10 99

10_98

10_100

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10 99 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X10,4,11,3 X16,11,17,12 X14,7,15,8 X8,15,9,16 X20,13,1,14 X12,19,13,20 X18,6,19,5 X2,10,3,9 X4,18,5,17
Gauss code	1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 6, -4, 5, -3, 10, -8, 7, -6
Dowker-Thistlethwaite code	6 10 18 14 2 16 20 8 4 12
Conway Notation	[.2.2.20.20]

Three dimensional invariants

Symmetry type Fully amphicheiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 14.3343 A-Polynomial See Data:10 99/A-polynomial

[edit Notes for 10 99's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 99's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-4t^{3}+10t^{2}-16t+19-16t^{-1}+10t^{-2}-4t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+4z^{6}+6z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{t^{4}-2t^{3}+3t^{2}-2t+1\right\}$
Determinant and Signature	{ 81, 0 }
Jones polynomial	$-q^{5}+3q^{4}-7q^{3}+10q^{2}-12q+15-12q^{-1}+10q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-4z^{4}a^{-2}+14z^{4}-6a^{2}z^{2}-6z^{2}a^{-2}+16z^{2}-4a^{2}-4a^{-2}+9$
Kauffman polynomial (db, data sources)	$2az^{9}+2z^{9}a^{-1}+5a^{2}z^{8}+5z^{8}a^{-2}+10z^{8}+5a^{3}z^{7}+5az^{7}+5z^{7}a^{-1}+5z^{7}a^{-3}+3a^{4}z^{6}-9a^{2}z^{6}-9z^{6}a^{-2}+3z^{6}a^{-4}-24z^{6}+a^{5}z^{5}-9a^{3}z^{5}-18az^{5}-18z^{5}a^{-1}-9z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}+9a^{2}z^{4}+9z^{4}a^{-2}-5z^{4}a^{-4}+28z^{4}-2a^{5}z^{3}+5a^{3}z^{3}+21az^{3}+21z^{3}a^{-1}+5z^{3}a^{-3}-2z^{3}a^{-5}+a^{4}z^{2}-8a^{2}z^{2}-8z^{2}a^{-2}+z^{2}a^{-4}-18z^{2}+a^{5}z-3a^{3}z-10az-10za^{-1}-3za^{-3}+za^{-5}+4a^{2}+4a^{-2}+9$
The A2 invariant	$-q^{14}+q^{12}-3q^{10}-q^{6}-q^{4}+6q^{2}+1+6q^{-2}-q^{-4}-q^{-6}-3q^{-10}+q^{-12}-q^{-14}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+9q^{72}-7q^{70}+14q^{66}-30q^{64}+46q^{62}-56q^{60}+42q^{58}-13q^{56}-38q^{54}+99q^{52}-139q^{50}+152q^{48}-109q^{46}+11q^{44}+103q^{42}-204q^{40}+232q^{38}-184q^{36}+63q^{34}+71q^{32}-177q^{30}+209q^{28}-138q^{26}+4q^{24}+117q^{22}-185q^{20}+147q^{18}-29q^{16}-127q^{14}+240q^{12}-255q^{10}+205q^{8}-47q^{6}-139q^{4}+285q^{2}-329+285q^{-2}-139q^{-4}-47q^{-6}+205q^{-8}-255q^{-10}+240q^{-12}-127q^{-14}-29q^{-16}+147q^{-18}-185q^{-20}+117q^{-22}+4q^{-24}-138q^{-26}+209q^{-28}-177q^{-30}+71q^{-32}+63q^{-34}-184q^{-36}+232q^{-38}-204q^{-40}+103q^{-42}+11q^{-44}-109q^{-46}+152q^{-48}-139q^{-50}+99q^{-52}-38q^{-54}-13q^{-56}+42q^{-58}-56q^{-60}+46q^{-62}-30q^{-64}+14q^{-66}-7q^{-70}+9q^{-72}-8q^{-74}+5q^{-76}-2q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_99.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-4q^{7}+3q^{5}-2q^{3}+3q+3q^{-1}-2q^{-3}+3q^{-5}-4q^{-7}+2q^{-9}-q^{-11}$
