10 93

10_92

10_94

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10 93 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X16,6,17,5 X20,8,1,7 X18,13,19,14 X14,9,15,10 X10,3,11,4 X4,11,5,12 X12,17,13,18 X8,20,9,19 X2,16,3,15
Gauss code	1, -10, 6, -7, 2, -1, 3, -9, 5, -6, 7, -8, 4, -5, 10, -2, 8, -4, 9, -3
Dowker-Thistlethwaite code	6 10 16 20 14 4 18 2 12 8
Conway Notation	[.3.20.2]

Minimum Braid Representative: Length is 11, width is 4. Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-8][-4] Hyperbolic Volume 13.0165 A-Polynomial See Data:10 93/A-polynomial

[edit Notes for 10 93's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for 10 93's four dimensional invariants]

Polynomial invariants

Alexander polynomial	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 2 t^3-8 t^2+15 t-17+15 t^{-1} -8 t^{-2} +2 t^{-3} }
Conway polynomial	$2z^{6}+4z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	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\}}
Determinant and Signature	{ 67, -2 }
Jones polynomial	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^4+3 q^3-5 q^2+8 q-10+11 q^{-1} -10 q^{-2} +9 q^{-3} -6 q^{-4} +3 q^{-5} - q^{-6} }
HOMFLY-PT polynomial (db, data sources)	$a^{2}z^{6}+z^{6}-a^{4}z^{4}+3a^{2}z^{4}-z^{4}a^{-2}+3z^{4}-2a^{4}z^{2}+3a^{2}z^{2}-2z^{2}a^{-2}+2z^{2}-a^{4}+2a^{2}$
Kauffman polynomial (db, data sources)	$2az^{9}+2z^{9}a^{-1}+6a^{2}z^{8}+3z^{8}a^{-2}+9z^{8}+9a^{3}z^{7}+5az^{7}-3z^{7}a^{-1}+z^{7}a^{-3}+9a^{4}z^{6}-9a^{2}z^{6}-13z^{6}a^{-2}-31z^{6}+6a^{5}z^{5}-17a^{3}z^{5}-29az^{5}-10z^{5}a^{-1}-4z^{5}a^{-3}+3a^{6}z^{4}-14a^{4}z^{4}-6a^{2}z^{4}+17z^{4}a^{-2}+28z^{4}+a^{7}z^{3}-4a^{5}z^{3}+7a^{3}z^{3}+25az^{3}+18z^{3}a^{-1}+5z^{3}a^{-3}+7a^{4}z^{2}+7a^{2}z^{2}-6z^{2}a^{-2}-6z^{2}+a^{5}z-a^{3}z-6az-6za^{-1}-2za^{-3}-a^{4}-2a^{2}$
The A2 invariant	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^{18}+q^{16}-q^{14}-q^{12}+2 q^{10}-q^8+3 q^6+1-2 q^{-2} +2 q^{-4} + q^{-10} - q^{-12} }
The G2 invariant	$q^{100}-2q^{98}+3q^{96}-4q^{94}+3q^{92}-2q^{90}-q^{88}+7q^{86}-11q^{84}+15q^{82}-18q^{80}+13q^{78}-6q^{76}-6q^{74}+23q^{72}-33q^{70}+39q^{68}-37q^{66}+22q^{64}-2q^{62}-26q^{60}+49q^{58}-63q^{56}+61q^{54}-43q^{52}+7q^{50}+37q^{48}-69q^{46}+80q^{44}-60q^{42}+11q^{40}+38q^{38}-70q^{36}+69q^{34}-26q^{32}-31q^{30}+85q^{28}-98q^{26}+64q^{24}+7q^{22}-85q^{20}+135q^{18}-133q^{16}+85q^{14}-5q^{12}-69q^{10}+123q^{8}-132q^{6}+97q^{4}-37q^{2}-33+80q^{-2}-93q^{-4}+70q^{-6}-15q^{-8}-39q^{-10}+75q^{-12}-78q^{-14}+39q^{-16}+27q^{-18}-87q^{-20}+113q^{-22}-92q^{-24}+32q^{-26}+43q^{-28}-95q^{-30}+111q^{-32}-86q^{-34}+36q^{-36}+15q^{-38}-54q^{-40}+62q^{-42}-47q^{-44}+24q^{-46}-q^{-48}-12q^{-50}+13q^{-52}-11q^{-54}+6q^{-56}-2q^{-58}+q^{-60}$

Further Quantum Invariants

Further quantum knot invariants for 10_93.

A1 Invariants.

