10 109

10_108

10_110

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10 109 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X10,4,11,3 X18,11,19,12 X16,7,17,8 X8,17,9,18 X20,15,1,16 X12,19,13,20 X14,6,15,5 X2,10,3,9 X4,14,5,13
Gauss code	1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 10, -8, 6, -4, 5, -3, 7, -6
Dowker-Thistlethwaite code	6 10 14 16 2 18 4 20 8 12
Conway Notation	[2.2.2.2]

Three dimensional invariants

Symmetry type Negative amphicheiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 14.9002 A-Polynomial See Data:10 109/A-polynomial

[edit Notes for 10 109's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $4$ Rasmussen s-Invariant 0

[edit Notes for 10 109's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-4t^{3}+10t^{2}-17t+21-17t^{-1}+10t^{-2}-4t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+4z^{6}+6z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 85, 0 }
Jones polynomial	$-q^{5}+3q^{4}-7q^{3}+11q^{2}-13q+15-13q^{-1}+11q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-4z^{4}a^{-2}+14z^{4}-6a^{2}z^{2}-6z^{2}a^{-2}+15z^{2}-3a^{2}-3a^{-2}+7$
Kauffman polynomial (db, data sources)	$2az^{9}+2z^{9}a^{-1}+5a^{2}z^{8}+5z^{8}a^{-2}+10z^{8}+5a^{3}z^{7}+6az^{7}+6z^{7}a^{-1}+5z^{7}a^{-3}+3a^{4}z^{6}-7a^{2}z^{6}-7z^{6}a^{-2}+3z^{6}a^{-4}-20z^{6}+a^{5}z^{5}-8a^{3}z^{5}-16az^{5}-16z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}+6a^{2}z^{4}+6z^{4}a^{-2}-5z^{4}a^{-4}+22z^{4}-2a^{5}z^{3}+4a^{3}z^{3}+13az^{3}+13z^{3}a^{-1}+4z^{3}a^{-3}-2z^{3}a^{-5}+2a^{4}z^{2}-7a^{2}z^{2}-7z^{2}a^{-2}+2z^{2}a^{-4}-18z^{2}+a^{5}z-a^{3}z-5az-5za^{-1}-za^{-3}+za^{-5}+3a^{2}+3a^{-2}+7$
The A2 invariant	$-q^{14}+q^{12}-3q^{10}+q^{8}-q^{4}+5q^{2}-1+5q^{-2}-q^{-4}+q^{-8}-3q^{-10}+q^{-12}-q^{-14}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+9q^{72}-8q^{70}+q^{68}+14q^{66}-32q^{64}+51q^{62}-62q^{60}+51q^{58}-20q^{56}-39q^{54}+113q^{52}-171q^{50}+187q^{48}-138q^{46}+19q^{44}+124q^{42}-249q^{40}+297q^{38}-239q^{36}+89q^{34}+90q^{32}-231q^{30}+264q^{28}-177q^{26}+13q^{24}+150q^{22}-232q^{20}+187q^{18}-37q^{16}-151q^{14}+301q^{12}-336q^{10}+248q^{8}-53q^{6}-170q^{4}+353q^{2}-417+353q^{-2}-170q^{-4}-53q^{-6}+248q^{-8}-336q^{-10}+301q^{-12}-151q^{-14}-37q^{-16}+187q^{-18}-232q^{-20}+150q^{-22}+13q^{-24}-177q^{-26}+264q^{-28}-231q^{-30}+90q^{-32}+89q^{-34}-239q^{-36}+297q^{-38}-249q^{-40}+124q^{-42}+19q^{-44}-138q^{-46}+187q^{-48}-171q^{-50}+113q^{-52}-39q^{-54}-20q^{-56}+51q^{-58}-62q^{-60}+51q^{-62}-32q^{-64}+14q^{-66}+q^{-68}-8q^{-70}+9q^{-72}-8q^{-74}+5q^{-76}-2q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_109.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-4q^{7}+4q^{5}-2q^{3}+2q+2q^{-1}-2q^{-3}+4q^{-5}-4q^{-7}+2q^{-9}-q^{-11}$
