10 48

10_47

10_49

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10 48 Quick Notes

10 48 Further Notes and Views

Knot presentations

Planar diagram presentation	X₆₂₇₁ X₈₄₉₃ X_14,6,15,5 X_20,15,1,16 X_16,9,17,10 X_18,11,19,12 X_10,17,11,18 X_12,19,13,20 X₂₈₃₇ X_4,14,5,13
Gauss code	1, -9, 2, -10, 3, -1, 9, -2, 5, -7, 6, -8, 10, -3, 4, -5, 7, -6, 8, -4
Dowker-Thistlethwaite code	6 8 14 2 16 18 4 20 10 12
Conway Notation	[41,3,2]

Minimum Braid Representative:

Length is 10, width is 3.

Braid index is 3.

A Morse Link Presentation:

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	[-6][-6]
Hyperbolic Volume	10.3789
A-Polynomial	See Data:10 48/A-polynomial

[edit Notes for 10 48'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 48's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-3t^{3}+6t^{2}-9t+11-9t^{-1}+6t^{-2}-3t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+5z^{6}+8z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 49, 0 }
Jones polynomial	$-q^{5}+2q^{4}-4q^{3}+6q^{2}-7q+9-7q^{-1}+6q^{-2}-4q^{-3}+2q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+7z^{6}-5a^{2}z^{4}-5z^{4}a^{-2}+18z^{4}-8a^{2}z^{2}-8z^{2}a^{-2}+20z^{2}-4a^{2}-4a^{-2}+9$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+3z^{8}a^{-2}+5z^{8}+2a^{3}z^{7}+z^{7}a^{-1}+3z^{7}a^{-3}+2a^{4}z^{6}-5a^{2}z^{6}-11z^{6}a^{-2}+2z^{6}a^{-4}-20z^{6}+a^{5}z^{5}-3a^{3}z^{5}-5az^{5}-11z^{5}a^{-1}-9z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}+9a^{2}z^{4}+18z^{4}a^{-2}-5z^{4}a^{-4}+37z^{4}-3a^{5}z^{3}-a^{3}z^{3}+12az^{3}+21z^{3}a^{-1}+8z^{3}a^{-3}-3z^{3}a^{-5}+2a^{4}z^{2}-11a^{2}z^{2}-13z^{2}a^{-2}+z^{2}a^{-4}-27z^{2}+2a^{5}z-7az-9za^{-1}-3za^{-3}+za^{-5}+4a^{2}+4a^{-2}+9$
The A2 invariant	$-q^{14}-2q^{10}+4q^{2}+1+4q^{-2}-2q^{-10}-q^{-14}$
The G2 invariant	$q^{80}-q^{78}+3q^{76}-4q^{74}+3q^{72}-q^{70}-3q^{68}+8q^{66}-11q^{64}+13q^{62}-11q^{60}+3q^{58}+4q^{56}-14q^{54}+21q^{52}-25q^{50}+21q^{48}-15q^{46}-3q^{44}+18q^{42}-33q^{40}+35q^{38}-29q^{36}+11q^{34}+8q^{32}-27q^{30}+32q^{28}-22q^{26}+4q^{24}+17q^{22}-28q^{20}+22q^{18}-2q^{16}-21q^{14}+41q^{12}-41q^{10}+34q^{8}-4q^{6}-24q^{4}+51q^{2}-55+51q^{-2}-23q^{-4}-5q^{-6}+34q^{-8}-41q^{-10}+44q^{-12}-25q^{-14}+3q^{-16}+19q^{-18}-30q^{-20}+23q^{-22}-5q^{-24}-17q^{-26}+32q^{-28}-32q^{-30}+14q^{-32}+7q^{-34}-31q^{-36}+41q^{-38}-40q^{-40}+21q^{-42}-q^{-44}-20q^{-46}+30q^{-48}-31q^{-50}+23q^{-52}-10q^{-54}-2q^{-56}+8q^{-58}-14q^{-60}+12q^{-62}-9q^{-64}+6q^{-66}-q^{-68}-2q^{-70}+3q^{-72}-3q^{-74}+2q^{-76}-q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_48.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+q^{9}-2q^{7}+2q^{5}-q^{3}+2q+2q^{-1}-q^{-3}+2q^{-5}-2q^{-7}+q^{-9}-q^{-11}$
2	$q^{32}-q^{30}-q^{28}+3q^{26}-2q^{24}-3q^{22}+6q^{20}-3q^{18}-7q^{16}+9q^{14}-10q^{10}+7q^{8}+4q^{6}-7q^{4}+2q^{2}+6+2q^{-2}-6q^{-4}+4q^{-6}+7q^{-8}-10q^{-10}+q^{-12}+9q^{-14}-8q^{-16}-3q^{-18}+7q^{-20}-3q^{-22}-4q^{-24}+3q^{-26}-q^{-30}+q^{-32}$
3	$-q^{63}+q^{61}+q^{59}-2q^{55}+2q^{51}-2q^{47}+2q^{45}+3q^{43}-4q^{41}-7q^{39}+6q^{37}+13q^{35}-6q^{33}-21q^{31}-2q^{29}+29q^{27}+10q^{25}-29q^{23}-23q^{21}+28q^{19}+29q^{17}-19q^{15}-35q^{13}+11q^{11}+30q^{9}-26q^{5}-5q^{3}+19q+16q^{-1}-7q^{-3}-23q^{-5}+5q^{-7}+28q^{-9}+5q^{-11}-34q^{-13}-10q^{-15}+35q^{-17}+20q^{-19}-33q^{-21}-26q^{-23}+22q^{-25}+30q^{-27}-12q^{-29}-28q^{-31}-2q^{-33}+23q^{-35}+7q^{-37}-15q^{-39}-11q^{-41}+6q^{-43}+10q^{-45}-2q^{-47}-5q^{-49}+q^{-51}+3q^{-53}-q^{-57}+q^{-61}-q^{-63}$