2	$q^{32}-2q^{30}+7q^{26}-9q^{24}-6q^{22}+23q^{20}-11q^{18}-23q^{16}+31q^{14}-q^{12}-34q^{10}+21q^{8}+13q^{6}-22q^{4}+3q^{2}+21+3q^{-2}-22q^{-4}+13q^{-6}+21q^{-8}-34q^{-10}-q^{-12}+31q^{-14}-23q^{-16}-11q^{-18}+23q^{-20}-6q^{-22}-9q^{-24}+7q^{-26}-2q^{-30}+q^{-32}$
3	$-q^{63}+2q^{61}-3q^{57}-q^{55}+8q^{53}+4q^{51}-20q^{49}-12q^{47}+37q^{45}+34q^{43}-48q^{41}-80q^{39}+49q^{37}+133q^{35}-19q^{33}-181q^{31}-39q^{29}+211q^{27}+108q^{25}-202q^{23}-177q^{21}+165q^{19}+209q^{17}-105q^{15}-229q^{13}+43q^{11}+202q^{9}+24q^{7}-172q^{5}-68q^{3}+130q+130q^{-1}-68q^{-3}-172q^{-5}+24q^{-7}+202q^{-9}+43q^{-11}-229q^{-13}-105q^{-15}+209q^{-17}+165q^{-19}-177q^{-21}-202q^{-23}+108q^{-25}+211q^{-27}-39q^{-29}-181q^{-31}-19q^{-33}+133q^{-35}+49q^{-37}-80q^{-39}-48q^{-41}+34q^{-43}+37q^{-45}-12q^{-47}-20q^{-49}+4q^{-51}+8q^{-53}-q^{-55}-3q^{-57}+2q^{-61}-q^{-63}$
4	$q^{104}-2q^{102}+3q^{98}-3q^{96}+2q^{94}-6q^{92}+4q^{90}+17q^{88}-12q^{86}-11q^{84}-38q^{82}+16q^{80}+100q^{78}+30q^{76}-53q^{74}-215q^{72}-94q^{70}+252q^{68}+329q^{66}+150q^{64}-488q^{62}-640q^{60}+20q^{58}+739q^{56}+964q^{54}-208q^{52}-1286q^{50}-985q^{48}+428q^{46}+1843q^{44}+953q^{42}-1060q^{40}-1997q^{38}-773q^{36}+1726q^{34}+2004q^{32}+67q^{30}-1963q^{28}-1781q^{26}+704q^{24}+2004q^{22}+1033q^{20}-1077q^{18}-1842q^{16}-271q^{14}+1275q^{12}+1312q^{10}-174q^{8}-1338q^{6}-831q^{4}+526q^{2}+1297+526q^{-2}-831q^{-4}-1338q^{-6}-174q^{-8}+1312q^{-10}+1275q^{-12}-271q^{-14}-1842q^{-16}-1077q^{-18}+1033q^{-20}+2004q^{-22}+704q^{-24}-1781q^{-26}-1963q^{-28}+67q^{-30}+2004q^{-32}+1726q^{-34}-773q^{-36}-1997q^{-38}-1060q^{-40}+953q^{-42}+1843q^{-44}+428q^{-46}-985q^{-48}-1286q^{-50}-208q^{-52}+964q^{-54}+739q^{-56}+20q^{-58}-640q^{-60}-488q^{-62}+150q^{-64}+329q^{-66}+252q^{-68}-94q^{-70}-215q^{-72}-53q^{-74}+30q^{-76}+100q^{-78}+16q^{-80}-38q^{-82}-11q^{-84}-12q^{-86}+17q^{-88}+4q^{-90}-6q^{-92}+2q^{-94}-3q^{-96}+3q^{-98}-2q^{-102}+q^{-104}$
5	$-q^{155}+2q^{153}-3q^{149}+3q^{147}+2q^{145}-4q^{143}-2q^{141}-q^{139}-4q^{137}+12q^{135}+25q^{133}-q^{131}-34q^{129}-56q^{127}-38q^{125}+47q^{123}+161q^{121}+168q^{119}-28q^{117}-311q^{115}-454q^{113}-203q^{111}+409q^{109}+956q^{107}+833q^{105}-200q^{103}-1506q^{101}-1950q^{99}-750q^{97}+1630q^{95}+3441q^{93}+2727q^{91}-748q^{89}-4624q^{87}-5553q^{85}-1809q^{83}+4581q^{81}+8584q^{79}+5970q^{77}-2476q^{75}-10507q^{73}-10990q^{71}-2036q^{69}+10210q^{67}+15522q^{65}+8175q^{63}-7130q^{61}-17994q^{59}-14553q^{57}+1623q^{55}+17617q^{53}+19518q^{51}+4895q^{49}-14319q^{47}-21827q^{45}-10943q^{43}+9175q^{41}+21198q^{39}+15066q^{37}-3453q^{35}-18189q^{33}-16820q^{31}-1429q^{29}+13860q^{27}+16264q^{25}+4923q^{23}-9432q^{21}-14396q^{19}-6737q^{17}+5644q^{15}+11929q^