10 104

10_103

10_105

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10 104 Quick Notes

10 104 Further Notes and Views

Knot presentations

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

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 14.1071 A-Polynomial See Data:10 104/A-polynomial

[edit Notes for 10 104'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 10 104's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-4t^{3}+9t^{2}-15t+19-15t^{-1}+9t^{-2}-4t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+4z^{6}+5z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 77, 0 }
Jones polynomial	$-q^{5}+3q^{4}-6q^{3}+10q^{2}-12q+13-12q^{-1}+10q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-4z^{4}a^{-2}+13z^{4}-5a^{2}z^{2}-5z^{2}a^{-2}+11z^{2}-a^{2}-a^{-2}+3$
Kauffman polynomial (db, data sources)	$2az^{9}+2z^{9}a^{-1}+4a^{2}z^{8}+5z^{8}a^{-2}+9z^{8}+4a^{3}z^{7}+2az^{7}+3z^{7}a^{-1}+5z^{7}a^{-3}+3a^{4}z^{6}-5a^{2}z^{6}-11z^{6}a^{-2}+3z^{6}a^{-4}-22z^{6}+a^{5}z^{5}-5a^{3}z^{5}-6az^{5}-12z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}+3a^{2}z^{4}+12z^{4}a^{-2}-6z^{4}a^{-4}+27z^{4}-2a^{5}z^{3}-a^{3}z^{3}+4az^{3}+13z^{3}a^{-1}+8z^{3}a^{-3}-2z^{3}a^{-5}+3a^{4}z^{2}-4a^{2}z^{2}-6z^{2}a^{-2}+2z^{2}a^{-4}-15z^{2}+a^{5}z+a^{3}z-2az-4za^{-1}-2za^{-3}+a^{2}+a^{-2}+3$
The A2 invariant	$-q^{14}+q^{12}-2q^{10}+2q^{8}+q^{6}-q^{4}+3q^{2}-3+3q^{-2}-q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+q^{-12}-q^{-14}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+8q^{72}-6q^{70}-2q^{68}+16q^{66}-29q^{64}+41q^{62}-43q^{60}+30q^{58}-6q^{56}-36q^{54}+82q^{52}-118q^{50}+121q^{48}-87q^{46}+9q^{44}+85q^{42}-164q^{40}+201q^{38}-162q^{36}+66q^{34}+57q^{32}-157q^{30}+181q^{28}-122q^{26}+15q^{24}+99q^{22}-156q^{20}+131q^{18}-27q^{16}-104q^{14}+206q^{12}-236q^{10}+168q^{8}-34q^{6}-123q^{4}+244q^{2}-286+243q^{-2}-119q^{-4}-35q^{-6}+168q^{-8}-234q^{-10}+212q^{-12}-111q^{-14}-17q^{-16}+126q^{-18}-158q^{-20}+111q^{-22}-q^{-24}-113q^{-26}+180q^{-28}-166q^{-30}+68q^{-32}+57q^{-34}-166q^{-36}+212q^{-38}-177q^{-40}+91q^{-42}+12q^{-44}-99q^{-46}+137q^{-48}-128q^{-50}+84q^{-52}-29q^{-54}-16q^{-56}+40q^{-58}-47q^{-60}+40q^{-62}-25q^{-64}+12q^{-66}+q^{-68}-7q^{-70}+7q^{-72}-7q^{-74}+4q^{-76}-2q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_104.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-3q^{7}+4q^{5}-2q^{3}+q+q^{-1}-2q^{-3}+4q^{-5}-3q^{-7}+2q^{-9}-q^{-11}$
2	$q^{32}-2q^{30}-q^{28}+6q^{26}-7q^{24}-5q^{22}+20q^{20}-10q^{18}-20q^{16}+30q^{14}-30q^{10}+20q^{8}+12q^{6}-21q^{4}+16-20q^{-4}+12q^{-6}+20q^{-8}-30q^{-10}+q^{-12}+30q^{-14}-21q^{-16}-10q^{-18}+21q^{-20}-5q^{-22}-9q^{-24}+6q^{-26}-2q^{-30}+q^{-32}$