Weight	Invariant
1	$-q^{13}+2q^{11}-3q^{9}+3q^{7}-q^{5}+q^{3}+q-2q^{-1}+3q^{-3}-2q^{-5}+2q^{-7}-q^{-9}$
2	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^{36}-2 q^{34}+q^{32}+3 q^{30}-8 q^{28}+5 q^{26}+6 q^{24}-15 q^{22}+8 q^{20}+11 q^{18}-17 q^{16}+q^{14}+15 q^{12}-7 q^{10}-10 q^8+10 q^6+7 q^4-12 q^2-1+16 q^{-2} -8 q^{-4} -12 q^{-6} +17 q^{-8} -17 q^{-12} +11 q^{-14} +8 q^{-16} -12 q^{-18} + q^{-20} +7 q^{-22} -3 q^{-24} -2 q^{-26} + q^{-28} }
3	$-q^{69}+2q^{67}-q^{65}-q^{63}+2q^{61}+q^{59}-3q^{57}-2q^{55}+9q^{53}-3q^{51}-17q^{49}+11q^{47}+30q^{45}-18q^{43}-52q^{41}+16q^{39}+77q^{37}-4q^{35}-89q^{33}-25q^{31}+87q^{29}+54q^{27}-59q^{25}-78q^{23}+20q^{21}+86q^{19}+21q^{17}-77q^{15}-54q^{13}+61q^{11}+75q^{9}-41q^{7}-85q^{5}+23q^{3}+87q-3q^{-1}-89q^{-3}-16q^{-5}+87q^{-7}+40q^{-9}-76q^{-11}-64q^{-13}+58q^{-15}+84q^{-17}-27q^{-19}-91q^{-21}-10q^{-23}+82q^{-25}+39q^{-27}-55q^{-29}-57q^{-31}+23q^{-33}+54q^{-35}+4q^{-37}-37q^{-39}-17q^{-41}+17q^{-43}+18q^{-45}-4q^{-47}-10q^{-49}-2q^{-51}+3q^{-53}+2q^{-55}-q^{-57}$
4	$q^{112}-2q^{110}+q^{108}+q^{106}-4q^{104}+5q^{102}-3q^{100}+2q^{98}-q^{96}-8q^{94}+21q^{92}-8q^{90}-15q^{88}-13q^{86}+7q^{84}+83q^{82}-4q^{80}-100q^{78}-109q^{76}+39q^{74}+277q^{72}+111q^{70}-233q^{68}-387q^{66}-74q^{64}+513q^{62}+458q^{60}-145q^{58}-678q^{56}-474q^{54}+410q^{52}+768q^{50}+309q^{48}-509q^{46}-793q^{44}-128q^{42}+550q^{40}+668q^{38}+91q^{36}-575q^{34}-553q^{32}-46q^{30}+541q^{28}+531q^{26}-77q^{24}-546q^{22}-431q^{20}+216q^{18}+576q^{16}+220q^{14}-390q^{12}-503q^{10}+49q^{8}+527q^{6}+321q^{4}-335q^{2}-542-56q^{-2}+526q^{-4}+485q^{-6}-221q^{-8}-627q^{-10}-325q^{-12}+383q^{-14}+694q^{-16}+135q^{-18}-498q^{-20}-634q^{-22}-59q^{-24}+613q^{-26}+526q^{-28}-18q^{-30}-584q^{-32}-504q^{-34}+125q^{-36}+498q^{-38}+436q^{-40}-112q^{-42}-477q^{-44}-304q^{-46}+63q^{-48}+395q^{-50}+255q^{-52}-81q^{-54}-257q^{-56}-221q^{-58}+62q^{-60}+186q^{-62}+129q^{-64}-7q^{-66}-133q^{-68}-75q^{-70}+4q^{-72}+60q^{-74}+56q^{-76}-9q^{-78}-24q^{-80}-23q^{-82}-3q^{-84}+13q^{-86}+5q^{-88}+2q^{-90}-3q^{-92}-2q^{-94}+q^{-96}$
5	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^{165}+2 q^{163}-q^{161}-q^{159}+4 q^{157}-3 q^{155}-3 q^{153}+4 q^{151}+q^{149}-3 q^{147}-q^{145}-2 q^{143}+4 q^{141}+17 q^{139}+4 q^{137}-38 q^{135}-51 q^{133}+11 q^{131}+107 q^{129}+122 q^{127}-11 q^{125}-243 q^{123}-313 q^{121}-4 q^{119}+491 q^{117}+650 q^{115}+115 q^{113}-809 q^{111}-1238 q^{109}-452 q^{107}+1165 q^{105}+2102 q^{103}+1111 q^{101}-1341 q^{99}-3132 q^{97}-2242 q^{95}+1087 q^{93}+4093 q^{91}+3762 q^{89}-189 q^{87}-4564 q^{85}-5343 q^{83}-1421 q^{81}+4137 q^{79}+6533 q^{77}+3434 q^{75}-2725 q^{73}-6744 q^{71}-5260 q^{69}+473 q^{67}+5755 q^{65}+6348 q^{63}+1950 q^{61}-3729 q^{59}-6235 q^{57}-3913 q^{55}+1169 q^{53}+5053 q^{51}+4933 q^{49}+1182 q^{47}-3207 q^{45}-4919 q^{43}-2825 q^{41}+1304 q^{39}+4200 q^{37}+3584 q^{35}+111 q^{33}-3238 q^{31}-3631 q^{29}-850 q^{27}+2482 q^{25}+3338 q^{23}+1013 q^{21}-2167 q^{19}-3097 q^{17}-875 q^{15}+2273 q^{13}+3155 q^{11}+796 q^9-2574 q^7-3589 q^5-1047 q^3+2801 q+4256 q^{-1} +1732 q^{-3} -2633 q^{-5} -4896 q^{-7} -2840 q^{-9} +1912 q^{-11} +5210 q^{-13} +4099 q^{-15} -572 q^{-17} -4877 q^{-19} -5195 q^{-21} -1220 q^{-23} +3785 q^{-25} +5716 q^{-27} +3068 q^{-29} -1949 q^{-31} -5348 q^{-33} -4535 q^{-35} -285 q^{-37} +4033 q^{-39} +5138 q^{-41} +2370 q^{-43} -1972 q^{-45} -4632 q^{-47} -3791 q^{-49} -286 q^{-51} +3173 q^{-53} +4111 q^{-55} +2099 q^{-57} -1171 q^{-59} -3339 q^{-61} -3014 q^{-63} -671 q^{-65} +1861 q^{-67} +2843 q^{-69} +1819 q^{-71} -260 q^{-73} -1892 q^{-75} -2060 q^{-77} -872 q^{-79} +691 q^{-81} +1552 q^{-83} +1277 q^{-85} +262 q^{-87} -733 q^{-89} -1082 q^{-91} -688 q^{-93} +51 q^{-95} +584 q^{-97} +638 q^{-99} +307 q^{-101} -134 q^{-103} -382 q^{-105} -329 q^{-107} -95 q^{-109} +117 q^{-111} +198 q^{-113} +145 q^{-115} +16 q^{-117} -75 q^{-119} -85 q^{-121} -43 q^{-123} +2 q^{-125} +30 q^{-127} +31 q^{-129} +8 q^{-131} -6 q^{-133} -8 q^{-135} -5 q^{-137} -2 q^{-139} +3 q^{-141} +2 q^{-143} - q^{-145} }

