10 91: Difference between revisions

Latest revision as of 17:57, 1 September 2005

10_90

10_92

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[edit 10 91 Quick Notes]

[edit 10 91 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_20,6,1,5 X_16,9,17,10 X_10,3,11,4 X_2,18,3,17 X_14,7,15,8 X_8,15,9,16 X_12,20,13,19 X_18,12,19,11 X_4,13,5,14
Gauss code	1, -5, 4, -10, 2, -1, 6, -7, 3, -4, 9, -8, 10, -6, 7, -3, 5, -9, 8, -2
Dowker-Thistlethwaite code	6 10 20 14 16 18 4 8 2 12
Conway Notation	[.3.2.20]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 10, width is 3,

Braid index is 3

[{3, 6}, {2, 5}, {1, 3}, {7, 14}, {6, 11}, {8, 13}, {9, 7}, {10, 8}, {4, 9}, {5, 10}, {14, 12}, {11, 2}, {13, 4}, {12, 1}]

[edit Notes on presentations of 10 91]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["10 91"];

In[4]:= PD[K]

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

Out[4]= X₆₂₇₁ X_20,6,1,5 X_16,9,17,10 X_10,3,11,4 X_2,18,3,17 X_14,7,15,8 X_8,15,9,16 X_12,20,13,19 X_18,12,19,11 X_4,13,5,14

In[5]:= GaussCode[K]

Out[5]= 1, -5, 4, -10, 2, -1, 6, -7, 3, -4, 9, -8, 10, -6, 7, -3, 5, -9, 8, -2

In[6]:= DTCode[K]

Out[6]= 6 10 20 14 16 18 4 8 2 12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [.3.2.20]

In[9]:= br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]= ${\textrm {BR}}(3,\{-1,-1,-1,2,-1,2,2,-1,2,2\})$

In[10]:= {First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]= { 3, 10, 3 }

In[11]:= Show[BraidPlot[br]]

Out[11]= -Graphics-

In[12]:= Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]= -Graphics-

In[13]:= ap = ArcPresentation[K]

Out[13]= ArcPresentation[{3, 6}, {2, 5}, {1, 3}, {7, 14}, {6, 11}, {8, 13}, {9, 7}, {10, 8}, {4, 9}, {5, 10}, {14, 12}, {11, 2}, {13, 4}, {12, 1}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Chiral
Unknotting number	1
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-6][-6]
Hyperbolic Volume	13.487
A-Polynomial	See Data:10 91/A-polynomial