3	$-q^{63}+2q^{61}+q^{59}-2q^{57}-3q^{55}+4q^{53}+5q^{51}-12q^{49}-9q^{47}+23q^{45}+25q^{43}-37q^{41}-58q^{39}+43q^{37}+106q^{35}-25q^{33}-152q^{31}-25q^{29}+187q^{27}+83q^{25}-184q^{23}-144q^{21}+150q^{19}+186q^{17}-100q^{15}-196q^{13}+42q^{11}+182q^{9}+15q^{7}-148q^{5}-63q^{3}+110q+105q^{-1}-68q^{-3}-143q^{-5}+21q^{-7}+176q^{-9}+30q^{-11}-195q^{-13}-84q^{-15}+195q^{-17}+135q^{-19}-162q^{-21}-176q^{-23}+106q^{-25}+190q^{-27}-44q^{-29}-164q^{-31}-16q^{-33}+124q^{-35}+46q^{-37}-72q^{-39}-50q^{-41}+29q^{-43}+36q^{-45}-8q^{-47}-20q^{-49}+2q^{-51}+8q^{-53}-3q^{-57}+2q^{-61}-q^{-63}$
4	$q^{104}-2q^{102}-q^{100}+2q^{98}-q^{96}+6q^{94}-4q^{92}-q^{90}+6q^{88}-13q^{86}+3q^{84}-17q^{82}+14q^{80}+59q^{78}-2q^{76}-37q^{74}-139q^{72}-39q^{70}+201q^{68}+216q^{66}+68q^{64}-404q^{62}-455q^{60}+99q^{58}+619q^{56}+712q^{54}-286q^{52}-1082q^{50}-708q^{48}+488q^{46}+1553q^{44}+648q^{42}-1031q^{40}-1655q^{38}-494q^{36}+1580q^{34}+1633q^{32}-98q^{30}-1731q^{28}-1431q^{26}+730q^{24}+1728q^{22}+789q^{20}-995q^{18}-1542q^{16}-151q^{14}+1115q^{12}+1070q^{10}-204q^{8}-1127q^{6}-653q^{4}+471q^{2}+1058+374q^{-2}-725q^{-4}-1067q^{-6}-73q^{-8}+1107q^{-10}+991q^{-12}-316q^{-14}-1535q^{-16}-788q^{-18}+970q^{-20}+1658q^{-22}+449q^{-24}-1609q^{-26}-1581q^{-28}+226q^{-30}+1774q^{-32}+1368q^{-34}-824q^{-36}-1739q^{-38}-798q^{-40}+938q^{-42}+1594q^{-44}+274q^{-46}-928q^{-48}-1106q^{-50}-124q^{-52}+874q^{-54}+645q^{-56}+2q^{-58}-577q^{-60}-442q^{-62}+123q^{-64}+301q^{-66}+244q^{-68}-75q^{-70}-200q^{-72}-65q^{-74}+21q^{-76}+100q^{-78}+24q^{-80}-34q^{-82}-16q^{-84}-14q^{-86}+18q^{-88}+6q^{-90}-6q^{-92}+q^{-94}-3q^{-96}+3q^{-98}-2q^{-102}+q^{-104}$
5	$-q^{155}+2q^{153}+q^{151}-2q^{149}+q^{147}-2q^{145}-6q^{143}+7q^{139}+4q^{137}+12q^{135}+10q^{133}-17q^{131}-36q^{129}-32q^{127}-3q^{125}+56q^{123}+116q^{121}+87q^{119}-66q^{117}-240q^{115}-284q^{113}-73q^{111}+348q^{109}+680q^{107}+494q^{105}-287q^{103}-1156q^{101}-1335q^{99}-316q^{97}+1447q^{95}+2588q^{93}+1730q^{91}-1032q^{89}-3792q^{87}-4020q^{85}-680q^{83}+4223q^{81}+6723q^{79}+3881q^{77}-3075q^{75}-8851q^{73}-8085q^{71}-149q^{69}+9311q^{67}+12280q^{65}+5034q^{63}-7480q^{61}-15006q^{59}-10508q^{57}+3365q^{55}+15430q^{53}+15145q^{51}+1960q^{49}-13320q^{47}-17683q^{45}-7211q^{43}+9314q^{41}+17762q^{39}+11109q^{37}-4611q^{35}-15671q^{33}-12973q^{31}+295q^{29}+12259q^{27}+12965q^{25}+2839q^{23}-8578q^{21}-11604q^{19}-4646q^{17}+5353q^{15}+9734q^{13}+5462q^{11}-2959q^{9}-8065q^{7}-5830q^{5}+1317q^{3}+6996q+6351q^{-1}-16q^{-3}-6591q^{-5}-7451q^{-7}-1423q^{-9}+6534q^{-11}+9215q^{-13}+3487q^{-15}-6274q^{-17}-11394q^{-19}-6417q^{-21}+5223q^{-23}+13382q^{-25}+