4	$q^{104}-q^{102}-q^{100}-q^{96}+4q^{94}-q^{88}-8q^{86}+4q^{84}+q^{82}+7q^{80}+8q^{78}-13q^{76}-6q^{74}-14q^{72}+10q^{70}+32q^{68}+5q^{66}-6q^{64}-50q^{62}-24q^{60}+40q^{58}+51q^{56}+46q^{54}-62q^{52}-94q^{50}-24q^{48}+61q^{46}+140q^{44}+15q^{42}-116q^{40}-130q^{38}-19q^{36}+174q^{34}+126q^{32}-41q^{30}-168q^{28}-120q^{26}+108q^{24}+156q^{22}+47q^{20}-109q^{18}-140q^{16}+19q^{14}+103q^{12}+77q^{10}-33q^{8}-96q^{6}-31q^{4}+46q^{2}+75+19q^{-2}-54q^{-4}-81q^{-6}+6q^{-8}+94q^{-10}+72q^{-12}-32q^{-14}-144q^{-16}-47q^{-18}+107q^{-20}+141q^{-22}+22q^{-24}-178q^{-26}-123q^{-28}+57q^{-30}+169q^{-32}+108q^{-34}-119q^{-36}-153q^{-38}-46q^{-40}+100q^{-42}+148q^{-44}-4q^{-46}-86q^{-48}-99q^{-50}-14q^{-52}+91q^{-54}+57q^{-56}+12q^{-58}-60q^{-60}-60q^{-62}+11q^{-64}+30q^{-66}+45q^{-68}-2q^{-70}-32q^{-72}-13q^{-74}-4q^{-76}+22q^{-78}+8q^{-80}-6q^{-82}-2q^{-84}-7q^{-86}+4q^{-88}+2q^{-90}-2q^{-92}+q^{-94}-2q^{-96}+q^{-98}-q^{-102}+q^{-104}$
5	$-q^{155}+q^{153}+q^{151}+q^{147}-q^{145}-4q^{143}-2q^{141}+2q^{139}+3q^{137}+7q^{135}+5q^{133}-7q^{131}-13q^{129}-9q^{127}+15q^{123}+25q^{121}+11q^{119}-17q^{117}-36q^{115}-32q^{113}+3q^{111}+48q^{109}+64q^{107}+24q^{105}-49q^{103}-100q^{101}-73q^{99}+24q^{97}+128q^{95}+151q^{93}+42q^{91}-132q^{89}-232q^{87}-157q^{85}+73q^{83}+293q^{81}+315q^{79}+69q^{77}-290q^{75}-471q^{73}-284q^{71}+170q^{69}+564q^{67}+550q^{65}+53q^{63}-556q^{61}-765q^{59}-357q^{57}+392q^{55}+893q^{53}+664q^{51}-149q^{49}-860q^{47}-878q^{45}-155q^{43}+713q^{41}+962q^{39}+398q^{37}-472q^{35}-901q^{33}-550q^{31}+238q^{29}+735q^{27}+573q^{25}-33q^{23}-536q^{21}-523q^{19}-76q^{17}+347q^{15}+409q^{13}+137q^{11}-204q^{9}-329q^{7}-150q^{5}+123q^{3}+271q+181q^{-1}-63q^{-3}-264q^{-5}-255q^{-7}+15q^{-9}+313q^{-11}+355q^{-13}+72q^{-15}-343q^{-17}-519q^{-19}-203q^{-21}+368q^{-23}+669q^{-25}+383q^{-27}-300q^{-29}-796q^{-31}-596q^{-33}+169q^{-35}+829q^{-37}+787q^{-39}+51q^{-41}-752q^{-43}-910q^{-45}-298q^{-47}+547q^{-49}+916q^{-51}+524q^{-53}-272q^{-55}-781q^{-57}-648q^{-59}-42q^{-61}+539q^{-63}+648q^{-65}+268q^{-67}-243q^{-69}-505q^{-71}-390q^{-73}-31q^{-75}+301q^{-77}+371q^{-79}+195q^{-81}-75q^{-83}-258q^{-85}-250q^{-87}-85q^{-89}+116q^{-91}+207q^{-93}+152q^{-95}+7q^{-97}-117q^{-99}-146q^{-101}-70q^{-103}+42q^{-105}+98q^{-107}+77q^{-109}+9q^{-111}-48q^{-113}-55q^{-115}-27q^{-117}+16q^{-119}+31q^{-121}+18q^{-123}-q^{-125}-11q^{-127}-13q^{-129}-5q^{-131}+7q^{-133}+5q^{-135}-q^{-141}-2q^{-143}+q^{-145}+2q^{-147}-q^{-149}+q^{-153}-q^{-155}$
6	$q^{216}-q^{214}-q^{212}-q^{208}+q^{206}+q^{204}+6q^{202}-4q^{198}-3q^{196}-8q^{194}-4q^{192}+q^{190}+19q^{188}+11q^{186}+2q^{184}-2q^{182}-20q^{180}-25q^{178}-19q^{176}+25q^{174}+30q^{172}+28q^{170}+23q^{168}-18q^{166}-55q^{164}-67q^{162}+3q^{160}+38q^{158}+69q^{156}+83q^{154}+17q^{152}-84q^{150}-156q^{148}-76q^{146}+10q^{144}+130q^{142}+229q^{140}+163q^{138}-49q^{136}-294q^{134}-323q^{132}-235q^{130}+59q^{128}+448q^{126}+602q^{124}+376q^{122}-171q^{120}-633q^{118}-922q^{116}-655q^{114}+174q^{112}+1036q^{110}+1402q^{108}+921q^{106}-115q^{104}-1445q^{102}-2079q^{100}-1427q^{98}+215q^{96}+1980q^{94}+2741q^{92}+2044q^{90}-251q^{88}-2665q^{86}-3637q^{84}-2439q^{82}+419q^{80}+3271q^{78}+4510q^{76}+2781q^{74}-794q^{72}-4072q^{70}-4962q^{68}-2786q^{66}+1209q^{64}+4754q^{62}+5147q^{60}+2357q^{58}-1920q^{56}-4923q^{54}-4810q^{52}-1698q^{50}+2545q^{48}+4784q^{46}+3960q^{44}+695q^{42}-2703q^{40}-4167q^{38}-2905q^{36}+178q^{34}+2628q^{32}+3193q^{30}+1665q^{28}-609q^{