{13}+7548q^{11}-2824q^{9}-9911q^{7}-7870q^{5}+865q^{3}+8536q+8536q^{-1}+865q^{-3}-7870q^{-5}-9911q^{-7}-2824q^{-9}+7548q^{-11}+11929q^{-13}+5644q^{-15}-6737q^{-17}-14396q^{-19}-9432q^{-21}+4923q^{-23}+16264q^{-25}+13860q^{-27}-1429q^{-29}-16820q^{-31}-18189q^{-33}-3453q^{-35}+15066q^{-37}+21198q^{-39}+9175q^{-41}-10943q^{-43}-21827q^{-45}-14319q^{-47}+4895q^{-49}+19518q^{-51}+17617q^{-53}+1623q^{-55}-14553q^{-57}-17994q^{-59}-7130q^{-61}+8175q^{-63}+15522q^{-65}+10210q^{-67}-2036q^{-69}-10990q^{-71}-10507q^{-73}-2476q^{-75}+5970q^{-77}+8584q^{-79}+4581q^{-81}-1809q^{-83}-5553q^{-85}-4624q^{-87}-748q^{-89}+2727q^{-91}+3441q^{-93}+1630q^{-95}-750q^{-97}-1950q^{-99}-1506q^{-101}-200q^{-103}+833q^{-105}+956q^{-107}+409q^{-109}-203q^{-111}-454q^{-113}-311q^{-115}-28q^{-117}+168q^{-119}+161q^{-121}+47q^{-123}-38q^{-125}-56q^{-127}-34q^{-129}-q^{-131}+25q^{-133}+12q^{-135}-4q^{-137}-q^{-139}-2q^{-141}-4q^{-143}+2q^{-145}+3q^{-147}-3q^{-149}+2q^{-153}-q^{-155}$
6	$q^{216}-2q^{214}+3q^{210}-3q^{208}-2q^{206}+12q^{202}-q^{200}-12q^{198}+4q^{196}-15q^{194}-12q^{192}+9q^{190}+61q^{188}+40q^{186}-27q^{184}-40q^{182}-126q^{180}-135q^{178}-20q^{176}+270q^{174}+390q^{172}+265q^{170}+29q^{168}-581q^{166}-1037q^{164}-953q^{162}+124q^{160}+1430q^{158}+2280q^{156}+2226q^{154}+150q^{152}-2922q^{150}-5333q^{148}-4570q^{146}-694q^{144}+5142q^{142}+10252q^{140}+9707q^{138}+2449q^{136}-9219q^{134}-17832q^{132}-18043q^{130}-6371q^{128}+13863q^{126}+30171q^{124}+31412q^{122}+11974q^{120}-19134q^{118}-46737q^{116}-50701q^{114}-21835q^{112}+26831q^{110}+68777q^{108}+74379q^{106}+35022q^{104}-35698q^{102}-95905q^{100}-104029q^{98}-47632q^{96}+47803q^{94}+125414q^{92}+135620q^{90}+58788q^{88}-63906q^{86}-158481q^{84}-162175q^{82}-63243q^{80}+82938q^{78}+189641q^{76}+180943q^{74}+57919q^{72}-107400q^{70}-211584q^{68}-185594q^{66}-42662q^{64}+131818q^{62}+222293q^{60}+173641q^{58}+15966q^{56}-149114q^{54}-216952q^{52}-147027q^{50}+14658q^{48}+157680q^{46}+195215q^{44}+107764q^{42}-40913q^{40}-153594q^{38}-161370q^{36}-65709q^{34}+60640q^{32}+137507q^{30}+119529q^{28}+28919q^{26}-70629q^{24}-114160q^{22}-78954q^{20}+1552q^{18}+72515q^{16}+87614q^{14}+44310q^{12}-24876q^{10}-70728q^{8}-64269q^{6}-14025q^{4}+44494q^{2}+69071+44494q^{-2}-14025q^{-4}-64269q^{-6}-70728q^{-8}-24876q^{-10}+44310q^{-12}+87614q^{-14}+72515q^{-16}+1552q^{-18}-78954q^{-20}-114160q^{-22}-70629q^{-24}+28919q^{-26}+119529q^{-28}+137507q^{-30}+60640q^{-32}-65709q^{-34}-161370q^{-36}-153594q^{-38}-40913q^{-40}+107764q^{-42}+195215q^{-44}+157680q^{-46}+14658q^{-48}-147027q^{-50}-216952q^{-52}-149114q^{-54}+15966q^{-56}+173641q^{-58}+222293q^{-60}+131818q^{-62}-42662q^{-64}-185594q^{-66}-211584q^{-68}-107400q^{-70}+57919q^{-72