10045q^{-27}-2878q^{-29}-14348q^{-31}-13827q^{-33}-870q^{-35}+13628q^{-37}+16794q^{-39}+5499q^{-41}-10758q^{-43}-17996q^{-45}-10100q^{-47}+6084q^{-49}+16818q^{-51}+13399q^{-53}-605q^{-55}-13207q^{-57}-14458q^{-59}-4393q^{-61}+8092q^{-63}+13040q^{-65}+7534q^{-67}-2798q^{-69}-9589q^{-71}-8378q^{-73}-1344q^{-75}+5448q^{-77}+7124q^{-79}+3503q^{-81}-1776q^{-83}-4729q^{-85}-3816q^{-87}-556q^{-89}+2305q^{-91}+2921q^{-93}+1441q^{-95}-578q^{-97}-1659q^{-99}-1356q^{-101}-269q^{-103}+680q^{-105}+875q^{-107}+431q^{-109}-133q^{-111}-412q^{-113}-323q^{-115}-59q^{-117}+155q^{-119}+169q^{-121}+60q^{-123}-35q^{-125}-62q^{-127}-40q^{-129}+q^{-131}+28q^{-133}+13q^{-135}-5q^{-137}-3q^{-139}-2q^{-141}-3q^{-143}+2q^{-145}+3q^{-147}-3q^{-149}+2q^{-153}-q^{-155}$
6	$q^{216}-2q^{214}-q^{212}+2q^{210}-q^{208}+2q^{206}+2q^{204}+10q^{202}-6q^{200}-17q^{198}-3q^{196}-13q^{194}-q^{192}+18q^{190}+64q^{188}+32q^{186}-30q^{184}-46q^{182}-110q^{180}-107q^{178}-16q^{176}+221q^{174}+288q^{172}+191q^{170}+22q^{168}-402q^{166}-719q^{164}-663q^{162}+120q^{160}+965q^{158}+1509q^{156}+1455q^{154}+40q^{152}-2039q^{150}-3613q^{148}-2902q^{146}-184q^{144}+3784q^{142}+7091q^{140}+6335q^{138}+887q^{136}-7324q^{134}-12778q^{132}-11978q^{130}-2741q^{128}+11913q^{126}+22565q^{124}+21343q^{122}+5211q^{120}-17867q^{118}-36180q^{116}-35286q^{114}-10285q^{112}+26717q^{110}+54694q^{108}+52634q^{106}+17382q^{104}-37270q^{102}-77786q^{100}-74642q^{98}-23577q^{96}+50689q^{94}+102814q^{92}+98204q^{90}+28659q^{88}-67117q^{86}-130261q^{84}-117497q^{82}-28938q^{80}+84770q^{78}+155408q^{76}+130572q^{74}+22279q^{72}-105255q^{70}-171854q^{68}-132481q^{66}-9272q^{64}+123887q^{62}+178331q^{60}+121446q^{58}-11339q^{56}-134837q^{54}-171214q^{52}-99656q^{50}+33358q^{48}+137568q^{46}+150892q^{44}+68755q^{42}-50367q^{40}-129519q^{38}-121518q^{36}-36909q^{34}+61371q^{32}+112138q^{30}+86762q^{28}+10286q^{26}-64462q^{24}-89710q^{22}-54145q^{20}+10755q^{18}+61572q^{16}+65831q^{14}+27076q^{12}-26133q^{10}-56386q^{8}-45327q^{6}-3919q^{4}+38752q^{2}+51908+28086q^{-2}-17504q^{-4}-51727q^{-6}-50221q^{-8}-10989q^{-10}+40669q^{-12}+67436q^{-14}+48458q^{-16}-9260q^{-18}-67262q^{-20}-85269q^{-22}-43241q^{-24}+35261q^{-26}+98116q^{-28}+100004q^{-30}+31346q^{-32}-66062q^{-34}-129418q^{-36}-108427q^{-38}-11799q^{-40}+100315q^{-42}+153682q^{-44}+107507q^{-46}-12011q^{-48}-131484q^{-50}-168070q^{-52}-97266q^{-54}+37826q^{-56}+151378q^{-58}+169818q^{-60}+81856q^{-62}-58799q^{-64}-158959q^{-66}-159612q^{-68}-62701q^{-70}+68653q^{-72}+152982q^{-74}+141879q^{-76}+45428q^{-78}-69353q^{-80}-135727q^{-82}-117868q^{-84}-34108q^{-86}+61500q^{-88}+112599q^{-90}+93187q^{-92}+25736q^{-94}-48170