A2 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^{18}+q^{16}-q^{14}-q^{12}+2 q^{10}-q^8+3 q^6+1-2 q^{-2} +2 q^{-4} + q^{-10} - q^{-12} }
1,1	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^{52}-4 q^{50}+8 q^{48}-12 q^{46}+20 q^{44}-32 q^{42}+44 q^{40}-56 q^{38}+75 q^{36}-98 q^{34}+116 q^{32}-140 q^{30}+167 q^{28}-198 q^{26}+222 q^{24}-242 q^{22}+251 q^{20}-222 q^{18}+160 q^{16}-60 q^{14}-77 q^{12}+238 q^{10}-398 q^8+532 q^6-619 q^4+666 q^2-640+566 q^{-2} -444 q^{-4} +288 q^{-6} -118 q^{-8} -58 q^{-10} +201 q^{-12} -314 q^{-14} +382 q^{-16} -388 q^{-18} +352 q^{-20} -284 q^{-22} +212 q^{-24} -136 q^{-26} +74 q^{-28} -38 q^{-30} +14 q^{-32} -4 q^{-34} + q^{-36} }
2,0	$q^{46}-q^{44}+2q^{40}-2q^{38}-2q^{36}+3q^{34}+q^{32}-5q^{30}-4q^{28}+7q^{26}+2q^{24}-11q^{22}+3q^{20}+9q^{18}-5q^{14}+3q^{12}+4q^{10}-3q^{8}+4q^{4}-q^{2}-3+7q^{-2}-7q^{-6}+q^{-8}+4q^{-10}-q^{-12}-7q^{-14}+2q^{-16}+8q^{-18}-4q^{-22}+q^{-24}+3q^{-26}-3q^{-30}-q^{-32}+q^{-34}$

A3 Invariants.