}+180943q^{-74}+189641q^{-76}+82938q^{-78}-63243q^{-80}-162175q^{-82}-158481q^{-84}-63906q^{-86}+58788q^{-88}+135620q^{-90}+125414q^{-92}+47803q^{-94}-47632q^{-96}-104029q^{-98}-95905q^{-100}-35698q^{-102}+35022q^{-104}+74379q^{-106}+68777q^{-108}+26831q^{-110}-21835q^{-112}-50701q^{-114}-46737q^{-116}-19134q^{-118}+11974q^{-120}+31412q^{-122}+30171q^{-124}+13863q^{-126}-6371q^{-128}-18043q^{-130}-17832q^{-132}-9219q^{-134}+2449q^{-136}+9707q^{-138}+10252q^{-140}+5142q^{-142}-694q^{-144}-4570q^{-146}-5333q^{-148}-2922q^{-150}+150q^{-152}+2226q^{-154}+2280q^{-156}+1430q^{-158}+124q^{-160}-953q^{-162}-1037q^{-164}-581q^{-166}+29q^{-168}+265q^{-170}+390q^{-172}+270q^{-174}-20q^{-176}-135q^{-178}-126q^{-180}-40q^{-182}-27q^{-184}+40q^{-186}+61q^{-188}+9q^{-190}-12q^{-192}-15q^{-194}+4q^{-196}-12q^{-198}-q^{-200}+12q^{-202}-2q^{-206}-3q^{-208}+3q^{-210}-2q^{-214}+q^{-216}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{14}+q^{12}-3q^{10}-q^{6}-q^{4}+6q^{2}+1+6q^{-2}-q^{-4}-q^{-6}-3q^{-10}+q^{-12}-q^{-14}$
1,1	$q^{44}-4q^{42}+12q^{40}-28q^{38}+58q^{36}-104q^{34}+178q^{32}-286q^{30}+420q^{28}-576q^{26}+752q^{24}-906q^{22}+987q^{20}-976q^{18}+850q^{16}-598q^{14}+186q^{12}+280q^{10}-796q^{8}+1266q^{6}-1651q^{4}+1934q^{2}-1990+1934q^{-2}-1651q^{-4}+1266q^{-6}-796q^{-8}+280q^{-10}+186q^{-12}-598q^{-14}+850q^{-16}-976q^{-18}+987q^{-20}-906q^{-22}+752q^{-24}-576q^{-26}+420q^{-28}-286q^{-30}+178q^{-32}-104q^{-34}+58q^{-36}-28q^{-38}+12q^{-40}-4q^{-42}+q^{-44}$
2,0	$q^{38}-q^{36}+4q^{32}-2q^{30}-4q^{28}+6q^{26}+4q^{24}-6q^{22}-6q^{20}+7q^{18}+q^{16}-20q^{14}-5q^{12}+5q^{10}-12q^{8}-6q^{6}+17q^{4}+16q^{2}+8+16q^{-2}+17q^{-4}-6q^{-6}-12q^{-8}+5q^{-10}-5q^{-12}-20q^{-14}+q^{-16}+7q^{-18}-6q^{-20}-6q^{-22}+4q^{-24}+6q^{-26}-4q^{-28}-2q^{-30}+4q^{-32}-q^{-36}+q^{-38}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-2q^{32}+q^{30}+5q^{28}-8q^{26}+2q^{24}+13q^{22}-19q^{20}+3q^{18}+19q^{16}-29q^{14}-5q^{12}+15q^{10}-22q^{8}-4q^{6}+20q^{4}+9q^{2}+8+9q^{-2}+20q^{-4}-4q^{-6}-22q^{-8}+15q^{-10}-5q^{-12}-29q^{-14}+19q^{-16}+3q^{-18}-19q^{-20}+13q^{-22}+2q^{-24}-8q^{-26}+5q^{-28}+q^{-30}-2q^{-32}+q^{-34}$
1,0,0	$-q^{17}+q^{15}-4q^{13}+q^{11}-5q^{9}+q^{7}-q^{5}+5q^{3}+5q+5q^{-1}+5q^{-3}-q^{-5}+q^{-7}-5q^{-9}+q^{-11}-4q^{-13}+q^{-15}-q^{-17}$
1,0,1	$q^{56}-4q^{54}+10q^{52}-14q^{50}+7q^{48}+21q^{46}-62q^{44}+95q^{42}-72q^{40}-35q^{38}+192q^{36}-316q^{34}+299q^{32}-66q^{30}-307q^{28}+653q^{26}-759q^{24}+519q^{22}+37q^{20}-667q^{18}+1029q^{16}-1024q^{14}+503q^{12}+58q^{10}-562q^{8}+643q^{6}-371q^{4}+141q^{2}+117+141q^{-2}-371q^{-4}+643q^{-6}-562q^{-8}+58q^{-10}+503q^{-12}-1024q^{-14}+1029q^{-16}-667q^{-18}+37q^{-20}+519q^{-22}-759q^{-24}+653q^{-26}-307q^{-28}-66q^{-30}+299q^{-32}-316