q^{-96}-85693q^{-98}-71766q^{-100}-20451q^{-102}+34132q^{-104}+60737q^{-106}+52052q^{-108}+17061q^{-110}-20425q^{-112}-40974q^{-114}-35931q^{-116}-13594q^{-118}+10413q^{-120}+25035q^{-122}+23690q^{-124}+10983q^{-126}-4816q^{-128}-14087q^{-130}-14363q^{-132}-7978q^{-134}+1259q^{-136}+7438q^{-138}+8505q^{-140}+4808q^{-142}+131q^{-144}-3439q^{-146}-4580q^{-148}-2905q^{-150}-329q^{-152}+1697q^{-154}+2045q^{-156}+1509q^{-158}+338q^{-160}-756q^{-162}-989q^{-164}-644q^{-166}-59q^{-168}+225q^{-170}+398q^{-172}+291q^{-174}+5q^{-176}-134q^{-178}-137q^{-180}-47q^{-182}-25q^{-184}+44q^{-186}+61q^{-188}+9q^{-190}-14q^{-192}-16q^{-194}+5q^{-196}-10q^{-198}-q^{-200}+11q^{-202}-2q^{-206}-3q^{-208}+3q^{-210}-2q^{-214}+q^{-216}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{14}+q^{12}-2q^{10}+2q^{8}+q^{6}-q^{4}+3q^{2}-3+3q^{-2}-q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+q^{-12}-q^{-14}$
1,1	$q^{44}-4q^{42}+12q^{40}-28q^{38}+54q^{36}-96q^{34}+154q^{32}-236q^{30}+341q^{28}-470q^{26}+610q^{24}-728q^{22}+816q^{20}-832q^{18}+740q^{16}-528q^{14}+208q^{12}+180q^{10}-618q^{8}+1040q^{6}-1381q^{4}+1618q^{2}-1706+1654q^{-2}-1453q^{-4}+1130q^{-6}-728q^{-8}+288q^{-10}+134q^{-12}-476q^{-14}+726q^{-16}-856q^{-18}+868q^{-20}-788q^{-22}+654q^{-24}-510q^{-26}+365q^{-28}-238q^{-30}+148q^{-32}-90q^{-34}+48q^{-36}-22q^{-38}+10q^{-40}-4q^{-42}+q^{-44}$
2,0	$q^{38}-q^{36}-q^{34}+2q^{32}-3q^{30}-4q^{28}+5q^{26}+3q^{24}-5q^{22}-2q^{20}+10q^{18}+6q^{16}-14q^{14}+2q^{12}+10q^{10}-8q^{8}-7q^{6}+7q^{4}+3q^{2}-8+4q^{-2}+8q^{-4}-4q^{-6}-6q^{-8}+12q^{-10}+2q^{-12}-13q^{-14}+7q^{-16}+9q^{-18}-4q^{-20}-7q^{-22}+3q^{-24}+4q^{-26}-5q^{-28}-3q^{-30}+3q^{-32}-q^{-36}+q^{-38}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-2q^{32}+q^{30}+4q^{28}-8q^{26}+2q^{24}+11q^{22}-17q^{20}+2q^{18}+19q^{16}-24q^{14}+q^{12}+20q^{10}-16q^{8}-2q^{6}+13q^{4}+q^{2}-4+13q^{-4}-3q^{-6}-16q^{-8}+19q^{-10}+q^{-12}-24q^{-14}+19q^{-16}+2q^{-18}-17q^{-20}+12q^{-22}+2q^{-24}-7q^{-26}+4q^{-28}-2q^{-32}+q^{-34}$
1,0,0	$-q^{17}+q^{15}-3q^{13}+3q^{11}-2q^{9}+3q^{7}-q^{5}+2q^{3}+2q^{-3}-q^{-5}+3q^{-7}-2q^{-9}+3q^{-11}-3q^{-13}+q^{-15}-q^{-17}$
1,0,1	$q^{56}-4q^{54}+10q^{52}-14q^{50}+7q^{48}+18q^{46}-57q^{44}+82q^{42}-65q^{40}-20q^{38}+149q^{36}-257q^{34}+246q^{32}-61q^{30}-242q^{28}+540q^{26}-622q^{24}+402q^{22}+63q^{20}-586q^{18}+887q^{16}-836q^{14}+431q^{12}+105q^{10}-503q^{8}+600q^{6}-388q^{4}+105q^{2}+46+67q^{-2}-326q^{-4}+506q^{-6}-412q^{-8}+21q^{-10}+497q^{-12}-860q^{-14}+894q^{-16}-552q^{-18}+18q^{-20}+456q^{-22}-679q^{-24}+576q^{-26}-275q^{-28}-51q^{-30}+247q^{-32}-269q^{-34}+170q^{-36}-38q^{-38}-49q^{-40}+72q^{-42}-54q^{-44}+20q^{-46}+4q^{-48}-10q^