[edit Notes for 10 91'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 91's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-4t^{3}+9t^{2}-14t+17-14t^{-1}+9t^{-2}-4t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+4z^{6}+5z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 73, 0 }
Jones polynomial	$-q^{5}+3q^{4}-6q^{3}+9q^{2}-11q+13-11q^{-1}+9q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-4z^{4}a^{-2}+13z^{4}-5a^{2}z^{2}-5z^{2}a^{-2}+12z^{2}-2a^{2}-2a^{-2}+5$
Kauffman polynomial (db, data sources)	$2az^{9}+2z^{9}a^{-1}+4a^{2}z^{8}+5z^{8}a^{-2}+9z^{8}+4a^{3}z^{7}+az^{7}+2z^{7}a^{-1}+5z^{7}a^{-3}+3a^{4}z^{6}-7a^{2}z^{6}-13z^{6}a^{-2}+3z^{6}a^{-4}-26z^{6}+a^{5}z^{5}-6a^{3}z^{5}-7az^{5}-13z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}+7a^{2}z^{4}+16z^{4}a^{-2}-6z^{4}a^{-4}+35z^{4}-2a^{5}z^{3}+9az^{3}+18z^{3}a^{-1}+9z^{3}a^{-3}-2z^{3}a^{-5}+2a^{4}z^{2}-7a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}-19z^{2}+a^{5}z-4az-6za^{-1}-3za^{-3}+2a^{2}+2a^{-2}+5$
The A2 invariant	$-q^{14}+q^{12}-2q^{10}+q^{8}-q^{4}+4q^{2}-1+4q^{-2}-q^{-4}+q^{-8}-2q^{-10}+q^{-12}-q^{-14}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+8q^{72}-5q^{70}-3q^{68}+16q^{66}-27q^{64}+36q^{62}-38q^{60}+23q^{58}-q^{56}-34q^{54}+70q^{52}-92q^{50}+95q^{48}-66q^{46}+4q^{44}+68q^{42}-130q^{40}+151q^{38}-122q^{36}+46q^{34}+44q^{32}-116q^{30}+138q^{28}-93q^{26}+9q^{24}+77q^{22}-122q^{20}+97q^{18}-19q^{16}-82q^{14}+161q^{12}-175q^{10}+132q^{8}-27q^{6}-94q^{4}+190q^{2}-223+188q^{-2}-92q^{-4}-27q^{-6}+132q^{-8}-176q^{-10}+165q^{-12}-89q^{-14}-10q^{-16}+93q^{-18}-126q^{-20}+87q^{-22}-3q^{-24}-84q^{-26}+137q^{-28}-124q^{-30}+54q^{-32}+40q^{-34}-124q^{-36}+160q^{-38}-142q^{-40}+74q^{-42}+7q^{-44}-77q^{-46}+108q^{-48}-102q^{-50}+72q^{-52}-28q^{-54}-10q^{-56}+32q^{-58}-42q^{-60}+36q^{-62}-23q^{-64}+12q^{-66}-6q^{-70}+7q^{-72}-7q^{-74}+4q^{-76}-2q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_91.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-3q^{7}+3q^{5}-2q^{3}+2q+2q^{-1}-2q^{-3}+3q^{-5}-3q^{-7}+2q^{-9}-q^{-11}$
2	$q^{32}-2q^{30}-q^{28}+6q^{26}-6q^{24}-5q^{22}+17q^{20}-8q^{18}-17q^{16}+24q^{14}-25q^{10}+17q^{8}+10q^{6}-18q^{4}+q^{2}+14+q^{-2}-17q^{-4}+10q^{-6}+17q^{-8}-25q^{-10}+q^{-12}+24q^{-14}-18q^{-16}-8q^{-18}+18q^{-20}-5q^{-22}-8q^{-24}+6q^{-26}-2q^{-30}+q^{-32}$
3	$-q^{63}+2q^{61}+q^{59}-2q^{57}-3q^{55}+3q^{53}+5q^{51}-9q^{49}-7q^{47}+18q^{45}+18q^{43}-28q^{41}-43q^{39}+32q^{37}+77q^{35}-19q^{33}-111q^{31}-16q^{29}+135q^{27}+61q^{25}-134q^{23}-107q^{21}+113q^{19}+136q^{17}-74q^{15}-147q^{13}+33q^{11}+134q^{9}+11q^{7}-114q^{5}-43q^{3}+82q+78q^{-1}-47q^{-3}-108q^{-5}+18q^{-7}+130q^{-9}+22q^{-11}-147q^{-13}-60q^{-15}+144q^{-17}+100q^{-19}-123q^{-21}-127q^{-23}+81q^{-25}+138q^{-27}-33q^{-29}-122q^{-31}-12q^{-33}+92q^{-35}+34q^{-37}-55q^{-39}-38q^{-41}+23q^{-43}+30q^{-45}-7q^{-47}-17q^{-49}+2q^{-51}+7q^{-53}-3q^{-57}+2q^{-61}-q^{-63}$
4	$q^{104}-2q^{102}-q^{100}+2q^{98}-q^{96}+6q^{94}-3q^{92}-q^{90}+3q^{88}-15q^{86}+4q^{84}-9q^{82}+19q^{80}+48q^{78}-16q^{76}-39q^{74}-104q^{72}-6q^{70}+168q^{68}+139q^{66}+21q^{64}-307q^{62}-297q^{60}+114q^{58}+432q^{56}+462q^{54}-242q^{52}-737q^{50}-435q^{48}+355q^{46}+1046q^{44}+398q^{42}-712q^{40}-1097q^{38}-314q^{36}+1080q^{34}+1080q^{32}-83q^{30}-1165q^{28}-957q^{26}+511q^{24}+1151q^{22}+518q^{20}-669q^{18}-1032q^{16}-79q^{14}+741q^{12}+699q^{10}-144q^{8}-745q^{6}-410q^{4}+316q^{2}+684+232q^{-2}-476q^{-4}-699q^{-6}-32q^{-8}+739q^{-10}+644q^{-12}-221q^{-14}-1039q^{-16}-505q^{-18}+672q^{-20}+1106q^{-22}+288q^{-24}-1105q^{-26}-1051q^{-28}+173q^{-30}+1194q^{-32}+915q^{-34}-570q^{-36}-1158q^{-38}-528q^{-40}+619q^{-42}+1068q^{-44}+183q^{-46}-600q^{-48}-745q^{-50}-112q^{-52}+579q^{-54}+439q^{-56}+35q^{-58}-384q^{-60}-324q^{-62}+69q^{-64}+201q^{-66}+194q^{-68}-39q^{-70}-150q^{-72}-56q^{-74}+6q^{-76}+81q^{-78}+24q^{-80}-26q^{-82}-13q^{-84}-15q^{-86}+15q^{-88}+6q^{-90}-5q^{-92}+q^{-94}-3q^{-96}+3q^{-98}-2q^{-102}+q^{-104}$