Weight	Invariant
0,1,0	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^{42}-2 q^{40}-q^{38}+6 q^{36}-3 q^{34}-7 q^{32}+10 q^{30}-q^{28}-12 q^{26}+11 q^{24}+2 q^{22}-13 q^{20}+8 q^{18}+4 q^{16}-9 q^{14}+3 q^{12}+4 q^{10}+2 q^8-5 q^6+2 q^4+11 q^2-8- q^{-2} +12 q^{-4} -8 q^{-6} -3 q^{-8} +10 q^{-10} -8 q^{-12} -2 q^{-14} +7 q^{-16} -6 q^{-18} + q^{-20} +2 q^{-22} -2 q^{-24} + q^{-26} }
1,0,0	$-q^{23}+q^{21}-2q^{19}+q^{17}-2q^{15}+2q^{13}-q^{11}+3q^{9}+q^{7}+q^{5}+q^{3}-q+q^{-1}-2q^{-3}+2q^{-5}-q^{-7}+2q^{-9}-q^{-11}+q^{-13}-q^{-15}$

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$-q^{42}+2q^{40}-3q^{38}+6q^{36}-9q^{34}+11q^{32}-16q^{30}+17q^{28}-18q^{26}+17q^{24}-14q^{22}+9q^{20}-8q^{16}+19q^{14}-25q^{12}+34q^{10}-36q^{8}+37q^{6}-34q^{4}+27q^{2}-20+9q^{-2}-8q^{-6}+15q^{-8}-18q^{-10}+20q^{-12}-18q^{-14}+17q^{-16}-12q^{-18}+9q^{-20}-6q^{-22}+2q^{-24}-q^{-26}$
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^{68}-2 q^{64}-2 q^{62}+q^{60}+6 q^{58}+3 q^{56}-6 q^{54}-9 q^{52}-q^{50}+13 q^{48}+8 q^{46}-9 q^{44}-15 q^{42}+17 q^{38}+8 q^{36}-13 q^{34}-15 q^{32}+7 q^{30}+17 q^{28}-17 q^{24}-5 q^{22}+14 q^{20}+10 q^{18}-10 q^{16}-10 q^{14}+7 q^{12}+11 q^{10}-5 q^8-11 q^6+5 q^4+15 q^2+1-17 q^{-2} -7 q^{-4} +16 q^{-6} +16 q^{-8} -8 q^{-10} -21 q^{-12} - q^{-14} +19 q^{-16} +10 q^{-18} -13 q^{-20} -15 q^{-22} +5 q^{-24} +14 q^{-26} + q^{-28} -9 q^{-30} -5 q^{-32} +5 q^{-34} +4 q^{-36} -2 q^{-38} -2 q^{-40} + q^{-44} }

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^{58}-2 q^{56}+q^{54}-2 q^{52}+6 q^{50}-6 q^{48}+5 q^{46}-9 q^{44}+13 q^{42}-12 q^{40}+11 q^{38}-15 q^{36}+14 q^{34}-12 q^{32}+10 q^{30}-9 q^{28}+3 q^{26}+q^{24}-6 q^{22}+12 q^{20}-17 q^{18}+22 q^{16}-24 q^{14}+29 q^{12}-28 q^{10}+30 q^8-25 q^6+25 q^4-19 q^2+17-7 q^{-2} +5 q^{-4} -6 q^{-8} +11 q^{-10} -13 q^{-12} +13 q^{-14} -17 q^{-16} +16 q^{-18} -15 q^{-20} +12 q^{-22} -11 q^{-24} +9 q^{-26} -6 q^{-28} +4 q^{-30} -2 q^{-32} + q^{-34} }

G2 Invariants.

Weight

Invariant

1,0

$q^{100}-2q^{98}+3q^{96}-4q^{94}+3q^{92}-2q^{90}-q^{88}+7q^{86}-11q^{84}+15q^{82}-18q^{80}+13q^{78}-6q^{76}-6q^{74}+23q^{72}-33q^{70}+39q^{68}-37q^{66}+22q^{64}-2q^{62}-26q^{60}+49q^{58}-63q^{56}+61q^{54}-43q^{52}+7q^{50}+37q^{48}-69q^{46}+80q^{44}-60q^{42}+11q^{40}+38q^{38}-70q^{36}+69q^{34}-26q^{32}-31q^{30}+85q^{28}-98q^{26}+64q^{24}+7q^{22}-85q^{20}+135q^{18}-133q^{16}+85q^{14}-5q^{12}-69q^{10}+123q^{8}-132q^{6}+97q^{4}-37q^{2}-33+80q^{-2}-93q^{-4}+70q^{-6}-15q^{-8}-39q^{-10}+75q^{-12}-78q^{-14}+39q^{-16}+27q^{-18}-87q^{-20}+113q^{-22}-92q^{-24}+32q^{-26}+43q^{-28}-95q^{-30}+111q^{-32}-86q^{-34}+36q^{-36}+15q^{-38}-54q^{-40}+62q^{-42}-47q^{-44}+24q^{-46}-q^{-48}-12q^{-50}+13q^{-52}-11q^{-54}+6q^{-56}-2q^{-58}+q^{-60}$

.