26}-2218q^{24}-2198q^{22}-708q^{20}+856q^{18}+1653q^{16}+1220q^{14}+152q^{12}-884q^{10}-1173q^{8}-592q^{6}+280q^{4}+911q^{2}+819+205q^{-2}-632q^{-4}-1041q^{-6}-761q^{-8}+175q^{-10}+1145q^{-12}+1373q^{-14}+696q^{-16}-683q^{-18}-1787q^{-20}-1894q^{-22}-502q^{-24}+1497q^{-26}+2684q^{-28}+2193q^{-30}+17q^{-32}-2435q^{-34}-3644q^{-36}-2256q^{-38}+828q^{-40}+3587q^{-42}+4179q^{-44}+1947q^{-46}-1702q^{-48}-4616q^{-50}-4393q^{-52}-1315q^{-54}+2668q^{-56}+5038q^{-58}+4195q^{-60}+737q^{-62}-3363q^{-64}-5078q^{-66}-3674q^{-68}-102q^{-70}+3396q^{-72}+4701q^{-74}+3177q^{-76}-304q^{-78}-3142q^{-80}-4005q^{-82}-2528q^{-84}+212q^{-86}+2603q^{-88}+3315q^{-90}+1973q^{-92}-77q^{-94}-1882q^{-96}-2452q^{-98}-1718q^{-100}-144q^{-102}+1274q^{-104}+1701q^{-106}+1378q^{-108}+391q^{-110}-617q^{-112}-1230q^{-114}-1091q^{-116}-476q^{-118}+167q^{-120}+751q^{-122}+870q^{-124}+585q^{-126}+q^{-128}-423q^{-130}-609q^{-132}-549q^{-134}-151q^{-136}+241q^{-138}+472q^{-140}+380q^{-142}+175q^{-144}-98q^{-146}-319q^{-148}-291q^{-150}-128q^{-152}+77q^{-154}+158q^{-156}+183q^{-158}+99q^{-160}-46q^{-162}-106q^{-164}-98q^{-166}-31q^{-168}+5q^{-170}+53q^{-172}+60q^{-174}+13q^{-176}-15q^{-178}-26q^{-180}-12q^{-182}-12q^{-184}+6q^{-186}+18q^{-188}+5q^{-190}-2q^{-192}-5q^{-194}+q^{-196}-4q^{-198}-q^{-200}+5q^{-202}-q^{-206}-2q^{-208}+q^{-210}-q^{-214}+q^{-216}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{14}-2q^{10}+4q^{2}+1+4q^{-2}-2q^{-10}-q^{-14}$
1,1	$q^{44}-2q^{42}+6q^{40}-12q^{38}+19q^{36}-28q^{34}+40q^{32}-52q^{30}+62q^{28}-74q^{26}+88q^{24}-100q^{22}+100q^{20}-100q^{18}+88q^{16}-64q^{14}+15q^{12}+30q^{10}-88q^{8}+142q^{6}-188q^{4}+228q^{2}-232+246q^{-2}-208q^{-4}+182q^{-6}-124q^{-8}+72q^{-10}-11q^{-12}-50q^{-14}+80q^{-16}-116q^{-18}+122q^{-20}-124q^{-22}+106q^{-24}-88q^{-26}+72q^{-28}-52q^{-30}+34q^{-32}-24q^{-34}+17q^{-36}-8q^{-38}+4q^{-40}-2q^{-42}+q^{-44}$
2,0	$q^{38}+q^{32}-q^{28}+q^{26}-3q^{22}-3q^{20}+q^{18}-q^{16}-7q^{14}-q^{12}+2q^{10}-3q^{8}-2q^{6}+7q^{4}+6q^{2}+4+7q^{-2}+8q^{-4}+q^{-6}-q^{-8}+4q^{-10}-q^{-12}-6q^{-14}-5q^{-20}-5q^{-22}-2q^{-28}+2q^{-32}+q^{-34}+q^{-38}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-q^{32}+q^{30}+2q^{28}-3q^{26}+q^{24}+3q^{22}-7q^{20}+4q^{16}-12q^{14}-4q^{12}+4q^{10}-7q^{8}+10q^{4}+8q^{2}+8+7q^{-2}+10q^{-4}-q^{-6}-7q^{-8}+3q^{-10}-4q^{-12}-12q^{-14}+4q^{-16}-7q^{-20}+4q^{-22}+q^{-24}-2q^{-26}+2q^{-28}-q^{-32}+q^{-34}$
1,0,0	$-q^{17}-3q^{13}-3q^{9}+q^{7}+4q^{3}+4q+4q^{-1}+4q^{-3}+q^{-7}-3q^{-9}-3q^{-13}-q^{-17}$
1,0,1	$q^{56}-2q^{54}+5q^{52}-5q^{50}+2q^{48}+5q^{46}-15q^{44}+20q^{42}-16q^{40}+2q^{38}+16q^{36}-31q^{34}+33q^{32}-10q^{30}-19q^{28}+53q^{26}-58q^{24}+34q^{22}+15q^{20}-86q^{18}+94q^{16}-109q^{14}+20q^{12}+19q^{10}-84q^{8}+87q^{6}-40q^{4}+45q^{2}+30+39q^{-2}-2q^{-4}+65q^{-6}-33q^{-8}-7q^{-10}+50q^{-12}-121q^{-14}+91q^{-16}-82q^{-18}-10q^{-20}+50q^{-22}-83q^{-24}+69q^{-26}-32q^{-28}-2q^{-30}+32q^{-32}-33q^{-34}+25q^{-36}-8q^{-38}-8q^{-40}+14q^{-42}-12q^{-44}+7q^{-46}+q^{-48}-3q^{-50}+3q^{-52}-2q^{-54}+q^{-56}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{40}+q^{36}+3q^{34}+q^{32}+4q^{28}-q^{26}-6q^{24}-2q^{22}-3q^{20}-14q^{18}-13q^{16}-6q^{14}-8q^{12}-11q^{10}+3q^{8}+16q^{6}+10q^{4}+20q^{2}+29+19q^{-2}+7q^{-4}+13q^{-6}-13q^{-10}-11q^{-12}-7q^{-14}-12q^{-16}-13q^{-18}-2q^{-20}-3q^{-24}+4q^{-28}+q^{-32}+2q^{-34}+q^{-40}$
1,0,0,0	$-q^{20}-3q^{16}-q^{14}-3q^{12}-2q^{10}+5q^{4}+4q^{2}+7+4q^{-2}+5q^{-4}-2q^{-10}-3q^{-12}-q^{-14}-3q^{-16}-q^{-20}$