q^{-34}+192q^{-36}-35q^{-38}-72q^{-40}+95q^{-42}-62q^{-44}+21q^{-46}+7q^{-48}-14q^{-50}+10q^{-52}-4q^{-54}+q^{-56}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{40}-q^{38}+5q^{34}-2q^{32}-3q^{30}+12q^{28}-q^{26}-11q^{24}+9q^{22}+8q^{20}-24q^{18}-18q^{16}+q^{14}-17q^{12}-34q^{10}+2q^{8}+28q^{6}-2q^{4}+23q^{2}+58+23q^{-2}-2q^{-4}+28q^{-6}+2q^{-8}-34q^{-10}-17q^{-12}+q^{-14}-18q^{-16}-24q^{-18}+8q^{-20}+9q^{-22}-11q^{-24}-q^{-26}+12q^{-28}-3q^{-30}-2q^{-32}+5q^{-34}-q^{-38}+q^{-40}$
1,0,0,0	$-q^{20}+q^{18}-4q^{16}-4q^{12}-3q^{10}-q^{6}+6q^{4}+4q^{2}+9+4q^{-2}+6q^{-4}-q^{-6}-3q^{-10}-4q^{-12}-4q^{-16}+q^{-18}-q^{-20}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2q^{32}-5q^{30}+9q^{28}-14q^{26}+20q^{24}-27q^{22}+29q^{20}-31q^{18}+27q^{16}-21q^{14}+9q^{12}+5q^{10}-20q^{8}+36q^{6}-46q^{4}+59q^{2}-58+59q^{-2}-46q^{-4}+36q^{-6}-20q^{-8}+5q^{-10}+9q^{-12}-21q^{-14}+27q^{-16}-31q^{-18}+29q^{-20}-27q^{-22}+20q^{-24}-14q^{-26}+9q^{-28}-5q^{-30}+2q^{-32}-q^{-34}$
1,0	$q^{56}-2q^{52}-2q^{50}+3q^{48}+7q^{46}-11q^{42}-8q^{40}+11q^{38}+20q^{36}-4q^{34}-27q^{32}-12q^{30}+25q^{28}+25q^{26}-15q^{24}-35q^{22}-5q^{20}+27q^{18}+11q^{16}-24q^{14}-21q^{12}+14q^{10}+22q^{8}-3q^{6}-17q^{4}+9q^{2}+27+9q^{-2}-17q^{-4}-3q^{-6}+22q^{-8}+14q^{-10}-21q^{-12}-24q^{-14}+11q^{-16}+27q^{-18}-5q^{-20}-35q^{-22}-15q^{-24}+25q^{-26}+25q^{-28}-12q^{-30}-27q^{-32}-4q^{-34}+20q^{-36}+11q^{-38}-8q^{-40}-11q^{-42}+7q^{-46}+3q^{-48}-2q^{-50}-2q^{-52}+q^{-56}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{46}-2q^{44}+3q^{42}-4q^{40}+8q^{38}-11q^{36}+13q^{34}-16q^{32}+22q^{30}-24q^{28}+23q^{26}-23q^{24}+22q^{22}-22q^{20}+4q^{18}-12q^{16}-7q^{14}+6q^{12}-27q^{10}+27q^{8}-28q^{6}+54q^{4}-32q^{2}+58-32q^{-2}+54q^{-4}-28q^{-6}+27q^{-8}-27q^{-10}+6q^{-12}-7q^{-14}-12q^{-16}+4q^{-18}-22q^{-20}+22q^{-22}-23q^{-24}+23q^{-26}-24q^{-28}+22q^{-30}-16q^{-32}+13q^{-34}-11q^{-36}+8q^{-38}-4q^{-40}+3q^{-42}-2q^{-44}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+5q^{76}-8q^{74}+9q^{72}-7q^{70}+14q^{66}-30q^{64}+46q^{62}-56q^{60}+42q^{58}-13q^{56}-38q^{54}+99q^{52}-139q^{50}+152q^{48}-109q^{46}+11q^{44}+103q^{42}-204q^{40}+232q^{38}-184q^{36}+63q^{34}+71q^{32}-177q^{30}+209q^{28}-138q^{26}+4q^{24}+117q^{22}-185q^{20}+147q^{18}-29q^{16}-127q^{14}+240q^{12}-255q^{10}+205q^{8}-47q^{6}-139q^{4}+285q^{2}-329+285q^{-2}-139q^{-4}-47q^{-6}+205q^{-8}-255q^{-10}+240q^{-12}-127q^{-14}-29q^{-16}+147q^{-18}-185q^{-20}+117q^{-22}+4q^{-24}-138q^{-26}+209q^{-28}-177q^{-30}+71q^{-32}+63q^{-34}-184q^{-36}+232q^{-38}-204q^{-40}+103q^{-42}+11q^{-44}-109q^{-46}+152q^{-48}-139q^{-50}+99q^{-52}-38q^{-54}-13q^{-56}+42q^{-58}-56q^{-60}+46q^{-62}-30q^{-64}+14q^{-66}-7q^{-70}+9q^{-72}-8q^{-74}+5q^{-76}-2q^{-78}+q^{-80}$