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 93"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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 2 t^3-8 t^2+15 t-17+15 t^{-1} -8 t^{-2} +2 t^{-3} }

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

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\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 67, -2 }

In[8]:=

Jones[K][q]

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

Out[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 -q^4+3 q^3-5 q^2+8 q-10+11 q^{-1} -10 q^{-2} +9 q^{-3} -6 q^{-4} +3 q^{-5} - q^{-6} }

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{2}z^{6}+z^{6}-a^{4}z^{4}+3a^{2}z^{4}-z^{4}a^{-2}+3z^{4}-2a^{4}z^{2}+3a^{2}z^{2}-2z^{2}a^{-2}+2z^{2}-a^{4}+2a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2az^{9}+2z^{9}a^{-1}+6a^{2}z^{8}+3z^{8}a^{-2}+9z^{8}+9a^{3}z^{7}+5az^{7}-3z^{7}a^{-1}+z^{7}a^{-3}+9a^{4}z^{6}-9a^{2}z^{6}-13z^{6}a^{-2}-31z^{6}+6a^{5}z^{5}-17a^{3}z^{5}-29az^{5}-10z^{5}a^{-1}-4z^{5}a^{-3}+3a^{6}z^{4}-14a^{4}z^{4}-6a^{2}z^{4}+17z^{4}a^{-2}+28z^{4}+a^{7}z^{3}-4a^{5}z^{3}+7a^{3}z^{3}+25az^{3}+18z^{3}a^{-1}+5z^{3}a^{-3}+7a^{4}z^{2}+7a^{2}z^{2}-6z^{2}a^{-2}-6z^{2}+a^{5}z-a^{3}z-6az-6za^{-1}-2za^{-3}-a^{4}-2a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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

Vassiliev invariants

V2 and V3:

(1, -1)

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 $4$ $-8$ 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 8} $-{\frac {34}{3}}$ $-{\frac {86}{3}}$ $-32$ 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{176}{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{160}{3}} $-72$ ${\frac {32}{3}}$ $32$ $-{\frac {136}{3}}$ $-{\frac {344}{3}}$ 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{751}{30}} ${\frac {698}{15}}$ $-{\frac {3658}{45}}$ $-{\frac {655}{18}}$ $-{\frac {2129}{30}}$