B2 Invariants.

Weight

Invariant

0,1

-q^{34}+q^{32}-3q^{30}+4q^{28}-5q^{26}+7q^{24}-9q^{22}+9q^{20}-10q^{18}+8q^{16}-6q^{14}+2q^{12}+2q^{10}-7q^{8}+12q^{6}-14q^{4}+20q^{2}-18+21q^{-2}-14q^{-4}+13q^{-6}-7q^{-8}+3q^{-10}+2q^{-12}-6q^{-14}+8q^{-16}-10q^{-18}+9q^{-20}-10q^{-22}+7q^{-24}-6q^{-26}+4q^{-28}-2q^{-30}+q^{-32}-q^{-34}

1,0

q^{56}-q^{52}-q^{50}+2q^{48}+3q^{46}-q^{44}-4q^{42}-2q^{40}+4q^{38}+6q^{36}-3q^{34}-9q^{32}-4q^{30}+7q^{28}+7q^{26}-6q^{24}-12q^{22}-3q^{20}+8q^{18}+4q^{16}-7q^{14}-6q^{12}+5q^{10}+8q^{8}+q^{6}-4q^{4}+5q^{2}+10+5q^{-2}-4q^{-4}+q^{-6}+7q^{-8}+5q^{-10}-6q^{-12}-7q^{-14}+3q^{-16}+8q^{-18}-2q^{-20}-12q^{-22}-7q^{-24}+7q^{-26}+8q^{-28}-4q^{-30}-10q^{-32}-3q^{-34}+7q^{-36}+4q^{-38}-2q^{-40}-4q^{-42}+3q^{-46}+q^{-48}-q^{-50}-q^{-52}+q^{-56}

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{46}-q^{44}+2q^{42}-2q^{40}+4q^{38}-4q^{36}+4q^{34}-6q^{32}+7q^{30}-8q^{28}+6q^{26}-8q^{24}+5q^{22}-9q^{20}-2q^{18}-6q^{16}-5q^{14}-10q^{10}+10q^{8}-5q^{6}+23q^{4}-4q^{2}+26-4q^{-2}+23q^{-4}-6q^{-6}+10q^{-8}-11q^{-10}-6q^{-14}-6q^{-16}-3q^{-18}-9q^{-20}+5q^{-22}-8q^{-24}+7q^{-26}-8q^{-28}+8q^{-30}-6q^{-32}+5q^{-34}-4q^{-36}+4q^{-38}-2q^{-40}+q^{-42}-q^{-44}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

q^{80}-q^{78}+3q^{76}-4q^{74}+3q^{72}-q^{70}-3q^{68}+8q^{66}-11q^{64}+13q^{62}-11q^{60}+3q^{58}+4q^{56}-14q^{54}+21q^{52}-25q^{50}+21q^{48}-15q^{46}-3q^{44}+18q^{42}-33q^{40}+35q^{38}-29q^{36}+11q^{34}+8q^{32}-27q^{30}+32q^{28}-22q^{26}+4q^{24}+17q^{22}-28q^{20}+22q^{18}-2q^{16}-21q^{14}+41q^{12}-41q^{10}+34q^{8}-4q^{6}-24q^{4}+51q^{2}-55+51q^{-2}-23q^{-4}-5q^{-6}+34q^{-8}-41q^{-10}+44q^{-12}-25q^{-14}+3q^{-16}+19q^{-18}-30q^{-20}+23q^{-22}-5q^{-24}-17q^{-26}+32q^{-28}-32q^{-30}+14q^{-32}+7q^{-34}-31q^{-36}+41q^{-38}-40q^{-40}+21q^{-42}-q^{-44}-20q^{-46}+30q^{-48}-31q^{-50}+23q^{-52}-10q^{-54}-2q^{-56}+8q^{-58}-14q^{-60}+12q^{-62}-9q^{-64}+6q^{-66}-q^{-68}-2q^{-70}+3q^{-72}-3q^{-74}+2q^{-76}-q^{-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.