5	$-q^{155}+2q^{153}+q^{151}-2q^{149}+q^{147}-2q^{145}-6q^{143}-q^{141}+7q^{139}+7q^{137}+14q^{135}+9q^{133}-21q^{131}-42q^{129}-34q^{127}+6q^{125}+67q^{123}+114q^{121}+68q^{119}-86q^{117}-230q^{115}-229q^{113}-10q^{111}+337q^{109}+541q^{107}+312q^{105}-327q^{103}-914q^{101}-900q^{99}-60q^{97}+1158q^{95}+1774q^{93}+986q^{91}-942q^{89}-2613q^{87}-2479q^{85}-132q^{83}+2956q^{81}+4253q^{79}+2154q^{77}-2281q^{75}-5656q^{73}-4835q^{71}+270q^{69}+6021q^{67}+7522q^{65}+2799q^{63}-4910q^{61}-9312q^{59}-6264q^{57}+2355q^{55}+9646q^{53}+9218q^{51}+970q^{49}-8359q^{47}-10865q^{45}-4266q^{43}+5890q^{41}+10936q^{39}+6697q^{37}-2954q^{35}-9629q^{33}-7863q^{31}+307q^{29}+7487q^{27}+7789q^{25}+1633q^{23}-5202q^{21}-6905q^{19}-2664q^{17}+3225q^{15}+5673q^{13}+3104q^{11}-1790q^{9}-4664q^{7}-3236q^{5}+875q^{3}+4027q+3531q^{-1}-167q^{-3}-3868q^{-5}-4244q^{-7}-626q^{-9}+3982q^{-11}+5379q^{-13}+1868q^{-15}-3940q^{-17}-6868q^{-19}-3681q^{-21}+3417q^{-23}+8205q^{-25}+5965q^{-27}-1999q^{-29}-8937q^{-31}-8380q^{-33}-285q^{-35}+8536q^{-37}+10289q^{-39}+3193q^{-41}-6785q^{-43}-11078q^{-45}-6071q^{-47}+3866q^{-49}+10342q^{-51}+8163q^{-53}-435q^{-55}-8081q^{-57}-8823q^{-59}-2710q^{-61}+4870q^{-63}+7921q^{-65}+4688q^{-67}-1549q^{-69}-5752q^{-71}-5199q^{-73}-1054q^{-75}+3147q^{-77}+4391q^{-79}+2384q^{-81}-845q^{-83}-2851q^{-85}-2536q^{-87}-611q^{-89}+1307q^{-91}+1917q^{-93}+1124q^{-95}-203q^{-97}-1060q^{-99}-1016q^{-101}-324q^{-103}+405q^{-105}+645q^{-107}+391q^{-109}-35q^{-111}-300q^{-113}-280q^{-115}-85q^{-117}+108q^{-119}+145q^{-121}+67q^{-123}-18q^{-125}-54q^{-127}-42q^{-129}-4q^{-131}+25q^{-133}+14q^{-135}-2q^{-137}-3q^{-139}-3q^{-141}-3q^{-143}+2q^{-145}+3q^{-147}-3q^{-149}+2q^{-153}-q^{-155}$
6	$q^{216}-2q^{214}-q^{212}+2q^{210}-q^{208}+2q^{206}+2q^{204}+10q^{202}-5q^{200}-17q^{198}-6q^{196}-15q^{194}+22q^{190}+66q^{188}+35q^{186}-26q^{184}-53q^{182}-116q^{180}-105q^{178}-10q^{176}+211q^{174}+268q^{172}+173q^{170}+q^{168}-370q^{166}-611q^{164}-514q^{162}+164q^{160}+824q^{158}+1177q^{156}+1011q^{154}-149q^{152}-1661q^{150}-2616q^{148}-1812q^{146}+264q^{144}+2928q^{142}+4822q^{140}+3798q^{138}-158q^{136}-5401q^{134}-8227q^{132}-6897q^{130}-564q^{128}+8415q^{126}+14040q^{124}+12017q^{122}+1420q^{120}-12120q^{118}-21811q^{116}-19675q^{114}-3864q^{112}+17532q^{110}+32243q^{108}+29010q^{106}+7550q^{104}-23624q^{102}-45160q^{100}-41132q^{98}-10684q^{96}+31190q^{94}+58901q^{92}+54223q^{90}+13506q^{88}-40305q^{86}-74157q^{84}-64849q^{82}-13776q^{80}+49834q^{78}+88138q^{76}+72211q^{74}+10264q^{72}-61061q^{70}-97033q^{68}-73330q^{66}-3373q^{64}+71176q^{62}+100362q^{60}+67117q^{58}-7919q^{56}-76638q^{54}-95921q^{52}-54917q^{50}+19905q^{48}+77417q^{46}+83879q^{44}+37456q^{42}-28664q^{40}-71967q^{38}-66850q^{36}-19669q^{34}+33992q^{32}+61163q^{30}+46783q^{28}+5305q^{26}-34721q^{24}-47784q^{22}-28341q^{20}+5852q^{18}+32118q^{16}+33749q^{14}+13526q^{12}-13663q^{10}-28520q^{8}-22157q^{6}-1010q^{4}+20049q^{2}+25575+12761q^{-2}-10407q^{-4}-26842q^{-6}-24716q^{-8}-3554q^{-10}+23004q^{-12}+35485q^{-14}+23930q^{-16}-7415q^{-18}-37620q^{-20}-45703q^{-22}-21294q^{-24}+21842q^{-26}+54957q^{-28}+54079q^{-30}+14814q^{-32}-38933q^{-34}-72716q^{-36}-58950