-30q^{22}+64q^{20}-112q^{18}+188q^{16}-284q^{14}+387q^{12}-482q^{10}+544q^{8}-558q^{6}+502q^{4}-366q^{2}+172+74q^{-2}-329q^{-4}+574q^{-6}-784q^{-8}+928q^{-10}-999q^{-12}+982q^{-14}-884q^{-16}+720q^{-18}-501q^{-20}+264q^{-22}-30q^{-24}-162q^{-26}+302q^{-28}-386q^{-30}+414q^{-32}-404q^{-34}+367q^{-36}-320q^{-38}+268q^{-40}-218q^{-42}+171q^{-44}-128q^{-46}+92q^{-48}-60q^{-50}+36q^{-52}-20q^{-54}+10q^{-56}-4q^{-58}+q^{-60}$
2,0	$q^{24}-q^{22}-q^{20}+4q^{18}+2q^{16}-4q^{14}+6q^{10}-9q^{6}+7q^{2}-4-6q^{-2}+9q^{-4}+q^{-6}-5q^{-8}+4q^{-10}+5q^{-12}-3q^{-14}-4q^{-16}+9q^{-18}-9q^{-22}+3q^{-24}+8q^{-26}-7q^{-28}-5q^{-30}+6q^{-32}+q^{-34}-3q^{-36}-2q^{-38}+3q^{-40}-2q^{-44}+2q^{-46}-q^{-50}+q^{-52}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{20}-2q^{18}+q^{16}+3q^{14}-7q^{12}+5q^{10}+5q^{8}-12q^{6}+13q^{4}+7q^{2}-14+12q^{-2}+7q^{-4}-17q^{-6}+2q^{-8}+5q^{-10}-9q^{-12}-4q^{-14}+2q^{-16}+8q^{-18}-7q^{-20}-2q^{-22}+19q^{-24}-10q^{-26}-8q^{-28}+18q^{-30}-8q^{-32}-10q^{-34}+12q^{-36}-2q^{-38}-6q^{-40}+5q^{-42}-2q^{-46}+q^{-48}$
1,0,0	$q^{9}-q^{7}+3q^{5}-q^{3}+4q-q^{-1}+2q^{-3}-2q^{-5}-q^{-7}-q^{-9}-2q^{-11}+q^{-13}-2q^{-15}+3q^{-17}-2q^{-19}+3q^{-21}-q^{-23}+q^{-25}$
1,0,1	$q^{34}-4q^{32}+10q^{30}-16q^{28}+13q^{26}+10q^{24}-48q^{22}+94q^{20}-101q^{18}+51q^{16}+64q^{14}-203q^{12}+293q^{10}-283q^{8}+147q^{6}+84q^{4}-309q^{2}+458-433q^{-2}+260q^{-4}-19q^{-6}-190q^{-8}+250q^{-10}-205q^{-12}+103q^{-14}-51q^{-16}+100q^{-18}-192q^{-20}+274q^{-22}-228q^{-24}+58q^{-26}+191q^{-28}-396q^{-30}+449q^{-32}-318q^{-34}+64q^{-36}+182q^{-38}-308q^{-40}+265q^{-42}-109q^{-44}-66q^{-46}+164q^{-48}-155q^{-50}+67q^{-52}+36q^{-54}-97q^{-56}+97q^{-58}-49q^{-60}-6q^{-62}+40q^{-64}-41q^{-66}+22q^{-68}-2q^{-70}-8q^{-72}+8q^{-74}-4q^{-76}+q^{-78}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{22}-q^{20}+3q^{16}-3q^{14}-q^{12}+5q^{10}-2q^{8}-2q^{6}+12q^{4}+4q^{2}-5+8q^{-2}+11q^{-4}-8q^{-6}-14q^{-8}+8q^{-10}-2q^{-12}-23q^{-14}-2q^{-16}+14q^{-18}-14q^{-20}-3q^{-22}+21q^{-24}+2q^{-26}-7q^{-28}+12q^{-30}+10q^{-32}-12q^{-34}-5q^{-36}+10q^{-38}-2q^{-40}-13q^{-42}+5q^{-44}+7q^{-46}-6q^{-48}-q^{-50}+4q^{-52}-q^{-54}-q^{-56}+q^{-58}$
1,0,0,0	$q^{10}-q^{8}+3q^{6}+3q^{2}+2+2q^{-4}-3q^{-6}-4q^{-10}-3q^{-14}+q^{-16}-q^{-18}+q^{-20}+2q^{-22}-q^{-24}+3q^{-26}-q^{-28}+q^{-30}$