In[3]:= K = Knot["10 48"];

In[4]:= Alexander[K][t]

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

Out[4]= $t^{4}-3t^{3}+6t^{2}-9t+11-9t^{-1}+6t^{-2}-3t^{-3}+t^{-4}$

In[5]:= Conway[K][z]

Out[5]= $z^{8}+5z^{6}+8z^{4}+4z^{2}+1$

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]= $\{1\}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 49, 0 }

In[8]:= Jones[K][q]

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

Out[8]= $-q^{5}+2q^{4}-4q^{3}+6q^{2}-7q+9-7q^{-1}+6q^{-2}-4q^{-3}+2q^{-4}-q^{-5}$

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= $z^{8}-a^{2}z^{6}-z^{6}a^{-2}+7z^{6}-5a^{2}z^{4}-5z^{4}a^{-2}+18z^{4}-8a^{2}z^{2}-8z^{2}a^{-2}+20z^{2}-4a^{2}-4a^{-2}+9$

In[10]:= Kauffman[K][a, z]

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

Out[10]= $az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+3z^{8}a^{-2}+5z^{8}+2a^{3}z^{7}+z^{7}a^{-1}+3z^{7}a^{-3}+2a^{4}z^{6}-5a^{2}z^{6}-11z^{6}a^{-2}+2z^{6}a^{-4}-20z^{6}+a^{5}z^{5}-3a^{3}z^{5}-5az^{5}-11z^{5}a^{-1}-9z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}+9a^{2}z^{4}+18z^{4}a^{-2}-5z^{4}a^{-4}+37z^{4}-3a^{5}z^{3}-a^{3}z^{3}+12az^{3}+21z^{3}a^{-1}+8z^{3}a^{-3}-3z^{3}a^{-5}+2a^{4}z^{2}-11a^{2}z^{2}-13z^{2}a^{-2}+z^{2}a^{-4}-27z^{2}+2a^{5}z-7az-9za^{-1}-3za^{-3}+za^{-5}+4a^{2}+4a^{-2}+9$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {...}

Vassiliev invariants

V₂ and V₃:

(4, 0)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

16

0

128

{\frac {392}{3}}

-{\frac {8}{3}}

0

32

-32

64

{\frac {2048}{3}}

0

{\frac {6272}{3}}

-{\frac {128}{3}}

{\frac {30182}{15}}

{\frac {6392}{15}}

{\frac {3128}{45}}

{\frac {586}{9}}

-{\frac {1738}{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 48. 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

1

7

3

1

-2

5

3

1

2

3

4

3

-1

1

5

3

2

-1

3

5

2

-3

3

4

-1

-5

1

3

2

-7

1

3

-2

-9

1

-11

1

-1

Integral Khovanov Homology

(db, data source)

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

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the

n+1

-dimensional representation of

sl(2)

)