q^{-38}-3884q^{-40}+58180q^{-42}+86227q^{-44}+58431q^{-46}-9267q^{-48}-75600q^{-50}-94134q^{-52}-52644q^{-54}+23575q^{-56}+86248q^{-58}+94811q^{-60}+44229q^{-62}-34894q^{-64}-89869q^{-66}-88817q^{-68}-33787q^{-70}+39509q^{-72}+85832q^{-74}+78917q^{-76}+24663q^{-78}-38975q^{-80}-75574q^{-82}-65579q^{-84}-19254q^{-86}+33738q^{-88}+62390q^{-90}+52059q^{-92}+15336q^{-94}-25610q^{-96}-47167q^{-98}-40662q^{-100}-12962q^{-102}+17501q^{-104}+33234q^{-106}+29980q^{-108}+11451q^{-110}-9670q^{-112}-22494q^{-114}-21164q^{-116}-9504q^{-118}+4152q^{-120}+13708q^{-122}+14400q^{-124}+7972q^{-126}-1388q^{-128}-7710q^{-130}-9017q^{-132}-6002q^{-134}-348q^{-136}+4107q^{-138}+5613q^{-140}+3770q^{-142}+821q^{-144}-1881q^{-146}-3184q^{-148}-2387q^{-150}-647q^{-152}+988q^{-154}+1497q^{-156}+1301q^{-158}+468q^{-160}-464q^{-162}-779q^{-164}-598q^{-166}-136q^{-168}+137q^{-170}+336q^{-172}+284q^{-174}+40q^{-176}-102q^{-178}-129q^{-180}-57q^{-182}-31q^{-184}+38q^{-186}+60q^{-188}+14q^{-190}-11q^{-192}-17q^{-194}+2q^{-196}-10q^{-198}+11q^{-202}-2q^{-206}-3q^{-208}+3q^{-210}-2q^{-214}+q^{-216}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{14}+q^{12}-2q^{10}+q^{8}-q^{4}+4q^{2}-1+4q^{-2}-q^{-4}+q^{-8}-2q^{-10}+q^{-12}-q^{-14}$
1,1	$q^{44}-4q^{42}+12q^{40}-28q^{38}+52q^{36}-86q^{34}+134q^{32}-200q^{30}+273q^{28}-356q^{26}+454q^{24}-534q^{22}+577q^{20}-576q^{18}+508q^{16}-358q^{14}+124q^{12}+156q^{10}-458q^{8}+750q^{6}-996q^{4}+1162q^{2}-1226+1198q^{-2}-1050q^{-4}+834q^{-6}-548q^{-8}+246q^{-10}+58q^{-12}-320q^{-14}+494q^{-16}-598q^{-18}+623q^{-20}-584q^{-22}+494q^{-24}-390q^{-26}+293q^{-28}-202q^{-30}+128q^{-32}-80q^{-34}+46q^{-36}-22q^{-38}+10q^{-40}-4q^{-42}+q^{-44}$
2,0	$q^{38}-q^{36}-q^{34}+2q^{32}-2q^{30}-3q^{28}+4q^{26}+3q^{24}-4q^{22}-3q^{20}+7q^{18}+3q^{16}-12q^{14}+q^{12}+8q^{10}-7q^{8}-6q^{6}+8q^{4}+4q^{2}-4+5q^{-2}+9q^{-4}-3q^{-6}-5q^{-8}+10q^{-10}+q^{-12}-11q^{-14}+4q^{-16}+6q^{-18}-5q^{-20}-6q^{-22}+3q^{-24}+3q^{-26}-4q^{-28}-2q^{-30}+3q^{-32}-q^{-36}+q^{-38}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-2q^{32}+q^{30}+4q^{28}-7q^{26}+2q^{24}+9q^{22}-15q^{20}+2q^{18}+15q^{16}-21q^{14}-q^{12}+16q^{10}-14q^{8}-2q^{6}+13q^{4}+3q^{2}+2q^{-2}+13q^{-4}-3q^{-6}-14q^{-8}+15q^{-10}-q^{-12}-21q^{-14}+15q^{-16}+2q^{-18}-15q^{-20}+10q^{-22}+2q^{-24}-6q^{-26}+4q^{-28}-2q^{-32}+q^{-34}$
1,0,0	$-q^{17}+q^{15}-3q^{13}+2q^{11}-3q^{9}+2q^{7}-q^{5}+3q^{3}+2q+2q^{-1}+3q^{-3}-q^{-5}+2q^{-7}-3q^{-9}+2q^{-11}-3q^{-13}+q^{-15}-q^{-17}$
1,0,1	$q^{56}-4q^{54}+10q^{52}-14q^{50}+7q^{48}+16q^{46}-49q^{44}+71q^{42}-56q^{40}-13q^{38}+107q^{36}-182q^{34}+175q^{32}-43q^{30}-161q^{28}+356q^{26}-413q^{24}+264q^{22}+51q^{20}-409q^{18}+607q^{16}-567q^{14}+273q^{12}+91q^{10}-359q^{8}+410q^{6}-263q^{4}+82q^{2}+28+54q^{-2}-201q^{-4}+340q^{-6}-272q^{-8}+25q^{-10}+331q^{-12}-591q^{-14}+606q^{-16}-387q^{-18}+6q^{-20}+308q^{-22}-460q^{-24}+390q^{-26}-188q^{-28}-33q^{-30}+176q^{-32}-192q^{-34}+126q^{-36}-29q^{-38}-42q^{-40}+61q^{-42}-46q^{-44}+18q^{-46}+4q^{-48}-10q^{-50}+8q^{-52}-4q^{-54}+q^{-56}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{40}-q^{38}+4q^{34}-2q^{32}-3q^{30}+8q^{28}-2q^{26}-11q^{24}+5q^{22}+6q^{20}-16q^{18}-9q^{16}+9q^{14}-2q^{12}-17q^{10}+7q^{8}+21q^{6}-8q^{4}+5q^{2}+29+4q^{-2}-11q^{-4}+18q^{-6}+4q^{-8}-19q^{-10}-5q^{-12}+8q^{-14}-8q^{-16}-15q^{-18}+7q^{-20}+7q^{-22}-8q^{-24}-q^{-26}+8q^{-28}-3q^{-30}-2q^{-32}+3q^{-34}-q^{-36}-q^{-38}+q^{-40}$
1,0,0,0	$-q^{20}+q^{18}-3q^{16}+q^{14}-2q^{12}-q^{10}+q^{8}-q^{6}+4q^{4}+q^{2}+5+q^{-2}+4q^{-4}-q^{-6}+q^{-8}-q^{-10}-2q^{-12}+q^{-14}-3q^{-16}+q^{-18}-q^{-20}$