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^{20}-2 q^{18}+5 q^{16}-9 q^{14}+13 q^{12}-17 q^{10}+21 q^8-22 q^6+23 q^4-17 q^2+12-2 q^{-2} -7 q^{-4} +19 q^{-6} -30 q^{-8} +37 q^{-10} -43 q^{-12} +42 q^{-14} -40 q^{-16} +32 q^{-18} -23 q^{-20} +12 q^{-22} - q^{-24} -8 q^{-26} +16 q^{-28} -20 q^{-30} +22 q^{-32} -20 q^{-34} +18 q^{-36} -14 q^{-38} +10 q^{-40} -7 q^{-42} +4 q^{-44} -2 q^{-46} + q^{-48} }
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^{34}-2 q^{30}-2 q^{28}+3 q^{26}+6 q^{24}-2 q^{22}-10 q^{20}-3 q^{18}+13 q^{16}+12 q^{14}-11 q^{12}-19 q^{10}+3 q^8+25 q^6+10 q^4-19 q^2-18+11 q^{-2} +22 q^{-4} + q^{-6} -20 q^{-8} -8 q^{-10} +13 q^{-12} +9 q^{-14} -11 q^{-16} -12 q^{-18} +8 q^{-20} +12 q^{-22} -6 q^{-24} -16 q^{-26} +3 q^{-28} +18 q^{-30} +3 q^{-32} -18 q^{-34} -8 q^{-36} +18 q^{-38} +16 q^{-40} -11 q^{-42} -21 q^{-44} +2 q^{-46} +22 q^{-48} +9 q^{-50} -15 q^{-52} -16 q^{-54} +3 q^{-56} +15 q^{-58} +5 q^{-60} -8 q^{-62} -8 q^{-64} + q^{-66} +6 q^{-68} +2 q^{-70} -2 q^{-72} -2 q^{-74} + q^{-78} }

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^{26}-2 q^{24}+3 q^{22}-5 q^{20}+8 q^{18}-11 q^{16}+12 q^{14}-15 q^{12}+19 q^{10}-17 q^8+19 q^6-13 q^4+18 q^2-6+6 q^{-2} +3 q^{-4} -6 q^{-6} +11 q^{-8} -24 q^{-10} +21 q^{-12} -32 q^{-14} +28 q^{-16} -38 q^{-18} +30 q^{-20} -30 q^{-22} +30 q^{-24} -21 q^{-26} +18 q^{-28} -8 q^{-30} +9 q^{-32} +5 q^{-34} -7 q^{-36} +9 q^{-38} -14 q^{-40} +18 q^{-42} -17 q^{-44} +13 q^{-46} -16 q^{-48} +15 q^{-50} -10 q^{-52} +8 q^{-54} -8 q^{-56} +6 q^{-58} -3 q^{-60} +2 q^{-62} -2 q^{-64} + q^{-66} }

G2 Invariants.

Weight

Invariant

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^{46}-2 q^{44}+5 q^{42}-9 q^{40}+10 q^{38}-9 q^{36}+17 q^{32}-34 q^{30}+51 q^{28}-55 q^{26}+35 q^{24}+5 q^{22}-60 q^{20}+111 q^{18}-124 q^{16}+97 q^{14}-28 q^{12}-55 q^{10}+125 q^8-149 q^6+116 q^4-36 q^2-49+109 q^{-2} -110 q^{-4} +59 q^{-6} +24 q^{-8} -87 q^{-10} +112 q^{-12} -89 q^{-14} +15 q^{-16} +69 q^{-18} -141 q^{-20} +167 q^{-22} -135 q^{-24} +54 q^{-26} +49 q^{-28} -143 q^{-30} +185 q^{-32} -173 q^{-34} +99 q^{-36} -95 q^{-40} +144 q^{-42} -127 q^{-44} +62 q^{-46} +25 q^{-48} -86 q^{-50} +94 q^{-52} -53 q^{-54} -19 q^{-56} +84 q^{-58} -110 q^{-60} +97 q^{-62} -39 q^{-64} -29 q^{-66} +84 q^{-68} -109 q^{-70} +99 q^{-72} -63 q^{-74} +18 q^{-76} +23 q^{-78} -53 q^{-80} +65 q^{-82} -58 q^{-84} +44 q^{-86} -20 q^{-88} -2 q^{-90} +17 q^{-92} -28 q^{-94} +26 q^{-96} -19 q^{-98} +11 q^{-100} -2 q^{-102} -3 q^{-104} +5 q^{-106} -6 q^{-108} +4 q^{-110} -2 q^{-112} + q^{-114} }

.