2	$q^{32}-2q^{30}+7q^{26}-10q^{24}-6q^{22}+26q^{20}-14q^{18}-27q^{16}+38q^{14}-38q^{10}+26q^{8}+16q^{6}-26q^{4}+21-26q^{-4}+16q^{-6}+26q^{-8}-38q^{-10}+38q^{-14}-27q^{-16}-14q^{-18}+26q^{-20}-6q^{-22}-10q^{-24}+7q^{-26}-2q^{-30}+q^{-32}$
3	$-q^{63}+2q^{61}-3q^{57}-q^{55}+9q^{53}+4q^{51}-23q^{49}-13q^{47}+44q^{45}+41q^{43}-62q^{41}-100q^{39}+65q^{37}+172q^{35}-28q^{33}-238q^{31}-52q^{29}+280q^{27}+139q^{25}-267q^{23}-226q^{21}+215q^{19}+277q^{17}-135q^{15}-291q^{13}+55q^{11}+264q^{9}+30q^{7}-219q^{5}-98q^{3}+162q+162q^{-1}-98q^{-3}-219q^{-5}+30q^{-7}+264q^{-9}+55q^{-11}-291q^{-13}-135q^{-15}+277q^{-17}+215q^{-19}-226q^{-21}-267q^{-23}+139q^{-25}+280q^{-27}-52q^{-29}-238q^{-31}-28q^{-33}+172q^{-35}+65q^{-37}-100q^{-39}-62q^{-41}+41q^{-43}+44q^{-45}-13q^{-47}-23q^{-49}+4q^{-51}+9q^{-53}-q^{-55}-3q^{-57}+2q^{-61}-q^{-63}$
4	$q^{104}-2q^{102}+3q^{98}-3q^{96}+2q^{94}-7q^{92}+4q^{90}+20q^{88}-11q^{86}-14q^{84}-47q^{82}+15q^{80}+121q^{78}+49q^{76}-62q^{74}-276q^{72}-140q^{70}+316q^{68}+462q^{66}+219q^{64}-653q^{62}-905q^{60}-12q^{58}+1048q^{56}+1376q^{54}-258q^{52}-1835q^{50}-1447q^{48}+613q^{46}+2634q^{44}+1386q^{42}-1520q^{40}-2875q^{38}-1093q^{36}+2454q^{34}+2872q^{32}+93q^{30}-2801q^{28}-2520q^{26}+986q^{24}+2865q^{22}+1476q^{20}-1528q^{18}-2602q^{16}-408q^{14}+1822q^{12}+1882q^{10}-243q^{8}-1912q^{6}-1218q^{4}+745q^{2}+1861+745q^{-2}-1218q^{-4}-1912q^{-6}-243q^{-8}+1882q^{-10}+1822q^{-12}-408q^{-14}-2602q^{-16}-1528q^{-18}+1476q^{-20}+2865q^{-22}+986q^{-24}-2520q^{-26}-2801q^{-28}+93q^{-30}+2872q^{-32}+2454q^{-34}-1093q^{-36}-2875q^{-38}-1520q^{-40}+1386q^{-42}+2634q^{-44}+613q^{-46}-1447q^{-48}-1835q^{-50}-258q^{-52}+1376q^{-54}+1048q^{-56}-12q^{-58}-905q^{-60}-653q^{-62}+219q^{-64}+462q^{-66}+316q^{-68}-140q^{-70}-276q^{-72}-62q^{-74}+49q^{-76}+121q^{-78}+15q^{-80}-47q^{-82}-14q^{-84}-11q^{-86}+20q^{-88}+4q^{-90}-7q^{-92}+2q^{-94}-3q^{-96}+3q^{-98}-2q^{-102}+q^{-104}$
5	$-q^{155}+2q^{153}-3q^{149}+3q^{147}+2q^{145}-4q^{143}-q^{141}-q^{139}-7q^{137}+11q^{135}+28q^{133}+4q^{131}-31q^{129}-63q^{127}-57q^{125}+36q^{123}+184q^{121}+222q^{119}+11q^{117}-352q^{115}-592q^{113}-343q^{111}+438q^{109}+1253q^{107}+1227q^{105}-88q^{103}-1972q^{101}-2832q^{99}-1334q^{97}+2094q^{95}+4991q^{93}+4286q^{91}-701q^{89}-6719q^{87}-8568q^{85}-3193q^{83}+6606q^{81}+13159q^{79}+9577q^{77}-3359q^{75}-16099q^{73}-17271q^{71}-3553q^{69}+15601q^{67}+24222q^{65}+12985q^{63}-10867q^{61}-27958q^{59}-22765q^{57}+2383q^{55}+27293q^{53}+30347q^{51}+7663q^{49}-22195q^{47}-33817q^{45}-16957q^{43}+14195q^{41}+32822q^{39}+23342q^{37}-5386q^{35}-28197q^{33}-26026q^{31}-2235q^{29}+21581q^{27}+25347q^{25}+7640q^{23}-14764q^{21}-22543q^{19}-10608q^{17}+8924q^{15}+18941q^{13}+11978q^{11}-4543q^{9}-15841q^{7}-12669q^{5}+1379q^{3}