$n$	$J_{n}$
2	$q^{15}-2q^{14}+q^{13}+4q^{12}-9q^{11}+2q^{10}+14q^{9}-19q^{8}-3q^{7}+31q^{6}-27q^{5}-14q^{4}+48q^{3}-30q^{2}-24q+56-26q^{-1}-28q^{-2}+47q^{-3}-15q^{-4}-25q^{-5}+30q^{-6}-5q^{-7}-16q^{-8}+14q^{-9}-q^{-10}-7q^{-11}+5q^{-12}-2q^{-14}+q^{-15}$
3	$-q^{30}+2q^{29}-q^{28}-q^{27}+5q^{25}-3q^{24}-7q^{23}+3q^{22}+17q^{21}-7q^{20}-24q^{19}-q^{18}+39q^{17}+9q^{16}-49q^{15}-27q^{14}+55q^{13}+51q^{12}-57q^{11}-75q^{10}+48q^{9}+104q^{8}-42q^{7}-120q^{6}+24q^{5}+143q^{4}-19q^{3}-143q^{2}-4q+159+4q^{-1}-140q^{-2}-28q^{-3}+138q^{-4}+30q^{-5}-110q^{-6}-47q^{-7}+92q^{-8}+46q^{-9}-62q^{-10}-48q^{-11}+41q^{-12}+40q^{-13}-23q^{-14}-29q^{-15}+10q^{-16}+21q^{-17}-8q^{-18}-10q^{-19}+3q^{-20}+8q^{-21}-5q^{-22}-3q^{-23}+2q^{-24}+4q^{-25}-3q^{-26}-q^{-27}+2q^{-29}-q^{-30}$
4	$q^{50}-2q^{49}+q^{48}+q^{47}-3q^{46}+4q^{45}-5q^{44}+5q^{43}+3q^{42}-14q^{41}+9q^{40}-9q^{39}+19q^{38}+17q^{37}-40q^{36}-28q^{34}+49q^{33}+64q^{32}-55q^{31}-19q^{30}-99q^{29}+49q^{28}+136q^{27}-10q^{26}+15q^{25}-204q^{24}-36q^{23}+149q^{22}+72q^{21}+167q^{20}-252q^{19}-182q^{18}+42q^{17}+106q^{16}+394q^{15}-191q^{14}-294q^{13}-138q^{12}+51q^{11}+594q^{10}-72q^{9}-328q^{8}-292q^{7}-46q^{6}+706q^{5}+32q^{4}-306q^{3}-380q^{2}-133q+733+105q^{-1}-250q^{-2}-409q^{-3}-210q^{-4}+668q^{-5}+168q^{-6}-140q^{-7}-383q^{-8}-294q^{-9}+509q^{-10}+199q^{-11}+16q^{-12}-274q^{-13}-342q^{-14}+281q^{-15}+151q^{-16}+143q^{-17}-107q^{-18}-294q^{-19}+88q^{-20}+40q^{-21}+157q^{-22}+24q^{-23}-169q^{-24}+9q^{-25}-45q^{-26}+87q^{-27}+56q^{-28}-61q^{-29}+14q^{-30}-56q^{-31}+23q^{-32}+30q^{-33}-17q^{-34}+25q^{-35}-29q^{-36}+q^{-37}+6q^{-38}-9q^{-39}+18q^{-40}-8q^{-41}-6q^{-44}+6q^{-45}-q^{-46}+q^{-47}-2q^{-49}+q^{-50}$
5	$-q^{75}+2q^{74}-q^{73}-q^{72}+3q^{71}-q^{70}-4q^{69}+3q^{68}-q^{66}+8q^{65}+q^{64}-16q^{63}-5q^{62}+2q^{61}+9q^{60}+27q^{59}+14q^{58}-31q^{57}-48q^{56}-26q^{55}+16q^{54}+84q^{53}+82q^{52}-10q^{51}-104q^{50}-138q^{49}-60q^{48}+113q^{47}+206q^{46}+135q^{45}-49q^{44}-229q^{43}-261q^{42}-52q^{41}+198q^{40}+318q^{39}+221q^{38}-53q^{37}-331q^{36}-384q^{35}-161q^{34}+203q^{33}+483q^{32}+458q^{31}+49q^{30}-493q^{29}-742q^{28}-403q^{27}+350q^{26}+967q^{25}+845q^{24}-101q^{23}-1111q^{22}-1248q^{21}-262q^{20}+1125q^{19}+1648q^{18}+635q^{17}-1069q^{16}-1908q^{15}-1027q^{14}+925q^{13}+2144q^{12}+1318q^{11}-783q^{10}-2209q^{9}-1598q^{8}+609q^{7}+2320q^{6}+1733q^{5}-500q^{4}-2251q^{3}-1896q^{2}+339q+2311+1934q^{-1}-256q^{-2}-2161q^{-3}-2044q^{-4}+66q^{-5}+2132q^{-6}+2059q^{-7}+85q^{-8}-1889q^{-9}-2106q^{-10}-357q^{-11}+1685q^{-12}+2046q^{-13}+588q^{-14}-1283q^{-15}-1944q^{-16}-854q^{-17}+897q^{-18}+1695q^{-19}+1017q^{-20}-413q^{-21}-1380q^{-22}-1103q^{-23}+29q^{-24}+972q^{-25}+1035q^{-26}+298q^{-27}-567q^{-28}-874q^{-29}-472q^{-30}+223q^{-31}+627q^{-32}+507q^{-33}+42q^{-34}-377q^{-35}-458q^{-36}-171q^{-37}+173q^{-38}+320q^{-39}+223q^{-40}-18q^{-41}-212q^{-42}-193q^{-43}-47q^{-44}+90q^{-45}+148q^{-46}+82q^{-47}-38q^{-48}-84q^{-49}-70q^{-50}-14q^{-51}+51q^{-52}+55q^{-53}+13q^{-54}-11q^{-55}-30q^{-56}-30q^{-57}+6q^{-58}+20q^{-59}+9q^{-60}+8q^{-61}-2q^{-62}-16q^{-63}-4q^{-64}+5q^{-65}+4q^{-67}+4q^{-68}-4q^{-69}-2q^{-70}+q^{-71}-q^{-72}+2q^{-74}-q^{-75}$
6	$q^{105}-2q^{104}+q^{103}+q^{102}-3q^{101}+q^{100}+q^{99}+6q^{98}-8q^{97}-2q^{96}+6q^{95}-9q^{94}+4q^{93}+8q^{92}+19q^{91}-20q^{90}-20q^{89}+6q^{88}-23q^{87}+15q^{86}+36q^{85}+66q^{84}-27q^{83}-68q^{82}-30q^{81}-90q^{80}+7q^{79}+96q^{78}+211q^{77}+57q^{76}-93q^{75}-111q^{74}-295q^{73}-156q^{72}+68q^{71}+432q^{70}+330q^{69}+112q^{68}-19q^{67}-526q^{66}-548q^{65}-330q^{64}+372q^{63}+516q^{62}+536q^{61}+565q^{60}-241q^{59}-667q^{58}-914q^{57}-271q^{56}-99q^{55}+397q^{54}+1178q^{53}+767q^{52}+320q^{51}-591q^{50}-698q^{49}-1517q^{48}-1177q^{47}+444q^{46}+1337q^{45}+2125q^{44}+1459q^{43}+644q^{42}-2229q^{41}-3568q^{40}-2296q^{39}-140q^{38}+2988q^{37}+4297q^{36}+4125q^{35}-705q^{34}-4873q^{33}-5794q^{32}-3712q^{31}+1584q^{30}+6012q^{29}+8225q^{28}+2753q^{27}-4030q^{26}-8164q^{25}-7695q^{24}-1494q^{23}+5789q^{22}+11139q^{21}+6402q^{20}-1798q^{19}-8756q^{18}-10454q^{17}-4578q^{16}+4401q^{15}+12348q^{14}+8854q^{13}+378q^{12}-8265q^{11}-11641q^{10}-6577q^{9}+3009q^{8}+12455q^{7}+9958q^{6}+1757q^{5}-7588q^{4}-11869q^{3}-7547q^{2}+2073q+12175+10367q^{-1}+2594q^{-