.

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 99"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-4t^{3}+10t^{2}-16t+19-16t^{-1}+10t^{-2}-4t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\left\{t^{4}-2t^{3}+3t^{2}-2t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 81, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+3q^{4}-7q^{3}+10q^{2}-12q+15-12q^{-1}+10q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-4z^{4}a^{-2}+14z^{4}-6a^{2}z^{2}-6z^{2}a^{-2}+16z^{2}-4a^{2}-4a^{-2}+9$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2az^{9}+2z^{9}a^{-1}+5a^{2}z^{8}+5z^{8}a^{-2}+10z^{8}+5a^{3}z^{7}+5az^{7}+5z^{7}a^{-1}+5z^{7}a^{-3}+3a^{4}z^{6}-9a^{2}z^{6}-9z^{6}a^{-2}+3z^{6}a^{-4}-24z^{6}+a^{5}z^{5}-9a^{3}z^{5}-18az^{5}-18z^{5}a^{-1}-9z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}+9a^{2}z^{4}+9z^{4}a^{-2}-5z^{4}a^{-4}+28z^{4}-2a^{5}z^{3}+5a^{3}z^{3}+21az^{3}+21z^{3}a^{-1}+5z^{3}a^{-3}-2z^{3}a^{-5}+a^{4}z^{2}-8a^{2}z^{2}-8z^{2}a^{-2}+z^{2}a^{-4}-18z^{2}+a^{5}z-3a^{3}z-10az-10za^{-1}-3za^{-3}+za^{-5}+4a^{2}+4a^{-2}+9$