+13741q+13741q^{-1}+1379q^{-3}-12669q^{-5}-15841q^{-7}-4543q^{-9}+11978q^{-11}+18941q^{-13}+8924q^{-15}-10608q^{-17}-22543q^{-19}-14764q^{-21}+7640q^{-23}+25347q^{-25}+21581q^{-27}-2235q^{-29}-26026q^{-31}-28197q^{-33}-5386q^{-35}+23342q^{-37}+32822q^{-39}+14195q^{-41}-16957q^{-43}-33817q^{-45}-22195q^{-47}+7663q^{-49}+30347q^{-51}+27293q^{-53}+2383q^{-55}-22765q^{-57}-27958q^{-59}-10867q^{-61}+12985q^{-63}+24222q^{-65}+15601q^{-67}-3553q^{-69}-17271q^{-71}-16099q^{-73}-3359q^{-75}+9577q^{-77}+13159q^{-79}+6606q^{-81}-3193q^{-83}-8568q^{-85}-6719q^{-87}-701q^{-89}+4286q^{-91}+4991q^{-93}+2094q^{-95}-1334q^{-97}-2832q^{-99}-1972q^{-101}-88q^{-103}+1227q^{-105}+1253q^{-107}+438q^{-109}-343q^{-111}-592q^{-113}-352q^{-115}+11q^{-117}+222q^{-119}+184q^{-121}+36q^{-123}-57q^{-125}-63q^{-127}-31q^{-129}+4q^{-131}+28q^{-133}+11q^{-135}-7q^{-137}-q^{-139}-q^{-141}-4q^{-143}+2q^{-145}+3q^{-147}-3q^{-149}+2q^{-153}-q^{-155}$
6	$q^{216}-2q^{214}+3q^{210}-3q^{208}-2q^{206}+12q^{202}-2q^{200}-12q^{198}+7q^{196}-14q^{194}-15q^{192}+4q^{190}+62q^{188}+45q^{186}-22q^{184}-28q^{182}-130q^{180}-166q^{178}-61q^{176}+268q^{174}+447q^{172}+364q^{170}+140q^{168}-596q^{166}-1263q^{164}-1334q^{162}-123q^{160}+1590q^{158}+2958q^{156}+3267q^{154}+805q^{152}-3423q^{150}-7276q^{148}-7082q^{146}-2166q^{144}+6351q^{142}+14700q^{140}+15389q^{138}+5690q^{136}-11964q^{134}-26799q^{132}-29393q^{130}-12890q^{128}+18819q^{126}+46840q^{124}+52118q^{122}+23431q^{120}-27179q^{118}-74733q^{116}-85092q^{114}-40712q^{112}+39679q^{110}+112342q^{108}+126233q^{106}+63203q^{104}-55078q^{102}-158976q^{100}-176923q^{98}-84719q^{96}+76363q^{94}+210285q^{92}+230455q^{90}+102841q^{88}-104731q^{86}-266784q^{84}-275385q^{82}-109284q^{80}+138428q^{78}+319539q^{76}+306278q^{74}+99016q^{72}-180371q^{70}-356738q^{68}-313153q^{66}-71885q^{64}+221829q^{62}+374479q^{60}+292206q^{58}+26369q^{56}-251667q^{54}-365377q^{52}-246762q^{50}+25519q^{48}+266682q^{46}+329262q^{44}+180904q^{42}-70774q^{40}-260866q^{38}-272938q^{36}-110226q^{34}+105095q^{32}+235464q^{30}+203724q^{28}+47380q^{26}-123571q^{24}-197604q^{22}-135835q^{20}+5058q^{18}+128690q^{16}+154331q^{14}+76682q^{12}-46056q^{10}-127127q^{8}-115073q^{6}-24477q^{4}+80664q^{2}+125109+80664q^{-2}-24477q^{-4}-115073q^{-6}-127127q^{-8}-46056q^{-10}+76682q^{-12}+154331q^{-14}+128690q^{-16}+5058q^{-18}-135835q^{-20}-197604q^{-22}-123571q^{-24}+47380q^{-26}+203724q^{-28}+235464q^{-30}+105095q^{-32}-110226q^{-34}-272938q^{-36}-260866q^{-38}-70774q^{-40}+180904q^{-42}+329262q^{-44}+266682q^{-46}+25519q^{-48}-246762q^{-50}-365377q^{-52}-251667q^{-54}+26369q^{-56}+292206q^{-58}+374479q^{-60}+221829q^{-62}-71885q^{-64}-313153q^{-66}-356738q^{-68}-180371q^{-70}+99016q^{-72}+306278q^{-74}+319539q^{-76}+138428