{-50}+8q^{-52}-4q^{-54}+q^{-56}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{40}-q^{38}+4q^{34}-3q^{32}-4q^{30}+9q^{28}-3q^{26}-13q^{24}+8q^{22}+9q^{20}-16q^{18}-7q^{16}+15q^{14}-19q^{10}+8q^{8}+22q^{6}-14q^{4}-q^{2}+29-2q^{-2}-17q^{-4}+19q^{-6}+5q^{-8}-21q^{-10}-3q^{-12}+14q^{-14}-6q^{-16}-15q^{-18}+10q^{-20}+10q^{-22}-10q^{-24}-2q^{-26}+9q^{-28}-4q^{-30}-3q^{-32}+3q^{-34}-q^{-36}-q^{-38}+q^{-40}$
1,0,0,0	$-q^{20}+q^{18}-3q^{16}+2q^{14}-q^{12}+2q^{8}-q^{6}+3q^{4}-q^{2}+3-q^{-2}+3q^{-4}-q^{-6}+2q^{-8}-q^{-12}+2q^{-14}-3q^{-16}+q^{-18}-q^{-20}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2q^{32}-5q^{30}+8q^{28}-12q^{26}+18q^{24}-23q^{22}+27q^{20}-28q^{18}+25q^{16}-18q^{14}+9q^{12}+4q^{10}-18q^{8}+32q^{6}-43q^{4}+51q^{2}-54+52q^{-2}-43q^{-4}+33q^{-6}-18q^{-8}+5q^{-10}+9q^{-12}-18q^{-14}+25q^{-16}-28q^{-18}+27q^{-20}-24q^{-22}+18q^{-24}-13q^{-26}+8q^{-28}-4q^{-30}+2q^{-32}-q^{-34}$
1,0	$q^{56}-2q^{52}-2q^{50}+3q^{48}+6q^{46}-q^{44}-10q^{42}-7q^{40}+10q^{38}+17q^{36}-3q^{34}-24q^{32}-11q^{30}+22q^{28}+24q^{26}-12q^{24}-30q^{22}-3q^{20}+28q^{18}+13q^{16}-20q^{14}-17q^{12}+13q^{10}+19q^{8}-6q^{6}-18q^{4}+3q^{2}+20+3q^{-2}-18q^{-4}-6q^{-6}+18q^{-8}+13q^{-10}-17q^{-12}-20q^{-14}+12q^{-16}+28q^{-18}-2q^{-20}-30q^{-22}-13q^{-24}+24q^{-26}+23q^{-28}-11q^{-30}-25q^{-32}-3q^{-34}+18q^{-36}+10q^{-38}-7q^{-40}-10q^{-42}+6q^{-46}+2q^{-48}-2q^{-50}-2q^{-52}+q^{-56}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{46}-2q^{44}+3q^{42}-4q^{40}+7q^{38}-10q^{36}+11q^{34}-14q^{32}+19q^{30}-22q^{28}+20q^{26}-21q^{24}+21q^{22}-18q^{20}+8q^{18}-6q^{16}+11q^{12}-21q^{10}+25q^{8}-28q^{6}+42q^{4}-38q^{2}+42-38q^{-2}+42q^{-4}-29q^{-6}+25q^{-8}-22q^{-10}+11q^{-12}-q^{-14}-6q^{-16}+7q^{-18}-18q^{-20}+21q^{-22}-21q^{-24}+21q^{-26}-22q^{-28}+20q^{-30}-14q^{-32}+12q^{-34}-10q^{-36}+7q^{-38}-4q^{-40}+2q^{-42}-2q^{-44}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+5q^{76}-8q^{74}+8q^{72}-6q^{70}-2q^{68}+16q^{66}-29q^{64}+41q^{62}-43q^{60}+30q^{58}-6q^{56}-36q^{54}+82q^{52}-118q^{50}+121q^{48}-87q^{46}+9q^{44}+85q^{42}-164q^{40}+201q^{38}-162q^{36}+66q^{34}+57q^{32}-157q^{30}+181q^{28}-122q^{26}+15q^{24}+99q^{22}-156q^{20}+131q^{18}-27q^{16}-104q^{14}+206q^{12}-236q^{10}+168q^{8}-34q^{6}-123q^{4}+244q^{2}-286+243q^{-2}-119q^{-4}-35q^{-6}+168q^{-8}-234q^{-10}+212q^{-12}-111q^{-14}-17q^{-16}+126q^{-18}-158q^{-20}+111q^{-22}-q^{-24}-113q^{-26}+180q^{-28}-166q^{-30}+68q^{-32}+57q^{-34}-166q^{-36}+212q^{-38}-177q^{-40}+91q^{-42}+12q^{-44}-99q^{-46}+137q^{-48}-128q^{-50}+84q^{-52}-29q^{-54}-16q^{-56}+40q^{-58}-47q^{-60}+40q^{-62}-25q^{-64}+12q^{-66}+q^{-68}-7q^{-70}+7q^{-72}-7q^{-74}+4q^{-76}-2q^{-78}+q^{-80}$