Vassiliev invariants

V2 and V3:

(4, 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 $16$ $0$ $128$ ${\frac {440}{3}}$ ${\frac {40}{3}}$ $0$ $0$ $0$ $0$ ${\frac {2048}{3}}$ $0$ ${\frac {7040}{3}}$ ${\frac {640}{3}}$ ${\frac {34622}{15}}$ ${\frac {984}{5}}$ ${\frac {29528}{45}}$ ${\frac {34}{9}}$ ${\frac {542}{15}}$

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 $t^{r}q^{j}$ 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 99. 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               5 2     3 3             7 5       -2 1           8 5         3 -1         5 8           3 -3       5 7             -2 -5     2 5               3 -7   1 5                 -4 -9   2                   2 -11 1                     -1

Integral Khovanov Homology (db, data source)

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

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, 99]]
Out[2]=	10
In[3]:=	PD[Knot[10, 99]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[16, 11, 17, 12], X[14, 7, 15, 8], X[8, 15, 9, 16], X[20, 13, 1, 14], X[12, 19, 13, 20], X[18, 6, 19, 5], X[2, 10, 3, 9], X[4, 18, 5, 17]]
In[4]:=	GaussCode[Knot[10, 99]]
Out[4]=	GaussCode[1, -9, 2, -10, 8, -1, 4, -5, 9, -2, 3, -7, 6, -4, 5, -3, 10, -8, 7, -6]
In[5]:=	BR[Knot[10, 99]]
Out[5]=	BR[3, {-1, -1, 2, -1, -1, 2, 2, -1, 2, 2}]
In[6]:=	alex = Alexander[Knot[10, 99]][t]
Out[6]=	-4 4 10 16 2 3 4 19 + t - -- + -- - -- - 16 t + 10 t - 4 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 99]][z]
Out[7]=	2 4 6 8 1 + 4 z + 6 z + 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 99]}
In[9]:=	{KnotDet[Knot[10, 99]], KnotSignature[Knot[10, 99]]}
Out[9]=	{81, 0}
In[10]:=	J=Jones[Knot[10, 99]][q]
Out[10]=	-5 3 7 10 12 2 3 4 5 15 - q + -- - -- + -- - -- - 12 q + 10 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, 99]}
In[12]:=	A2Invariant[Knot[10, 99]][q]
Out[12]=	-14 -12 3 -6 -4 6 2 4 6 10 12 1 - q + q - --- - q - q + -- + 6 q - q - q - 3 q + q - 10 2 q q 14 q
In[13]:=	Kauffman[Knot[10, 99]][a, z]
Out[13]=	2 4 2 z 3 z 10 z 3 5 2 z 9 + -- + 4 a + -- - --- - ---- - 10 a z - 3 a z + a z - 18 z + -- - 2 5 3 a 4 a a a a 2 3 3 3 8 z 2 2 4 2 2 z 5 z 21 z 3 3 3 ---- - 8 a z + a z - ---- + ---- + ----- + 21 a z + 5 a z - 2 5 3 a a a a 4 4 5 5 5 3 4 5 z 9 z 2 4 4 4 z 9 z 2 a z + 28 z - ---- + ---- + 9 a z - 5 a z + -- - ---- - 4 2 5 3 a a a a 5 6 6 18 z 5 3 5 5 5 6 3 z 9 z 2 6 ----- - 18 a z - 9 a z + a z - 24 z + ---- - ---- - 9 a z + a 4 2 a a 7 7 8 4 6 5 z 5 z 7 3 7 8 5 z 2 8 3 a z + ---- + ---- + 5 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, 99]], Vassiliev[3][Knot[10, 99]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 99]][q, t]
Out[15]=	8 1 2 1 5 2 5 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 5 3 3 2 5 2 5 3 7 3 ---- + --- + 5 q t + 7 q t + 5 q t + 5 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