B2 Invariants.

Weight

Invariant

0,1

-q^{34}+2q^{32}-5q^{30}+8q^{28}-11q^{26}+16q^{24}-21q^{22}+23q^{20}-24q^{18}+21q^{16}-15q^{14}+7q^{12}+4q^{10}-16q^{8}+28q^{6}-37q^{4}+45q^{2}-46+46q^{-2}-37q^{-4}+29q^{-6}-16q^{-8}+5q^{-10}+7q^{-12}-15q^{-14}+21q^{-16}-24q^{-18}+23q^{-20}-22q^{-22}+16q^{-24}-12q^{-26}+8q^{-28}-4q^{-30}+2q^{-32}-q^{-34}

1,0

q^{56}-2q^{52}-2q^{50}+3q^{48}+6q^{46}-q^{44}-9q^{42}-6q^{40}+9q^{38}+15q^{36}-4q^{34}-21q^{32}-9q^{30}+19q^{28}+20q^{26}-11q^{24}-26q^{22}-3q^{20}+23q^{18}+11q^{16}-17q^{14}-15q^{12}+11q^{10}+17q^{8}-4q^{6}-15q^{4}+4q^{2}+18+4q^{-2}-15q^{-4}-4q^{-6}+16q^{-8}+11q^{-10}-15q^{-12}-17q^{-14}+10q^{-16}+23q^{-18}-2q^{-20}-26q^{-22}-12q^{-24}+20q^{-26}+20q^{-28}-9q^{-30}-22q^{-32}-4q^{-34}+16q^{-36}+9q^{-38}-6q^{-40}-9q^{-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}-9q^{36}+10q^{34}-13q^{32}+17q^{30}-19q^{28}+17q^{26}-18q^{24}+17q^{22}-16q^{20}+5q^{18}-6q^{16}-2q^{14}+8q^{12}-19q^{10}+22q^{8}-23q^{6}+39q^{4}-30q^{2}+40-30q^{-2}+39q^{-4}-24q^{-6}+22q^{-8}-20q^{-10}+8q^{-12}-3q^{-14}-6q^{-16}+4q^{-18}-16q^{-20}+17q^{-22}-18q^{-24}+18q^{-26}-19q^{-28}+18q^{-30}-13q^{-32}+11q^{-34}-9q^{-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}-5q^{70}-3q^{68}+16q^{66}-27q^{64}+36q^{62}-38q^{60}+23q^{58}-q^{56}-34q^{54}+70q^{52}-92q^{50}+95q^{48}-66q^{46}+4q^{44}+68q^{42}-130q^{40}+151q^{38}-122q^{36}+46q^{34}+44q^{32}-116q^{30}+138q^{28}-93q^{26}+9q^{24}+77q^{22}-122q^{20}+97q^{18}-19q^{16}-82q^{14}+161q^{12}-175q^{10}+132q^{8}-27q^{6}-94q^{4}+190q^{2}-223+188q^{-2}-92q^{-4}-27q^{-6}+132q^{-8}-176q^{-10}+165q^{-12}-89q^{-14}-10q^{-16}+93q^{-18}-126q^{-20}+87q^{-22}-3q^{-24}-84q^{-26}+137q^{-28}-124q^{-30}+54q^{-32}+40q^{-34}-124q^{-36}+160q^{-38}-142q^{-40}+74q^{-42}+7q^{-44}-77q^{-46}+108q^{-48}-102q^{-50}+72q^{-52}-28q^{-54}-10q^{-56}+32q^{-58}-42q^{-60}+36q^{-62}-23q^{-64}+12q^{-66}-6q^{-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.

In[3]:= K = Knot["10 91"];

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

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

Out[4]= $t^{4}-4t^{3}+9t^{2}-14t+17-14t^{-1}+9t^{-2}-4t^{-3}+t^{-4}$

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

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

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]= { 73, 0 }

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

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

Out[8]= $-q^{5}+3q^{4}-6q^{3}+9q^{2}-11q+13-11q^{-1}+9q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$

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

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

Out[9]= $z^{8}-a^{2}z^{6}-z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-4z^{4}a^{-2}+13z^{4}-5a^{2}z^{2}-5z^{2}a^{-2}+12z^{2}-2a^{2}-2a^{-2}+5$

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

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

Out[10]= $2az^{9}+2z^{9}a^{-1}+4a^{2}z^{8}+5z^{8}a^{-2}+9z^{8}+4a^{3}z^{7}+az^{7}+2z^{7}a^{-1}+5z^{7}a^{-3}+3a^{4}z^{6}-7a^{2}z^{6}-13z^{6}a^{-2}+3z^{6}a^{-4}-26z^{6}+a^{5}z^{5}-6a^{3}z^{5}-7az^{5}-13z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}+7a^{2}z^{4}+16z^{4}a^{-2}-6z^{4}a^{-4}+35z^{4}-2a^{5}z^{3}+9az^{3}+18z^{3}a^{-1}+9z^{3}a^{-3}-2z^{3}a^{-5}+2a^{4}z^{2}-7a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}-19z^{2}+a^{5}z-4az-6za^{-1}-3za^{-3}+2a^{2}+2a^{-2}+5$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {10_43,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091.