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 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 $\chi$ (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 $j-2r=s+1$ or $j-2r=s-1$, where $s=$-2 is the signature of 10 93. 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 9                     1 -1 7                   2   2 5                 3 1   -2 3               5 2     3 1             5 3       -2 -1           6 5         1 -3         5 6           1 -5       4 5             -1 -7     2 5               3 -9   1 4                 -3 -11   2                   2 -13 1                     -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{13}-3q^{12}-q^{11}+11q^{10}-9q^{9}-14q^{8}+31q^{7}-6q^{6}-42q^{5}+48q^{4}+11q^{3}-71q^{2}+52q+35-88q^{-1}+41q^{-2}+54q^{-3}-85q^{-4}+21q^{-5}+57q^{-6}-63q^{-7}+7q^{-8}+39q^{-9}-35q^{-10}+4q^{-11}+16q^{-12}-14q^{-13}+3q^{-14}+3q^{-15}-3q^{-16}+q^{-17}$
3	$-q^{27}+3q^{26}+q^{25}-5q^{24}-9q^{23}+9q^{22}+23q^{21}-6q^{20}-43q^{19}-11q^{18}+64q^{17}+44q^{16}-74q^{15}-91q^{14}+66q^{13}+138q^{12}-31q^{11}-183q^{10}-15q^{9}+202q^{8}+80q^{7}-209q^{6}-137q^{5}+190q^{4}+196q^{3}-162q^{2}-240q+117+282q^{-1}-72q^{-2}-304q^{-3}+9q^{-4}+326q^{-5}+44q^{-6}-318q^{-7}-106q^{-8}+303q^{-9}+142q^{-10}-253q^{-11}-172q^{-12}+205q^{-13}+161q^{-14}-140q^{-15}-139q^{-16}+93q^{-17}+97q^{-18}-55q^{-19}-58q^{-20}+32q^{-21}+29q^{-22}-21q^{-23}-10q^{-24}+13q^{-25}+q^{-26}-7q^{-27}+2q^{-28}+2q^{-29}-3q^{-31}+3q^{-32}-q^{-33}$
4	$q^{46}-3q^{45}-q^{44}+5q^{43}+3q^{42}+9q^{41}-19q^{40}-21q^{39}+4q^{38}+18q^{37}+74q^{36}-15q^{35}-77q^{34}-75q^{33}-40q^{32}+200q^{31}+121q^{30}-20q^{29}-199q^{28}-323q^{27}+164q^{26}+297q^{25}+316q^{24}-59q^{23}-655q^{22}-203q^{21}+124q^{20}+681q^{19}+489q^{18}-593q^{17}-576q^{16}-505q^{15}+601q^{14}+1055q^{13}-49q^{12}-489q^{11}-1177q^{10}+26q^{9}+1191q^{8}+584q^{7}+70q^{6}-1488q^{5}-682q^{4}+889q^{3}+990q^{2}+776q-1447-1264q^{-1}+403q^{-2}+1197q^{-3}+1432q^{-4}-1241q^{-5}-1742q^{-6}-149q^{-7}+1310q^{-8}+2042q^{-9}-885q^{-10}-2102q^{-11}-796q^{-12}+1195q^{-13}+2511q^{-14}-277q^{-15}-2092q^{-16}-1383q^{-17}+688q^{-18}+2489q^{-19}+389q^{-20}-1515q^{-21}-1501q^{-22}+10q^{-23}+1824q^{-24}+673q^{-25}-697q^{-26}-1042q^{-27}-348q^{-28}+940q^{-29}+469q^{-30}-164q^{-31}-439q^{-32}-293q^{-33}+353q^{-34}+156q^{-35}-10q^{-36}-95q^{-37}-127q^{-38}+115q^{-39}+8q^{-40}-q^{-41}+q^{-42}-40q^{-43}+39q^{-44}-12q^{-45}-3q^{-46}+8q^{-47}-11q^{-48}+10q^{-49}-5q^{-50}+3q^{-52}-3q^{-53}+q^{-54}$
5	$-q^{70}+3q^{69}+q^{68}-5q^{67}-3q^{66}-3q^{65}+q^{64}+17q^{63}+24q^{62}-6q^{61}-31q^{60}-48q^{59}-41q^{58}+27q^{57}+115q^{56}+123q^{55}+22q^{54}-129q^{53}-253q^{52}-207q^{51}+62q^{50}+371q^{49}+463q^{48}+202q^{47}-307q^{46}-740q^{45}-677q^{44}-23q^{43}+812q^{42}+1197q^{41}+708q^{40}-465q^{39}-1538q^{38}-1586q^{37}-376q^{36}+1365q^{35}+2340q^{34}+1614q^{33}-514q^{32}-2568q^{31}-2908q^{30}-978q^{29}+2015q^{28}+3782q^{27}+2756q^{26}-556q^{25}-3846q^{24}-4437q^{23}-1490q^{22}+2941q^{21}+5416q^{20}+3786q^{19}-1078q^{18}-5542q^{17}-5808q^{16}-1309q^{15}+4603q^{14}+7185q^{13}+3939q^{12}-2894q^{11}-7739q^{10}-6314q^{9}+628q^{8}+7503q^{7}+8244q^{6}+1777q^{5}-6628q^{4}-9612q^{3}-4124q^{2}+5447q+10507+6142q^{-1}-4104q^{-2}-11067q^{-3}-7972q^{-4}+2905q^{-5}+11522q^{-6