q^{-78}-109284q^{-80}-275385q^{-82}-266784q^{-84}-104731q^{-86}+102841q^{-88}+230455q^{-90}+210285q^{-92}+76363q^{-94}-84719q^{-96}-176923q^{-98}-158976q^{-100}-55078q^{-102}+63203q^{-104}+126233q^{-106}+112342q^{-108}+39679q^{-110}-40712q^{-112}-85092q^{-114}-74733q^{-116}-27179q^{-118}+23431q^{-120}+52118q^{-122}+46840q^{-124}+18819q^{-126}-12890q^{-128}-29393q^{-130}-26799q^{-132}-11964q^{-134}+5690q^{-136}+15389q^{-138}+14700q^{-140}+6351q^{-142}-2166q^{-144}-7082q^{-146}-7276q^{-148}-3423q^{-150}+805q^{-152}+3267q^{-154}+2958q^{-156}+1590q^{-158}-123q^{-160}-1334q^{-162}-1263q^{-164}-596q^{-166}+140q^{-168}+364q^{-170}+447q^{-172}+268q^{-174}-61q^{-176}-166q^{-178}-130q^{-180}-28q^{-182}-22q^{-184}+45q^{-186}+62q^{-188}+4q^{-190}-15q^{-192}-14q^{-194}+7q^{-196}-12q^{-198}-2q^{-200}+12q^{-202}-2q^{-206}-3q^{-208}+3q^{-210}-2q^{-214}+q^{-216}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{14}+q^{12}-3q^{10}+q^{8}-q^{4}+5q^{2}-1+5q^{-2}-q^{-4}+q^{-8}-3q^{-10}+q^{-12}-q^{-14}$
1,1	$q^{44}-4q^{42}+12q^{40}-28q^{38}+60q^{36}-116q^{34}+204q^{32}-334q^{30}+514q^{28}-738q^{26}+976q^{24}-1190q^{22}+1334q^{20}-1352q^{18}+1186q^{16}-840q^{14}+312q^{12}+332q^{10}-1030q^{8}+1690q^{6}-2220q^{4}+2582q^{2}-2694+2582q^{-2}-2220q^{-4}+1690q^{-6}-1030q^{-8}+332q^{-10}+312q^{-12}-840q^{-14}+1186q^{-16}-1352q^{-18}+1334q^{-20}-1190q^{-22}+976q^{-24}-738q^{-26}+514q^{-28}-334q^{-30}+204q^{-32}-116q^{-34}+60q^{-36}-28q^{-38}+12q^{-40}-4q^{-42}+q^{-44}$
2,0	$q^{38}-q^{36}+4q^{32}-3q^{30}-5q^{28}+7q^{26}+3q^{24}-9q^{22}-7q^{20}+9q^{18}+5q^{16}-19q^{14}+2q^{12}+14q^{10}-8q^{8}-6q^{6}+12q^{4}+7q^{2}-6+7q^{-2}+12q^{-4}-6q^{-6}-8q^{-8}+14q^{-10}+2q^{-12}-19q^{-14}+5q^{-16}+9q^{-18}-7q^{-20}-9q^{-22}+3q^{-24}+7q^{-26}-5q^{-28}-3q^{-30}+4q^{-32}-q^{-36}+q^{-38}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-2q^{32}+q^{30}+5q^{28}-9q^{26}+2q^{24}+15q^{22}-22q^{20}+2q^{18}+23q^{16}-32q^{14}-q^{12}+22q^{10}-22q^{8}-3q^{6}+19q^{4}+4q^{2}+4q^{-2}+19q^{-4}-3q^{-6}-22q^{-8}+22q^{-10}-q^{-12}-32q^{-14}+23q^{-16}+2q^{-18}-22q^{-20}+15q^{-22}+2q^{-24}-9q^{-26}+5q^{-28}+q^{-30}-2q^{-32}+q^{-34}$
1,0,0	$-q^{17}+q^{15}-4q^{13}+2q^{11}-4q^{9}+2q^{7}-q^{5}+4q^{3}+3q+3q^{-1}+4q^{-3}-q^{-5}+2q^{-7}-4q^{-9}+2q^{-11}-4q^{-13}+q^{-15}-q^{-17}$
1,0,1	$q^{56}-4q^{54}+10q^{52}-14q^{50}+7q^{48}+23q^{46}-72q^{44}+110q^{42}-85q^{40}-44q^{38}+252q^{36}-425q^{34}+405q^{32}-93q^{30}-430q^{28}+923q^{26}-1076q^{24}+723q^{22}+48q^{20}-926q^{18}+1449q^{16}-1414q^{14}+762q^{12}+89q^{10}-756q^{8}+914q^{6}-568q^{4}+142q^{2}+113+142q^{-2}-568q^{-4}+914q^{-6}-756q^{-8}+89q^{-10}+762q^{-12}-1414q^{-14}+1449q^{-16}-926q^{-18}+48q^{-20}+723q^{-22}-1076q^{-24}+923q^{-26}-430q^{-28}-93q^{-30}+405q^{-32}-425q^{-34}+252q^{-36}-44q^{-38}-85