2}-6974q^{-3}-11777q^{-4}-8178q^{-5}+1201q^{-6}+11594q^{-7}+10656q^{-8}+3630q^{-9}-5906q^{-10}-11344q^{-11}-8975q^{-12}-363q^{-13}+10104q^{-14}+10636q^{-15}+5239q^{-16}-3632q^{-17}-9816q^{-18}-9540q^{-19}-2813q^{-20}+7021q^{-21}+9374q^{-22}+6703q^{-23}-234q^{-24}-6551q^{-25}-8716q^{-26}-5052q^{-27}+2778q^{-28}+6262q^{-29}+6590q^{-30}+2769q^{-31}-2274q^{-32}-5926q^{-33}-5445q^{-34}-767q^{-35}+2267q^{-36}+4414q^{-37}+3659q^{-38}+1004q^{-39}-2351q^{-40}-3716q^{-41}-2006q^{-42}-585q^{-43}+1556q^{-44}+2461q^{-45}+1976q^{-46}+63q^{-47}-1421q^{-48}-1309q^{-49}-1346q^{-50}-209q^{-51}+819q^{-52}+1324q^{-53}+697q^{-54}-91q^{-55}-273q^{-56}-865q^{-57}-575q^{-58}-43q^{-59}+495q^{-60}+430q^{-61}+198q^{-62}+189q^{-63}-318q^{-64}-349q^{-65}-197q^{-66}+106q^{-67}+136q^{-68}+110q^{-69}+218q^{-70}-73q^{-71}-137q^{-72}-131q^{-73}+7q^{-74}+16q^{-75}+24q^{-76}+138q^{-77}-q^{-78}-36q^{-79}-65q^{-80}-7q^{-81}-15q^{-82}-11q^{-83}+68q^{-84}+11q^{-85}+q^{-86}-24q^{-87}-2q^{-88}-13q^{-89}-16q^{-90}+24q^{-91}+5q^{-92}+6q^{-93}-6q^{-94}+2q^{-95}-4q^{-96}-8q^{-97}+6q^{-98}+2q^{-100}-q^{-101}+q^{-102}-2q^{-104}+q^{-105}$
7	$-q^{140}+2q^{139}-q^{138}-q^{137}+3q^{136}-q^{135}-q^{134}-3q^{133}-q^{132}+10q^{131}-3q^{130}-5q^{129}+5q^{128}-5q^{127}-2q^{126}-7q^{125}-2q^{124}+33q^{123}+6q^{122}-15q^{121}-6q^{120}-30q^{119}-8q^{118}-14q^{117}+2q^{116}+93q^{115}+57q^{114}-38q^{112}-127q^{111}-83q^{110}-65q^{109}-7q^{108}+220q^{107}+237q^{106}+167q^{105}+13q^{104}-302q^{103}-337q^{102}-363q^{101}-225q^{100}+291q^{99}+559q^{98}+685q^{97}+501q^{96}-178q^{95}-589q^{94}-977q^{93}-1017q^{92}-230q^{91}+444q^{90}+1187q^{89}+1506q^{88}+813q^{87}+97q^{86}-988q^{85}-1804q^{84}-1439q^{83}-921q^{82}+249q^{81}+1515q^{80}+1716q^{79}+1815q^{78}+921q^{77}-482q^{76}-1102q^{75}-2133q^{74}-2218q^{73}-1363q^{72}-599q^{71}+1375q^{70}+2838q^{69}+3296q^{68}+3448q^{67}+1124q^{66}-1978q^{65}-4695q^{64}-6860q^{63}-5112q^{62}-943q^{61}+4263q^{60}+9705q^{59}+10291q^{58}+6218q^{57}-1433q^{56}-11012q^{55}-15404q^{54}-13089q^{53}-4249q^{52}+9591q^{51}+19270q^{50}+20798q^{49}+12334q^{48}-5240q^{47}-20874q^{46}-27868q^{45}-21771q^{44}-1941q^{43}+19532q^{42}+33263q^{41}+31503q^{40}+11041q^{39}-15545q^{38}-36280q^{37}-40119q^{36}-20861q^{35}+9319q^{34}+36757q^{33}+47008q^{32}+30303q^{31}-2078q^{30}-35127q^{29}-51586q^{28}-38441q^{27}-5362q^{26}+32074q^{25}+54193q^{24}+44726q^{23}+11958q^{22}-28330q^{21}-54978q^{20}-49189q^{19}-17473q^{18}+24703q^{17}+54826q^{16}+51883q^{15}+21366q^{14}-21480q^{13}-53812q^{12}-53399q^{11}-24259q^{10}+19016q^{9}+52980q^{8}+54019q^{7}+25786q^{6}-17175q^{5}-51798q^{4}-54289q^{3}-27166q^{2}+15814q+51271+54457q^{-1}+27899q^{-2}-14603q^{-3}-50289q^{-4}-54737q^{-5}-29332q^{-6}+13058q^{-7}+49578q^{-8}+55178q^{-9}+30840q^{-10}-10821q^{-11}-47832q^{-12}-55467q^{-13}-33341q^{-14}+7352q^{-15}+45276q^{-16}+55326q^{-17}+36057q^{-18}-2630q^{-19}-40824q^{-20}-54014q^{-21}-39011q^{-22}-3416q^{-23}+34612q^{-24}+51036q^{-25}+41114q^{-26}+10162q^{-27}-26394q^{-28}-45825q^{-29}-41831q^{-30}-16736q^{-31}+16973q^{-32}+38327q^{-33}+40149q^{-34}+22061q^{-35}-7075q^{-36}-29046q^{-37}-36015q^{-38}-25090q^{-39}-1708q^{-40}+18893q^{-41}+29386q^{-42}+25272q^{-43}+8488q^{-44}-9186q^{-45}-21429q^{-46}-22652q^{-47}-12248q^{-48}+1252q^{-49}+13144q^{-50}+17871q^{-51}+13049q^{-52}+4179q^{-53}-5889q^{-54}-12248q^{-55}-11427q^{-56}-6715q^{-57}+624q^{-58}+6782q^{-59}+8299q^{-60}+6977q^{-61}+2498q^{-62}-2556q^{-63}-4992q^{-64}-5650q^{-65}-3506q^{-66}-121q^{-67}+2082q^{-68}+3731q^{-69}+3284q^{-70}+1360q^{-71}-242q^{-72}-1989q^{-73}-2326q^{-74}-1505q^{-75}-724q^{-76}+703q^{-77}+1368q^{-78}+1181q^{-79}+932q^{-80}-39q^{-81}-613q^{-82}-694q^{-83}-809q^{-84}-200q^{-85}+209q^{-86}+302q^{-87}+536q^{-88}+224q^{-89}-5q^{-90}-111q^{-91}-356q^{-92}-122q^{-93}+10q^{-94}-8q^{-95}+178q^{-96}+80q^{-97}+29q^{-98}+8q^{-99}-164q^{-100}-30q^{-101}+34q^{-102}-12q^{-103}+79q^{-104}+20q^{-105}+16q^{-106}+16q^{-107}-99q^{-108}-33q^{-109}+15q^{-110}-4q^{-111}+47q^{-112}+12q^{-113}+18q^{-114}+27q^{-115}-46q^{-116}-26q^{-117}-5q^{-118}-10q^{-119}+18q^{-120}+4q^{-121}+10q^{-122}+21q^{-123}-12q^{-124}-9q^{-125}-4q^{-126}-8q^{-127}+5q^{-128}-q^{-129}+2q^{-130}+8q^{-131}-2q^{-132}-2q^{-133}-2q^{-135}+q^{-136}-q^{-137}+2q^{-139}-q^{-140}$