}+9512q^{-7}-1745q^{-8}-11949q^{-9}-11120q^{-10}+683q^{-11}+12452q^{-12}+12692q^{-13}+580q^{-14}-12805q^{-15}-14452q^{-16}-2098q^{-17}+12845q^{-18}+16041q^{-19}+4053q^{-20}-12189q^{-21}-17348q^{-22}-6227q^{-23}+10751q^{-24}+17753q^{-25}+8442q^{-26}-8438q^{-27}-17228q^{-28}-10111q^{-29}+5669q^{-30}+15431q^{-31}+10948q^{-32}-2759q^{-33}-12830q^{-34}-10704q^{-35}+387q^{-36}+9698q^{-37}+9464q^{-38}+1260q^{-39}-6671q^{-40}-7605q^{-41}-2009q^{-42}+4140q^{-43}+5542q^{-44}+2039q^{-45}-2296q^{-46}-3654q^{-47}-1678q^{-48}+1134q^{-49}+2213q^{-50}+1149q^{-51}-505q^{-52}-1202q^{-53}-687q^{-54}+197q^{-55}+596q^{-56}+363q^{-57}-76q^{-58}-278q^{-59}-152q^{-60}+38q^{-61}+101q^{-62}+54q^{-63}-6q^{-64}-46q^{-65}-19q^{-66}+23q^{-67}+5q^{-68}-8q^{-69}+7q^{-70}-4q^{-71}-6q^{-72}+10q^{-73}-q^{-74}-7q^{-75}+5q^{-76}-3q^{-78}+3q^{-79}-q^{-80}$
6	$q^{99}-3q^{98}-q^{97}+5q^{96}+3q^{95}+3q^{94}-7q^{93}+q^{92}-20q^{91}-22q^{90}+18q^{89}+32q^{88}+54q^{87}+17q^{86}+25q^{85}-87q^{84}-163q^{83}-103q^{82}-13q^{81}+173q^{80}+231q^{79}+399q^{78}+107q^{77}-285q^{76}-558q^{75}-671q^{74}-345q^{73}+78q^{72}+1123q^{71}+1287q^{70}+906q^{69}-75q^{68}-1313q^{67}-2108q^{66}-2284q^{65}-210q^{64}+1619q^{63}+3332q^{62}+3403q^{61}+1731q^{60}-1471q^{59}-5153q^{58}-5186q^{57}-3550q^{56}+1116q^{55}+5626q^{54}+8418q^{53}+6419q^{52}-429q^{51}-6301q^{50}-11243q^{49}-9559q^{48}-2996q^{47}+7650q^{46}+14542q^{45}+13180q^{44}+6295q^{43}-7167q^{42}-17038q^{41}-19757q^{40}-9175q^{39}+6499q^{38}+19415q^{37}+24979q^{36}+14471q^{35}-4302q^{34}-24547q^{33}-29339q^{32}-19470q^{31}+2206q^{30}+27183q^{29}+36305q^{28}+25837q^{27}-3605q^{26}-29444q^{25}-42448q^{24}-31230q^{23}+2436q^{22}+35598q^{21}+50409q^{20}+32157q^{19}-2674q^{18}-41619q^{17}-57393q^{16}-35657q^{15}+8980q^{14}+51329q^{13}+60034q^{12}+35865q^{11}-16628q^{10}-61270q^{9}-66026q^{8}-28022q^{7}+30543q^{6}+68012q^{5}+67282q^{4}+17310q^{3}-46396q^{2}-79226q-58840+2410q^{-1}+60590q^{-2}+84289q^{-3}+45773q^{-4}-26053q^{-5}-80175q^{-6}-77915q^{-7}-20665q^{-8}+49855q^{-9}+92073q^{-10}+64708q^{-11}-10744q^{-12}-79734q^{-13}-90771q^{-14}-36351q^{-15}+44212q^{-16}+100459q^{-17}+80423q^{-18}-484q^{-19}-83866q^{-20}-106456q^{-21}-52531q^{-22}+40469q^{-23}+112681q^{-24}+101514q^{-25}+15029q^{-26}-85458q^{-27}-125207q^{-28}-77564q^{-29}+25482q^{-30}+117454q^{-31}+124960q^{-32}+43660q^{-33}-68995q^{-34}-131746q^{-35}-104791q^{-36}-7096q^{-37}+98093q^{-38}+132025q^{-39}+74597q^{-40}-31220q^{-41}-109773q^{-42}-112822q^{-43}-41784q^{-44}+55848q^{-45}+108329q^{-46}+84731q^{-47}+7736q^{-48}-65639q^{-49}-90391q^{-50}-55019q^{-51}+14490q^{-52}+65121q^{-53}+66556q^{-54}+25304q^{-55}-24938q^{-56}-52471q^{-57}-43133q^{-58}-5939q^{-59}+27722q^{-60}+36934q^{-61}+21177q^{-62}-4029q^{-63}-21967q^{-64}-22953q^{-65}-7771q^{-66}+8304q^{-67}+14801q^{-68}+10516q^{-69}+1149q^{-70}-6742q^{-71}-8844q^{-72}-3789q^{-73}+1900q^{-74}+4399q^{-75}+3493q^{-76}+881q^{-77}-1556q^{-78}-2623q^{-79}-1076q^{-80}+450q^{-81}+964q^{-82}+782q^{-83}+256q^{-84}-251q^{-85}-638q^{-86}-161q^{-87}+145q^{-88}+128q^{-89}+98q^{-90}+41q^{-91}-4q^{-92}-140q^{-93}+8q^{-94}+47q^{-95}-8q^{-96}+2q^{-97}+q^{-98}+19q^{-99}-32q^{-100}+9q^{-101}+13q^{-102}-12q^{-103}+2q^{-104}-2q^{-105}+7q^{-106}-5q^{-107}+3q^{-109}-3q^{-110}+q^{-111}$
7	Not Available