q^{-40}+110q^{-42}-72q^{-44}+23q^{-46}+7q^{-48}-14q^{-50}+10q^{-52}-4q^{-54}+q^{-56}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{40}-q^{38}+5q^{34}-3q^{32}-4q^{30}+13q^{28}-3q^{26}-15q^{24}+10q^{22}+9q^{20}-25q^{18}-14q^{16}+13q^{14}-7q^{12}-28q^{10}+9q^{8}+30q^{6}-14q^{4}+6q^{2}+46+6q^{-2}-14q^{-4}+30q^{-6}+9q^{-8}-28q^{-10}-7q^{-12}+13q^{-14}-14q^{-16}-25q^{-18}+9q^{-20}+10q^{-22}-15q^{-24}-3q^{-26}+13q^{-28}-4q^{-30}-3q^{-32}+5q^{-34}-q^{-38}+q^{-40}$
1,0,0,0	$-q^{20}+q^{18}-4q^{16}+q^{14}-3q^{12}-2q^{10}+q^{8}-q^{6}+5q^{4}+2q^{2}+7+2q^{-2}+5q^{-4}-q^{-6}+q^{-8}-2q^{-10}-3q^{-12}+q^{-14}-4q^{-16}+q^{-18}-q^{-20}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2q^{32}-5q^{30}+9q^{28}-15q^{26}+22q^{24}-29q^{22}+34q^{20}-36q^{18}+31q^{16}-24q^{14}+11q^{12}+4q^{10}-22q^{8}+41q^{6}-53q^{4}+66q^{2}-66+66q^{-2}-53q^{-4}+41q^{-6}-22q^{-8}+4q^{-10}+11q^{-12}-24q^{-14}+31q^{-16}-36q^{-18}+34q^{-20}-29q^{-22}+22q^{-24}-15q^{-26}+9q^{-28}-5q^{-30}+2q^{-32}-q^{-34}$
1,0	$q^{56}-2q^{52}-2q^{50}+3q^{48}+7q^{46}-12q^{42}-9q^{40}+12q^{38}+22q^{36}-3q^{34}-31q^{32}-15q^{30}+28q^{28}+29q^{26}-16q^{24}-39q^{22}-4q^{20}+34q^{18}+15q^{16}-26q^{14}-22q^{12}+17q^{10}+24q^{8}-6q^{6}-22q^{4}+6q^{2}+27+6q^{-2}-22q^{-4}-6q^{-6}+24q^{-8}+17q^{-10}-22q^{-12}-26q^{-14}+15q^{-16}+34q^{-18}-4q^{-20}-39q^{-22}-16q^{-24}+29q^{-26}+28q^{-28}-15q^{-30}-31q^{-32}-3q^{-34}+22q^{-36}+12q^{-38}-9q^{-40}-12q^{-42}+7q^{-46}+3q^{-48}-2q^{-50}-2q^{-52}+q^{-56}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{46}-2q^{44}+3q^{42}-4q^{40}+8q^{38}-12q^{36}+14q^{34}-17q^{32}+24q^{30}-28q^{28}+26q^{26}-27q^{24}+26q^{22}-24q^{20}+8q^{18}-11q^{16}-3q^{14}+11q^{12}-28q^{10}+31q^{8}-34q^{6}+56q^{4}-43q^{2}+58-43q^{-2}+56q^{-4}-34q^{-6}+31q^{-8}-28q^{-10}+11q^{-12}-3q^{-14}-11q^{-16}+8q^{-18}-24q^{-20}+26q^{-22}-27q^{-24}+26q^{-26}-28q^{-28}+24q^{-30}-17q^{-32}+14q^{-34}-12q^{-36}+8q^{-38}-4q^{-40}+3q^{-42}-2q^{-44}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+5q^{76}-8q^{74}+9q^{72}-8q^{70}+q^{68}+14q^{66}-32q^{64}+51q^{62}-62q^{60}+51q^{58}-20q^{56}-39q^{54}+113q^{52}-171q^{50}+187q^{48}-138q^{46}+19q^{44}+124q^{42}-249q^{40}+297q^{38}-239q^{36}+89q^{34}+90q^{32}-231q^{30}+264q^{28}-177q^{26}+13q^{24}+150q^{22}-232q^{20}+187q^{18}-37q^{16}-151q^{14}+301q^{12}-336q^{10}+248q^{8}-53q^{6}-170q^{4}+353q^{2}-417+353q^{-2}-170q^{-4}-53q^{-6}+248q^{-8}-336q^{-10}+301q^{-12}-151q^{-14}-37q^{-16}+187q^{-18}-232q^{-20}+150q^{-22}+13q^{-24}-177q^{-26}+264q^{-28}-231q^{-30}+90q^{-32}+89q^{-34}-239q^{-36}+297q^{-38}-249q^{-40}+124q^{-42}+19q^{-44}-138q^{-46}+187q^{-48}-171q^{-50}+113q^{-52}-39q^{-54}-20q^{-56}+51q^{-58}-62q^{-60}+51q^{-62}-32q^{-64}+14q^{-66}+q^{-68}-8q^{-70}+9q^{-72}-8q^{-74}+5q^{-76}-2q^{-78}+q^{-80}$