In[3]:= K = Knot["10 91"];

In[4]:= {A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[4]= { $t^{4}-4t^{3}+9t^{2}-14t+17-14t^{-1}+9t^{-2}-4t^{-3}+t^{-4}$ , $-q^{5}+3q^{4}-6q^{3}+9q^{2}-11q+13-11q^{-1}+9q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$ }

In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]= {}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {10_43,}

Vassiliev invariants

V₂ and V₃:

(2, 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

8

0

32

{\frac {28}{3}}

-{\frac {76}{3}}

0

32

-32

64

{\frac {256}{3}}

0

{\frac {224}{3}}

-{\frac {608}{3}}

{\frac {511}{15}}

{\frac {812}{5}}

-{\frac {13556}{45}}

-{\frac {175}{9}}

-{\frac {1649}{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 91. 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

7

4

1

-3

5

2

3

6

4

-2

1

7

5

2

-1

5

7

2

-3

4

6

-2

-5

2

5

3

-7

1

4

-3

-9

2

-11

1

-1

Integral Khovanov Homology

(db, data source)

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

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the

n+1

-dimensional representation of

sl(2)

)

$n$	$J_{n}$
2	$q^{15}-3q^{14}+2q^{13}+7q^{12}-17q^{11}+5q^{10}+30q^{9}-43q^{8}-5q^{7}+72q^{6}-66q^{5}-31q^{4}+114q^{3}-73q^{2}-58q+132-60q^{-1}-71q^{-2}+113q^{-3}-32q^{-4}-64q^{-5}+71q^{-6}-7q^{-7}-40q^{-8}+30q^{-9}+2q^{-10}-15q^{-11}+8q^{-12}+q^{-13}-3q^{-14}+q^{-15}$
3	$-q^{30}+3q^{29}-2q^{28}-3q^{27}+2q^{26}+10q^{25}-7q^{24}-22q^{23}+12q^{22}+47q^{21}-14q^{20}-83q^{19}-5q^{18}+136q^{17}+44q^{16}-187q^{15}-115q^{14}+225q^{13}+215q^{12}-244q^{11}-323q^{10}+229q^{9}+438q^{8}-200q^{7}-527q^{6}+142q^{5}+607q^{4}-92q^{3}-639q^{2}+16q+668+33q^{-1}-635q^{-2}-109q^{-3}+597q^{-4}+158q^{-5}-512q^{-6}-210q^{-7}+417q^{-8}+231q^{-9}-302q^{-10}-233q^{-11}+197q^{-12}+204q^{-13}-107q^{-14}-159q^{-15}+46q^{-16}+109q^{-17}-15q^{-18}-63q^{-19}+q^{-20}+34q^{-21}-17q^{-23}+q^{-24}+9q^{-25}-2q^{-26}-3q^{-27}-q^{-28}+3q^{-29}-q^{-30}$
4	$q^{50}-3q^{49}+2q^{48}+3q^{47}-6q^{46}+5q^{45}-9q^{44}+13q^{43}+12q^{42}-36q^{41}+7q^{40}-22q^{39}+63q^{38}+69q^{37}-111q^{36}-55q^{35}-116q^{34}+174q^{33}+302q^{32}-104q^{31}-187q^{30}-509q^{29}+114q^{28}+721q^{27}+300q^{26}-47q^{25}-1200q^{24}-519q^{23}+866q^{22}+1083q^{21}+838q^{20}-1649q^{19}-1666q^{18}+236q^{17}+1671q^{16}+2323q^{15}-1370q^{14}-2687q^{13}-988q^{12}+1617q^{11}+3716q^{10}-552q^{9}-3121q^{8}-2165q^{7}+1083q^{6}+4534q^{5}+313q^{4}-3026q^{3}-2936q^{2}+416q+4757+1021q^{-1}-2574q^{-2}-3304q^{-3}-310q^{-4}+4422q^{-5}+1622q^{-6}-1731q^{-7}-3262q^{-8}-1130q^{-9}+3469q^{-10}+1985q^{-11}-544q^{-12}-2629q^{-13}-1770q^{-14}+2001q^{-15}+1777q^{-16}+538q^{-17}-1466q^{-18}-1770q^{-19}+607q^{-20}+994q^{-21}+923q^{-22}-356q^{-23}-1122q^{-24}-84q^{-25}+204q^{-26}+621q^{-27}+139q^{-28}-418q^{-29}-114q^{-30}-114q^{-31}+210q^{-32}+129q^{-33}-90q^{-34}+4q^{-35}-85q^{-36}+36q^{-37}+31q^{-38}-25q^{-39}+27q^{-40}-21q^{-41}+7q^{-42}+3q^{-43}-12q^{-44}+8q^{-45}-3q^{-46}+3q^{-47}+q^{-48}-3q^{-49}+q^{-50}$
5	$-q^{75}+3q^{74}-2q^{73}-3q^{72}+6q^{71}-q^{70}-6q^{69}+3q^{68}-2q^{67}-2q^{66}+22q^{65}+10q^{64}-35q^{63}-35q^{62}-14q^{61}+34q^{60}+107q^{59}+88q^{58}-72q^{57}-228q^{56}-209q^{55}+14q^{54}+372q^{53}+514q^{52}+182q^{51}-468q^{50}-938q^{49}-678q^{48}+328q^{47}+1371q^{46}+1509q^{45}+325q^{44}-1548q^{43}-2596q^{42}-1597q^{41}+1056q^{40}+3515q^{39}+3554q^{38}+459q^{37}-3840q^{36}-5798q^{35}-3089q^{34}+2962q^{33}+7757q^{32}+6696q^{31}-607q^{30}-8849q^{29}-10669q^{28}-3151q^{27}+8499q^{26}+14342q^{25}+7991q^{24}-6670q^{23}-17145q^{22}-13088q^{21}+3492q^{20}+18635q^{19}+17969q^{18}+426q^{17}-18898q^{16}-21909q^{15}-4603q^{14}+18078q^{13}+24907q^{12}+8390q^{11}-16658q^{10}-26697q^{9}-11701q^{8}+14891q^{7}+27835q^{6}+14198q^{5}-13147q^{4}-28094q^{3}-16309q^{2}+11273q+28211+17899q^{-1}-9449q^{-2}-27598q^{-3}-19461q^{-4}+7162q^{-5}+26783q^{-6}+20773q^{-7}-4555q^{-8}-25029q^{-9}-21909q^{-10}+1273q^{-11}+22542q^{-12}+22476q^{-13}+2280q^{-14}-18873q^{-15}-22211q^{-16}-5907q^{-17}+14372q^{-18}+20710q^{-19}+8955q^{-20}-9222q^{-21}-17972q^{-22}-10953q^{-23}+4216q^{-24}+14111q^{-25}+11461q^{-26}+107q^{-27}-9724q^{-28}-10525q^{-29}-3075q^{-30}+5492q^{-31}+8413q^{-32}+4509q^{-33}-2015q^{-34}-5802q^{-35}-4576q^{-36}-259q^{-37}+3308q^{-38}+3688q^{-39}+1360q^{-40}-1367q^{-41}-2477q^{-42}-1556q^{-43}+220q^{-44}+1341q^{-45}+1226q^{-46}+304q^{-47}-549q^{-48}-768q^{-49}-396q^{-50}+123q^{-51}+386q^{-52}+290q^{-53}+38q^{-54}-129q^{-55}-167q^{-56}-81q^{-57}+39q^{-58}+71q^{-59}+37q^{-60}+15q^{-61}-13q^{-62}-35q^{-63}-8q^{-64}+10q^{-65}-3q^{-66}+7q^{-67}+8q^{-68}-5q^{-69}-3q^{-70}+3q^{-71}-3q^{-72}-q^{-73}+3q^{-74}-q^{-75}$