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, 93]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 6, 17, 5], X[20, 8, 1, 7], X[18, 13, 19, 14], X[14, 9, 15, 10], X[10, 3, 11, 4], X[4, 11, 5, 12], X[12, 17, 13, 18], X[8, 20, 9, 19], X[2, 16, 3, 15]]
In[3]:=	GaussCode[Knot[10, 93]]
Out[3]=	GaussCode[1, -10, 6, -7, 2, -1, 3, -9, 5, -6, 7, -8, 4, -5, 10, -2, 8, -4, 9, -3]
In[4]:=	DTCode[Knot[10, 93]]
Out[4]=	DTCode[6, 10, 16, 20, 14, 4, 18, 2, 12, 8]
In[5]:=	br = BR[Knot[10, 93]]
Out[5]=	BR[4, {-1, -1, 2, -1, -1, 2, -1, 2, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 93]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 93]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 93]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 93]][t]
Out[10]=	2 8 15 2 3 -17 + -- - -- + -- + 15 t - 8 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 93]][z]
Out[11]=	2 4 6 1 + z + 4 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 93]}
In[13]:=	{KnotDet[Knot[10, 93]], KnotSignature[Knot[10, 93]]}
Out[13]=	{67, -2}
In[14]:=	Jones[Knot[10, 93]][q]
Out[14]=	-6 3 6 9 10 11 2 3 4 -10 - q + -- - -- + -- - -- + -- + 8 q - 5 q + 3 q - q 5 4 3 2 q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 93]}
In[16]:=	A2Invariant[Knot[10, 93]][q]
Out[16]=	-18 -16 -14 -12 2 -8 3 2 4 10 12 1 - q + q - q - q + --- - q + -- - 2 q + 2 q + q - q 10 6 q q
In[17]:=	HOMFLYPT[Knot[10, 93]][a, z]
Out[17]=	2 4 2 4 2 2 z 2 2 4 2 4 z 2 4 2 a - a + 2 z - ---- + 3 a z - 2 a z + 3 z - -- + 3 a z - 2 2 a a 4 4 6 2 6 a z + z + a z
In[18]:=	Kauffman[Knot[10, 93]][a, z]
Out[18]=	2 2 4 2 z 6 z 3 5 2 6 z 2 2 -2 a - a - --- - --- - 6 a z - a z + a z - 6 z - ---- + 7 a z + 3 a 2 a a 3 3 4 2 5 z 18 z 3 3 3 5 3 7 3 7 a z + ---- + ----- + 25 a z + 7 a z - 4 a z + a z + 3 a a 4 5 5 4 17 z 2 4 4 4 6 4 4 z 10 z 28 z + ----- - 6 a z - 14 a z + 3 a z - ---- - ----- - 2 3 a a a 6 5 3 5 5 5 6 13 z 2 6 4 6 29 a z - 17 a z + 6 a z - 31 z - ----- - 9 a z + 9 a z + 2 a 7 7 8 9 z 3 z 7 3 7 8 3 z 2 8 2 z 9 -- - ---- + 5 a z + 9 a z + 9 z + ---- + 6 a z + ---- + 2 a z 3 a 2 a a a
In[19]:=	{Vassiliev[2][Knot[10, 93]], Vassiliev[3][Knot[10, 93]]}
Out[19]=	{1, -1}
In[20]:=	Kh[Knot[10, 93]][q, t]
Out[20]=	6 6 1 2 1 4 2 5 4 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 q q t q t q t q t q t q t q t 5 5 5 t 2 3 2 3 3 5 3 ---- + ---- + --- + 5 q t + 3 q t + 5 q t + 2 q t + 3 q t + 5 3 q q t q t 5 4 7 4 9 5 q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 93], 2][q]
Out[21]=	-17 3 3 3 14 16 4 35 39 7 63 35 + q - --- + --- + --- - --- + --- + --- - --- + -- + -- - -- + 16 15 14 13 12 11 10 9 8 7 q q q q q q q q q q 57 21 85 54 41 88 2 3 4 5 -- + -- - -- + -- + -- - -- + 52 q - 71 q + 11 q + 48 q - 42 q - 6 5 4 3 2 q q q q q q 6 7 8 9 10 11 12 13 6 q + 31 q - 14 q - 9 q + 11 q - q - 3 q + q

See/edit the Rolfsen_Splice_Template.