.

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["10 109"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-4t^{3}+10t^{2}-17t+21-17t^{-1}+10t^{-2}-4t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 85, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+3q^{4}-7q^{3}+11q^{2}-13q+15-13q^{-1}+11q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-4z^{4}a^{-2}+14z^{4}-6a^{2}z^{2}-6z^{2}a^{-2}+15z^{2}-3a^{2}-3a^{-2}+7$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2az^{9}+2z^{9}a^{-1}+5a^{2}z^{8}+5z^{8}a^{-2}+10z^{8}+5a^{3}z^{7}+6az^{7}+6z^{7}a^{-1}+5z^{7}a^{-3}+3a^{4}z^{6}-7a^{2}z^{6}-7z^{6}a^{-2}+3z^{6}a^{-4}-20z^{6}+a^{5}z^{5}-8a^{3}z^{5}-16az^{5}-16z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}+6a^{2}z^{4}+6z^{4}a^{-2}-5z^{4}a^{-4}+22z^{4}-2a^{5}z^{3}+4a^{3}z^{3}+13az^{3}+13z^{3}a^{-1}+4z^{3}a^{-3}-2z^{3}a^{-5}+2a^{4}z^{2}-7a^{2}z^{2}-7z^{2}a^{-2}+2z^{2}a^{-4}-18z^{2}+a^{5}z-a^{3}z-5az-5za^{-1}-za^{-3}+za^{-5}+3a^{2}+3a^{-2}+7$