Computer Talk

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[10, 48]]
Out[2]=	PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[20, 15, 1, 16], X[16, 9, 17, 10], X[18, 11, 19, 12], X[10, 17, 11, 18], X[12, 19, 13, 20], X[2, 8, 3, 7], X[4, 14, 5, 13]]
In[3]:=	GaussCode[Knot[10, 48]]
Out[3]=	GaussCode[1, -9, 2, -10, 3, -1, 9, -2, 5, -7, 6, -8, 10, -3, 4, -5, 7, -6, 8, -4]
In[4]:=	DTCode[Knot[10, 48]]
Out[4]=	DTCode[6, 8, 14, 2, 16, 18, 4, 20, 10, 12]
In[5]:=	br = BR[Knot[10, 48]]
Out[5]=	BR[3, {-1, -1, -1, -1, 2, 2, -1, 2, 2, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 48]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 48]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[10, 48]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 4, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 48]][t]
Out[10]=	-4 3 6 9 2 3 4 11 + t - -- + -- - - - 9 t + 6 t - 3 t + t 3 2 t t t
In[11]:=	Conway[Knot[10, 48]][z]
Out[11]=	2 4 6 8 1 + 4 z + 8 z + 5 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 48]}
In[13]:=	{KnotDet[Knot[10, 48]], KnotSignature[Knot[10, 48]]}
Out[13]=	{49, 0}
In[14]:=	Jones[Knot[10, 48]][q]
Out[14]=	-5 2 4 6 7 2 3 4 5 9 - q + -- - -- + -- - - - 7 q + 6 q - 4 q + 2 q - q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 48]}
In[16]:=	A2Invariant[Knot[10, 48]][q]
Out[16]=	-14 2 4 2 10 14 1 - q - --- + -- + 4 q - 2 q - q 10 2 q q
In[17]:=	HOMFLYPT[Knot[10, 48]][a, z]
Out[17]=	2 4 4 2 2 8 z 2 2 4 5 z 2 4 9 - -- - 4 a + 20 z - ---- - 8 a z + 18 z - ---- - 5 a z + 2 2 2 a a a 6 6 z 2 6 8 7 z - -- - a z + z 2 a
In[18]:=	Kauffman[Knot[10, 48]][a, z]
Out[18]=	2 2 4 2 z 3 z 9 z 5 2 z 13 z 9 + -- + 4 a + -- - --- - --- - 7 a z + 2 a z - 27 z + -- - ----- - 2 5 3 a 4 2 a a a a a 3 3 3 2 2 4 2 3 z 8 z 21 z 3 3 3 11 a z + 2 a z - ---- + ---- + ----- + 12 a z - a z - 5 3 a a a 4 4 5 5 5 3 4 5 z 18 z 2 4 4 4 z 9 z 3 a z + 37 z - ---- + ----- + 9 a z - 5 a z + -- - ---- - 4 2 5 3 a a a a 5 6 6 11 z 5 3 5 5 5 6 2 z 11 z 2 6 ----- - 5 a z - 3 a z + a z - 20 z + ---- - ----- - 5 a z + a 4 2 a a 7 7 8 9 4 6 3 z z 3 7 8 3 z 2 8 z 9 2 a z + ---- + -- + 2 a z + 5 z + ---- + 2 a z + -- + a z 3 a 2 a a a
In[19]:=	{Vassiliev[2][Knot[10, 48]], Vassiliev[3][Knot[10, 48]]}
Out[19]=	{4, 0}
In[20]:=	Kh[Knot[10, 48]][q, t]
Out[20]=	5 1 1 1 3 1 3 3 - + 5 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 4 3 3 3 2 5 2 5 3 7 3 ---- + --- + 3 q t + 4 q t + 3 q t + 3 q t + q t + 3 q t + 3 q t q t 7 4 9 4 11 5 q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 48], 2][q]
Out[21]=	-15 2 5 7 -10 14 16 5 30 25 15 47 56 + q - --- + --- - --- - q + -- - -- - -- + -- - -- - -- + -- - 14 12 11 9 8 7 6 5 4 3 q q q q q q q q q q 28 26 2 3 4 5 6 7 -- - -- - 24 q - 30 q + 48 q - 14 q - 27 q + 31 q - 3 q - 2 q q 8 9 10 11 12 13 14 15 19 q + 14 q + 2 q - 9 q + 4 q + q - 2 q + q

See/edit the Rolfsen_Splice_Template.

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10_47

10_49

10 48

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

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