6	$q^{105}-3q^{104}+2q^{103}+3q^{102}-6q^{101}+q^{100}+2q^{99}+12q^{98}-14q^{97}-8q^{96}+15q^{95}-25q^{94}+7q^{93}+27q^{92}+58q^{91}-36q^{90}-77q^{89}-11q^{88}-97q^{87}+34q^{86}+169q^{85}+302q^{84}+16q^{83}-276q^{82}-284q^{81}-559q^{80}-147q^{79}+484q^{78}+1234q^{77}+849q^{76}-80q^{75}-793q^{74}-2194q^{73}-1887q^{72}-313q^{71}+2537q^{70}+3551q^{69}+2869q^{68}+1050q^{67}-3700q^{66}-6342q^{65}-5967q^{64}-478q^{63}+4858q^{62}+9191q^{61}+10410q^{60}+2728q^{59}-7034q^{58}-15523q^{57}-14134q^{56}-6802q^{55}+7861q^{54}+23234q^{53}+23849q^{52}+11495q^{51}-12269q^{50}-29867q^{49}-37265q^{48}-19839q^{47}+16729q^{46}+45406q^{45}+52441q^{44}+24454q^{43}-19536q^{42}-65917q^{41}-72831q^{40}-29596q^{39}+35411q^{38}+89040q^{37}+88092q^{36}+34718q^{35}-59002q^{34}-119154q^{33}-102892q^{32}-19619q^{31}+87988q^{30}+143067q^{29}+113841q^{28}-8420q^{27}-127717q^{26}-165565q^{25}-95838q^{24}+46498q^{23}+161601q^{22}+180174q^{21}+59278q^{20}-99921q^{19}-193612q^{18}-157902q^{17}-8568q^{16}+147835q^{15}+213957q^{14}+113025q^{13}-60656q^{12}-192734q^{11}-191017q^{10}-51704q^{9}+123426q^{8}+222040q^{7}+143230q^{6}-29311q^{5}-181179q^{4}-203498q^{3}-78262q^{2}+102264q+219914+159665q^{-1}-6143q^{-2}-168365q^{-3}-209024q^{-4}-99321q^{-5}+81117q^{-6}+213551q^{-7}+174522q^{-8}+21046q^{-9}-148142q^{-10}-210655q^{-11}-125587q^{-12}+46924q^{-13}+194108q^{-14}+187585q^{-15}+61072q^{-16}-106664q^{-17}-196275q^{-18}-152758q^{-19}-6737q^{-20}+146927q^{-21}+182468q^{-22}+104375q^{-23}-40544q^{-24}-149852q^{-25}-159220q^{-26}-64249q^{-27}+72105q^{-28}+141464q^{-29}+123658q^{-30}+28175q^{-31}-74816q^{-32}-125975q^{-33}-93435q^{-34}-2444q^{-35}+71507q^{-36}+99955q^{-37}+64147q^{-38}-3491q^{-39}-64028q^{-40}-77508q^{-41}-40748q^{-42}+7897q^{-43}+48882q^{-44}+54839q^{-45}+30361q^{-46}-10217q^{-47}-36791q^{-48}-36070q^{-49}-19814q^{-50}+7008q^{-51}+24391q^{-52}+26333q^{-53}+11319q^{-54}-5617q^{-55}-14610q^{-56}-16581q^{-57}-7703q^{-58}+2995q^{-59}+10522q^{-60}+9183q^{-61}+4074q^{-62}-1070q^{-63}-5984q^{-64}-5680q^{-65}-2630q^{-66}+1543q^{-67}+2850q^{-68}+2744q^{-69}+1756q^{-70}-741q^{-71}-1724q^{-72}-1606q^{-73}-351q^{-74}+186q^{-75}+668q^{-76}+952q^{-77}+214q^{-78}-212q^{-79}-446q^{-80}-185q^{-81}-167q^{-82}+8q^{-83}+274q^{-84}+117q^{-85}+29q^{-86}-75q^{-87}-13q^{-88}-72q^{-89}-49q^{-90}+53q^{-91}+22q^{-92}+18q^{-93}-12q^{-94}+14q^{-95}-11q^{-96}-18q^{-97}+9q^{-98}+3q^{-100}-3q^{-101}+3q^{-102}+q^{-103}-3q^{-104}+q^{-105}$
7	$-q^{140}+3q^{139}-2q^{138}-3q^{137}+6q^{136}-q^{135}-2q^{134}-8q^{133}-q^{132}+24q^{131}-5q^{130}-12q^{129}+9q^{128}-13q^{127}-10q^{126}-28q^{125}+107q^{123}+46q^{122}-22q^{121}-43q^{120}-146q^{119}-98q^{118}-102q^{117}+34q^{116}+439q^{115}+423q^{114}+214q^{113}-160q^{112}-761q^{111}-853q^{110}-796q^{109}-210q^{108}+1276q^{107}+2062q^{106}+2114q^{105}+966q^{104}-1551q^{103}-3331q^{102}-4503q^{101}-3647q^{100}+359q^{99}+4648q^{98}+8226q^{97}+8480q^{96}+3476q^{95}-3482q^{94}-11393q^{93}-15909q^{92}-12223q^{91}-2920q^{90}+11191q^{89}+23390q^{88}+25430q^{87}+17552q^{86}-1768q^{85}-24994q^{84}-39645q^{83}-41137q^{82}-22049q^{81}+12339q^{80}+45867q^{79}+67945q^{78}+61275q^{77}+23695q^{76}-30157q^{75}-85168q^{74}-109777q^{73}-86340q^{72}-20184q^{71}+73631q^{70}+149589q^{69}+167802q^{68}+112693q^{67}-13360q^{66}-156006q^{65}-248112q^{64}-240273q^{63}-106294q^{62}+102642q^{61}+295626q^{60}+380223q^{59}+282191q^{58}+28014q^{57}-278825q^{56}-498532q^{55}-491176q^{54}-234937q^{53}+174113q^{52}+557006q^{51}+697131q^{50}+498890q^{49}+23069q^{48}-528379q^{47}-859848q^{46}-783000q^{45}-295636q^{44}+401905q^{43}+946025q^{42}+1045961q^{41}+611059q^{40}-189095q^{39}-940980q^{38}-1251817q^{37}-927406q^{36}-81393q^{35}+847907q^{34}+1379085q^{33}+1208434q^{32}+372926q^{31}-688799q^{30}-1424863q^{29}-1428063q^{28}-649734q^{27}+493157q^{26}+1401264q^{25}+1577280q^{24}+885628q^{23}-292144q^{22}-1329981q^{21}-1660056q^{20}-1067682q^{19}+109185q^{18}+1235684q^{17}+1691414q^{16}+1194554q^{15}+40634q^{14}-1138216q^{13}-1688281q^{12}-1275815q^{11}-155218q^{10}+1052033q^{9}+1670034q^{8}+1324510q^{7}+237205q^{6}-982038q^{5}-1647416q^{4}-1356453q^{3}-300176q^{2}+926937q+1631196+1384848q^{-1}+354411q^{-2}-878768q^{-3}-1619062q^{-4}-1419481q^{-5}-417387q^{-6}+825039q^{-7}+1609342q^{-8}+1464942q^{-9}+497790q^{-10}-751578q^{-11}-1587939q^{-12}-1517934q^{-13}-605726q^{-14}+643232q^{-15}+1541628q^{-16}+1569085q^{-17}+738971q^{-18}-490362q^{-19}-1451106q^{-20}-1600154q^{-21}-889095q^{-22}+289369q^{-23}+1302550q^{-24}+1589981q^{-25}+1035402q^{-26}-49707q^{-27}-1088279q^{-28}-1516805q^{-29}-1151186q^{-30}-207003q^{-31}+814744q^{-32}+1366862q^{-33}+1206888q^{-34}+448642q^{-35}-502674q^{-36}-1140697q^{-37}-1180870q^{-38}-638782q^{-39}+187587q^{-40}+855626q^{-41}+1064750q^{-42}+747466q^{-43}+90940q^{-44}-545478q^{-45}-871645q^{-46}-759540q^{-47}-295978q^{-48}+252223q^{-49}+630712q^{-50}+680531q^{-51}+407382q^{-52}-14127q^{-53}-382931q^{-54}-536608q^{-55}-424989q^{-56}-143221q^{-57}+167515q^{-58}+364088q^{-59}+368655q^{-60}+215994q^{-61}-10817q^{-62}-201155q^{-63}-271514q^{-64}-217968q^{-65}-77732q^{-66}+74444q^{-67}+166205q^{-68}+175493q^{-69}+107793q^{-70}+5076q^{-71}-78580q^{-72}-116915q^{-73}-98125q^{-74}-40710q^{-75}+19953q^{-76}+62796q^{-77}+70393q^{-78}+46335q^{-79}+10071q^{-80}-24744q^{-81}-41254q^{-82}-36242q^{-83}-19084q^{-84}+3547q^{-85}+19079q^{-86}+22466q^{-87}+17185q^{-88}+4852q^{-89}-6219q^{-90}-11397q^{-91}-11531q^{-92}-5868q^{-93}+431q^{-94}+4384q^{-95}+6340q^{-96}+4485q^{-97}+1356q^{-98}-1230q^{-99}-3135q^{-100}-2534q^{-101}-1169q^{-102}-75q^{-103}+1210q^{-104}+1319q^{-105}+912q^{-106}+321q^{-107}-593q^{-108}-618q^{-109}-388q^{-110}-264q^{-111}+134q^{-112}+228q^{-113}+277q^{-114}+241q^{-115}-99q^{-116}-138q^{-117}-84q^{-118}-96q^{-119}+5q^{-121}+52q^{-122}+97q^{-123}-7q^{-124}-24q^{-125}-12q^{-126}-22q^{-127}+2q^{-128}-12q^{-129}+22q^{-131}+q^{-132}-4q^{-133}-3q^{-135}+3q^{-136}-3q^{-137}-q^{-138}+3q^{-139}-q^{-140}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