Vassiliev invariants

V2 and V3:

(3, 0)

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 $12$ $0$ $72$ $62$ $-14$ $0$ $0$ $0$ $0$ $288$ $0$ $744$ $-168$ ${\frac {6831}{10}}$ ${\frac {1298}{5}}$ $-{\frac {3538}{15}}$ ${\frac {209}{6}}$ $-{\frac {1169}{10}}$

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=$0 is the signature of 10 109. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                     1 -1 9                   2   2 7                 5 1   -4 5               6 2     4 3             7 5       -2 1           8 6         2 -1         6 8           2 -3       5 7             -2 -5     2 6               4 -7   1 5                 -4 -9   2                   2 -11 1                     -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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, 109]]
Out[2]=	10
In[3]:=	PD[Knot[10, 109]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[18, 11, 19, 12], X[16, 7, 17, 8], X[8, 17, 9, 18], X[20, 15, 1, 16], X[12, 19, 13, 20], X[14, 6, 15, 5], X[2, 10, 3, 9], X[4, 14, 5, 13]]
In[4]:=	GaussCode[Knot[10, 109]]
Out[4]=	GaussCode[1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 10, -8, 6, -4, 5, -3, 7, -6]
In[5]:=	BR[Knot[10, 109]]
Out[5]=	BR[3, {-1, -1, 2, -1, 2, 2, -1, -1, 2, 2}]
In[6]:=	alex = Alexander[Knot[10, 109]][t]
Out[6]=	-4 4 10 17 2 3 4 21 + t - -- + -- - -- - 17 t + 10 t - 4 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 109]][z]
Out[7]=	2 4 6 8 1 + 3 z + 6 z + 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 109]}
In[9]:=	{KnotDet[Knot[10, 109]], KnotSignature[Knot[10, 109]]}
Out[9]=	{85, 0}
In[10]:=	J=Jones[Knot[10, 109]][q]
Out[10]=	-5 3 7 11 13 2 3 4 5 15 - q + -- - -- + -- - -- - 13 q + 11 q - 7 q + 3 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 81], Knot[10, 109]}
In[12]:=	A2Invariant[Knot[10, 109]][q]
Out[12]=	-14 -12 3 -8 -4 5 2 4 8 10 -1 - q + q - --- + q - q + -- + 5 q - q + q - 3 q + 10 2 q q 12 14 q - q
In[13]:=	Kauffman[Knot[10, 109]][a, z]
Out[13]=	2 3 2 z z 5 z 3 5 2 2 z 7 + -- + 3 a + -- - -- - --- - 5 a z - a z + a z - 18 z + ---- - 2 5 3 a 4 a a a a 2 3 3 3 7 z 2 2 4 2 2 z 4 z 13 z 3 3 3 ---- - 7 a z + 2 a z - ---- + ---- + ----- + 13 a z + 4 a z - 2 5 3 a a a a 4 4 5 5 5 3 4 5 z 6 z 2 4 4 4 z 8 z 2 a z + 22 z - ---- + ---- + 6 a z - 5 a z + -- - ---- - 4 2 5 3 a a a a 5 6 6 16 z 5 3 5 5 5 6 3 z 7 z 2 6 ----- - 16 a z - 8 a z + a z - 20 z + ---- - ---- - 7 a z + a 4 2 a a 7 7 8 4 6 5 z 6 z 7 3 7 8 5 z 2 8 3 a z + ---- + ---- + 6 a z + 5 a z + 10 z + ---- + 5 a z + 3 a 2 a a 9 2 z 9 ---- + 2 a z a
In[14]:=	{Vassiliev[2][Knot[10, 109]], Vassiliev[3][Knot[10, 109]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 109]][q, t]
Out[15]=	8 1 2 1 5 2 6 5 - + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 7 6 3 3 2 5 2 5 3 7 3 ---- + --- + 6 q t + 7 q t + 5 q t + 6 q t + 2 q t + 5 q t + 3 q t q t 7 4 9 4 11 5 q t + 2 q t + q t