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 94"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+4t^{3}-9t^{2}+14t-15+14t^{-1}-9t^{-2}+4t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 71, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-3q^{6}+6q^{5}-9q^{4}+11q^{3}-12q^{2}+11q-8+6q^{-1}-3q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-13z^{4}a^{-2}+4z^{4}a^{-4}+4z^{4}-12z^{2}a^{-2}+5z^{2}a^{-4}+5z^{2}-4a^{-2}+2a^{-4}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+5z^{8}a^{-4}+4z^{8}+3az^{7}+3z^{7}a^{-3}+6z^{7}a^{-5}+a^{2}z^{6}-27z^{6}a^{-2}-9z^{6}a^{-4}+5z^{6}a^{-6}-12z^{6}-9az^{5}-11z^{5}a^{-1}-15z^{5}a^{-3}-10z^{5}a^{-5}+3z^{5}a^{-7}-3a^{2}z^{4}+31z^{4}a^{-2}+10z^{4}a^{-4}-6z^{4}a^{-6}+z^{4}a^{-8}+11z^{4}+6az^{3}+10z^{3}a^{-1}+16z^{3}a^{-3}+9z^{3}a^{-5}-3z^{3}a^{-7}+2a^{2}z^{2}-18z^{2}a^{-2}-6z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}a^{-8}-7z^{2}-az-3za^{-1}-5za^{-3}-3za^{-5}+4a^{-2}+2a^{-4}+3$

Vassiliev invariants

V2 and V3:

(-2, -2)

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 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 -8} 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 -16} 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 32} 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{212}{3}} ${\frac {172}{3}}$ 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 128} 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{896}{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{320}{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 80} 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{256}{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 128} 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}} 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{1376}{3}} $-{\frac {1351}{15}}$ 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{9644}{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{48124}{45}} 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{2167}{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 -\frac{4951}{15}}

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=} 2 is the signature of 10 94. 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										2		-2
11									4	1		3
9								5	2			-3
7							6	4				2
5						6	5					-1
3					5	6						-1
1				4	7							3
-1			2	4								-2
-3		1	4									3
-5		2										-2
-7	1											1

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, 94]]
Out[2]=	10
In[3]:=	PD[Knot[10, 94]]
Out[3]=	PD[X[6, 2, 7, 1], X[2, 8, 3, 7], X[18, 12, 19, 11], X[14, 5, 15, 6], X[20, 14, 1, 13], X[8, 15, 9, 16], X[10, 4, 11, 3], X[16, 9, 17, 10], X[4, 17, 5, 18], X[12, 20, 13, 19]]
In[4]:=	GaussCode[Knot[10, 94]]
Out[4]=	GaussCode[1, -2, 7, -9, 4, -1, 2, -6, 8, -7, 3, -10, 5, -4, 6, -8, 9, -3, 10, -5]
In[5]:=	BR[Knot[10, 94]]
Out[5]=	BR[3, {1, 1, 1, -2, 1, 1, -2, -2, 1, -2}]
In[6]:=	alex = Alexander[Knot[10, 94]][t]
Out[6]=	-4 4 9 14 2 3 4 -15 - t + -- - -- + -- + 14 t - 9 t + 4 t - t 3 2 t t t
In[7]:=	Conway[Knot[10, 94]][z]
Out[7]=	2 4 6 8 1 - 2 z - 5 z - 4 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 94]}
In[9]:=	{KnotDet[Knot[10, 94]], KnotSignature[Knot[10, 94]]}
Out[9]=	{71, 2}
In[10]:=	J=Jones[Knot[10, 94]][q]
Out[10]=	-3 3 6 2 3 4 5 6 7 -8 + q - -- + - + 11 q - 12 q + 11 q - 9 q + 6 q - 3 q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 41], Knot[10, 94]}
In[12]:=	A2Invariant[Knot[10, 94]][q]
Out[12]=	-8 -6 2 2 4 6 8 10 14 16 1 + q - q + -- + 2 q - 3 q + 2 q - 3 q + q - q + 2 q - 4 q 18 20 q + q
In[13]:=	Kauffman[Knot[10, 94]][a, z]
Out[13]=	2 2 2 2 2 4 3 z 5 z 3 z 2 z 2 z 6 z 18 z 3 + -- + -- - --- - --- - --- - a z - 7 z - -- + ---- - ---- - ----- + 4 2 5 3 a 8 6 4 2 a a a a a a a a 3 3 3 3 4 4 2 2 3 z 9 z 16 z 10 z 3 4 z 6 z 2 a z - ---- + ---- + ----- + ----- + 6 a z + 11 z + -- - ---- + 7 5 3 a 8 6 a a a a a 4 4 5 5 5 5 10 z 31 z 2 4 3 z 10 z 15 z 11 z 5 ----- + ----- - 3 a z + ---- - ----- - ----- - ----- - 9 a z - 4 2 7 5 3 a a a a a a 6 6 6 7 7 6 5 z 9 z 27 z 2 6 6 z 3 z 7 8 12 z + ---- - ---- - ----- + a z + ---- + ---- + 3 a z + 4 z + 6 4 2 5 3 a a a a a 8 8 9 9 5 z 9 z 2 z 2 z ---- + ---- + ---- + ---- 4 2 3 a a a a
In[14]:=	{Vassiliev[2][Knot[10, 94]], Vassiliev[3][Knot[10, 94]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[10, 94]][q, t]
Out[15]=	3 1 2 1 4 2 4 4 q 7 q + 5 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 6 q t + 6 q t + 5 q t + 6 q t + 4 q t + 5 q t + 2 q t + 11 4 11 5 13 5 15 6 4 q t + q t + 2 q t + q t