.

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["10 104"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-4t^{3}+9t^{2}-15t+19-15t^{-1}+9t^{-2}-4t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 77, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+3q^{4}-6q^{3}+10q^{2}-12q+13-12q^{-1}+10q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-4z^{4}a^{-2}+13z^{4}-5a^{2}z^{2}-5z^{2}a^{-2}+11z^{2}-a^{2}-a^{-2}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2az^{9}+2z^{9}a^{-1}+4a^{2}z^{8}+5z^{8}a^{-2}+9z^{8}+4a^{3}z^{7}+2az^{7}+3z^{7}a^{-1}+5z^{7}a^{-3}+3a^{4}z^{6}-5a^{2}z^{6}-11z^{6}a^{-2}+3z^{6}a^{-4}-22z^{6}+a^{5}z^{5}-5a^{3}z^{5}-6az^{5}-12z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}+3a^{2}z^{4}+12z^{4}a^{-2}-6z^{4}a^{-4}+27z^{4}-2a^{5}z^{3}-a^{3}z^{3}+4az^{3}+13z^{3}a^{-1}+8z^{3}a^{-3}-2z^{3}a^{-5}+3a^{4}z^{2}-4a^{2}z^{2}-6z^{2}a^{-2}+2z^{2}a^{-4}-15z^{2}+a^{5}z+a^{3}z-2az-4za^{-1}-2za^{-3}+a^{2}+a^{-2}+3$

Vassiliev invariants

V2 and V3:

(1, 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 $4$ $0$ $8$ $-{\frac {82}{3}}$ $-{\frac {110}{3}}$ $0$ $32$ $-32$ $64$ ${\frac {32}{3}}$ $0$ $-{\frac {328}{3}}$ $-{\frac {440}{3}}$ $-{\frac {4769}{30}}$ ${\frac {338}{15}}$ $-{\frac {7138}{45}}$ $-{\frac {895}{18}}$ $-{\frac {1409}{30}}$

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 10 104. 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									4	1		-3
5								6	2			4
3							6	4				-2
1						7	6					1
-1					6	7						1
-3				4	6							-2
-5			2	6								4
-7		1	4									-3
-9		2										2
-11	1											-1

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, 104]]
Out[2]=	10
In[3]:=	PD[Knot[10, 104]]
Out[3]=	PD[X[6, 2, 7, 1], X[16, 4, 17, 3], X[18, 9, 19, 10], X[14, 7, 15, 8], X[20, 13, 1, 14], X[8, 17, 9, 18], X[10, 19, 11, 20], X[12, 6, 13, 5], X[4, 12, 5, 11], X[2, 16, 3, 15]]
In[4]:=	GaussCode[Knot[10, 104]]
Out[4]=	GaussCode[1, -10, 2, -9, 8, -1, 4, -6, 3, -7, 9, -8, 5, -4, 10, -2, 6, -3, 7, -5]
In[5]:=	BR[Knot[10, 104]]
Out[5]=	BR[3, {-1, -1, -1, 2, 2, -1, 2, -1, 2, 2}]
In[6]:=	alex = Alexander[Knot[10, 104]][t]
Out[6]=	-4 4 9 15 2 3 4 19 + t - -- + -- - -- - 15 t + 9 t - 4 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 104]][z]
Out[7]=	2 4 6 8 1 + z + 5 z + 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 104]}
In[9]:=	{KnotDet[Knot[10, 104]], KnotSignature[Knot[10, 104]]}
Out[9]=	{77, 0}
In[10]:=	J=Jones[Knot[10, 104]][q]
Out[10]=	-5 3 6 10 12 2 3 4 5 13 - q + -- - -- + -- - -- - 12 q + 10 q - 6 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, 71], Knot[10, 104]}
In[12]:=	A2Invariant[Knot[10, 104]][q]
Out[12]=	-14 -12 2 2 -6 -4 3 2 4 6 8 -3 - q + q - --- + -- + q - q + -- + 3 q - q + q + 2 q - 10 8 2 q q q 10 12 14 2 q + q - q
In[13]:=	Kauffman[Knot[10, 104]][a, z]
Out[13]=	2 2 -2 2 2 z 4 z 3 5 2 2 z 6 z 3 + a + a - --- - --- - 2 a z + a z + a z - 15 z + ---- - ---- - 3 a 4 2 a a a 3 3 3 2 2 4 2 2 z 8 z 13 z 3 3 3 5 3 4 a z + 3 a z - ---- + ---- + ----- + 4 a z - a z - 2 a z + 5 3 a a a 4 4 5 5 5 4 6 z 12 z 2 4 4 4 z 11 z 12 z 27 z - ---- + ----- + 3 a z - 6 a z + -- - ----- - ----- - 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 3 z 11 z 2 6 4 6 6 a z - 5 a z + a z - 22 z + ---- - ----- - 5 a z + 3 a z + 4 2 a a 7 7 8 9 5 z 3 z 7 3 7 8 5 z 2 8 2 z 9 ---- + ---- + 2 a z + 4 a z + 9 z + ---- + 4 a z + ---- + 2 a z 3 a 2 a a a
In[14]:=	{Vassiliev[2][Knot[10, 104]], Vassiliev[3][Knot[10, 104]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 104]][q, t]
Out[15]=	7 1 2 1 4 2 6 4 - + 7 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 6 6 3 3 2 5 2 5 3 7 3 ---- + --- + 6 q t + 6 q t + 4 q t + 6 q t + 2 q t + 4 q t + 3 q t q t 7 4 9 4 11 5 q t + 2 q t + q t