Back to the top.

10_90

10_92

10 91: Difference between revisions

Latest revision as of 17:57, 1 September 2005

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

Modifying This Page

Navigation menu

Page actions

Page actions

Personal tools

Navigation

Search

Tools

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+<!-- This page was generated from the splice base [[Rolfsen_Splice_Base]]. Please do not edit!
+<!-- You probably want to edit the template referred to immediately below. (See [[Category:Knot Page Template]].)
+<!-- This page itself was created by running [[Media:KnotPageSpliceRobot.nb]] on [[Rolfsen_Splice_Base]]. -->
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+<!--  -->
+{{Rolfsen Knot Page|
+n = 10 |
+k = 91 |
+KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-5,4,-10,2,-1,6,-7,3,-4,9,-8,10,-6,7,-3,5,-9,8,-2/goTop.html |
+braid_table     = <table cellspacing=0 cellpadding=0 border=0>
+<tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr>
+<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]]</td></tr>
+<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]]</td></tr>
+</table> |
+braid_crossings = 10 |
+braid_width     = 3 |
+braid_index     = 3 |
+same_alexander  =  |
+same_jones      = [[10_43]],  |
+khovanov_table  = <table border=1>
+<tr align=center>
+<td width=13.3333%><table cellpadding=0 cellspacing=0>
+  <tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
+<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
+<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
+</table></td>
+  <td width=6.66667%>-5</td    ><td width=6.66667%>-4</td    ><td width=6.66667%>-3</td    ><td width=6.66667%>-2</td    ><td width=6.66667%>-1</td    ><td width=6.66667%>0</td    ><td width=6.66667%>1</td    ><td width=6.66667%>2</td    ><td width=6.66667%>3</td    ><td width=6.66667%>4</td    ><td width=6.66667%>5</td    ><td width=13.3333%>&chi;</td></tr>
+<tr align=center><td>11</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td>-1</td></tr>
+<tr align=center><td>9</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>2</td><td bgcolor=yellow>&nbsp;</td><td>2</td></tr>
+<tr align=center><td>7</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>4</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>-3</td></tr>
+<tr align=center><td>5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>5</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>3</td></tr>
+<tr align=center><td>3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>6</td><td bgcolor=yellow>4</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-2</td></tr>
+<tr align=center><td>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>7</td><td bgcolor=yellow>5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>2</td></tr>
+<tr align=center><td>-1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>5</td><td bgcolor=yellow>7</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>2</td></tr>
+<tr align=center><td>-3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>4</td><td bgcolor=yellow>6</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-2</td></tr>
+<tr align=center><td>-5</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>2</td><td bgcolor=yellow>5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>3</td></tr>
+<tr align=center><td>-7</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>4</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-3</td></tr>
+<tr align=center><td>-9</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>2</td></tr>
+<tr align=center><td>-11</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
+</table> |
+coloured_jones_2 = <math>q^{15}-3 q^{14}+2 q^{13}+7 q^{12}-17 q^{11}+5 q^{10}+30 q^9-43 q^8-5 q^7+72 q^6-66 q^5-31 q^4+114 q^3-73 q^2-58 q+132-60 q^{-1} -71 q^{-2} +113 q^{-3} -32 q^{-4} -64 q^{-5} +71 q^{-6} -7 q^{-7} -40 q^{-8} +30 q^{-9} +2 q^{-10} -15 q^{-11} +8 q^{-12} + q^{-13} -3 q^{-14} + q^{-15} </math> |
+coloured_jones_3 = <math>-q^{30}+3 q^{29}-2 q^{28}-3 q^{27}+2 q^{26}+10 q^{25}-7 q^{24}-22 q^{23}+12 q^{22}+47 q^{21}-14 q^{20}-83 q^{19}-5 q^{18}+136 q^{17}+44 q^{16}-187 q^{15}-115 q^{14}+225 q^{13}+215 q^{12}-244 q^{11}-323 q^{10}+229 q^9+438 q^8-200 q^7-527 q^6+142 q^5+607 q^4-92 q^3-639 q^2+16 q+668+33 q^{-1} -635 q^{-2} -109 q^{-3} +597 q^{-4} +158 q^{-5} -512 q^{-6} -210 q^{-7} +417 q^{-8} +231 q^{-9} -302 q^{-10} -233 q^{-11} +197 q^{-12} +204 q^{-13} -107 q^{-14} -159 q^{-15} +46 q^{-16} +109 q^{-17} -15 q^{-18} -63 q^{-19} + q^{-20} +34 q^{-21} -17 q^{-23} + q^{-24} +9 q^{-25} -2 q^{-26} -3 q^{-27} - q^{-28} +3 q^{-29} - q^{-30} </math> |
+coloured_jones_4 = <math>q^{50}-3 q^{49}+2 q^{48}+3 q^{47}-6 q^{46}+5 q^{45}-9 q^{44}+13 q^{43}+12 q^{42}-36 q^{41}+7 q^{40}-22 q^{39}+63 q^{38}+69 q^{37}-111 q^{36}-55 q^{35}-116 q^{34}+174 q^{33}+302 q^{32}-104 q^{31}-187 q^{30}-509 q^{29}+114 q^{28}+721 q^{27}+300 q^{26}-47 q^{25}-1200 q^{24}-519 q^{23}+866 q^{22}+1083 q^{21}+838 q^{20}-1649 q^{19}-1666 q^{18}+236 q^{17}+1671 q^{16}+2323 q^{15}-1370 q^{14}-2687 q^{13}-988 q^{12}+1617 q^{11}+3716 q^{10}-552 q^9-3121 q^8-2165 q^7+1083 q^6+4534 q^5+313 q^4-3026 q^3-2936 q^2+416 q+4757+1021 q^{-1} -2574 q^{-2} -3304 q^{-3} -310 q^{-4} +4422 q^{-5} +1622 q^{-6} -1731 q^{-7} -3262 q^{-8} -1130 q^{-9} +3469 q^{-10} +1985 q^{-11} -544 q^{-12} -2629 q^{-13} -1770 q^{-14} +2001 q^{-15} +1777 q^{-16} +538 q^{-17} -1466 q^{-18} -1770 q^{-19} +607 q^{-20} +994 q^{-21} +923 q^{-22} -356 q^{-23} -1122 q^{-24} -84 q^{-25} +204 q^{-26} +621 q^{-27} +139 q^{-28} -418 q^{-29} -114 q^{-30} -114 q^{-31} +210 q^{-32} +129 q^{-33} -90 q^{-34} +4 q^{-35} -85 q^{-36} +36 q^{-37} +31 q^{-38} -25 q^{-39} +27 q^{-40} -21 q^{-41} +7 q^{-42} +3 q^{-43} -12 q^{-44} +8 q^{-45} -3 q^{-46} +3 q^{-47} + q^{-48} -3 q^{-49} + q^{-50} </math> |
+coloured_jones_5 = <math>-q^{75}+3 q^{74}-2 q^{73}-3 q^{72}+6 q^{71}-q^{70}-6 q^{69}+3 q^{68}-2 q^{67}-2 q^{66}+22 q^{65}+10 q^{64}-35 q^{63}-35 q^{62}-14 q^{61}+34 q^{60}+107 q^{59}+88 q^{58}-72 q^{57}-228 q^{56}-209 q^{55}+14 q^{54}+372 q^{53}+514 q^{52}+182 q^{51}-468 q^{50}-938 q^{49}-678 q^{48}+328 q^{47}+1371 q^{46}+1509 q^{45}+325 q^{44}-1548 q^{43}-2596 q^{42}-1597 q^{41}+1056 q^{40}+3515 q^{39}+3554 q^{38}+459 q^{37}-3840 q^{36}-5798 q^{35}-3089 q^{34}+2962 q^{33}+7757 q^{32}+6696 q^{31}-607 q^{30}-8849 q^{29}-10669 q^{28}-3151 q^{27}+8499 q^{26}+14342 q^{25}+7991 q^{24}-6670 q^{23}-17145 q^{22}-13088 q^{21}+3492 q^{20}+18635 q^{19}+17969 q^{18}+426 q^{17}-18898 q^{16}-21909 q^{15}-4603 q^{14}+18078 q^{13}+24907 q^{12}+8390 q^{11}-16658 q^{10}-26697 q^9-11701 q^8+14891 q^7+27835 q^6+14198 q^5-13147 q^4-28094 q^3-16309 q^2+11273 q+28211+17899 q^{-1} -9449 q^{-2} -27598 q^{-3} -19461 q^{-4} +7162 q^{-5} +26783 q^{-6} +20773 q^{-7} -4555 q^{-8} -25029 q^{-9} -21909 q^{-10} +1273 q^{-11} +22542 q^{-12} +22476 q^{-13} +2280 q^{-14} -18873 q^{-15} -22211 q^{-16} -5907 q^{-17} +14372 q^{-18} +20710 q^{-19} +8955 q^{-20} -9222 q^{-21} -17972 q^{-22} -10953 q^{-23} +4216 q^{-24} +14111 q^{-25} +11461 q^{-26} +107 q^{-27} -9724 q^{-28} -10525 q^{-29} -3075 q^{-30} +5492 q^{-31} +8413 q^{-32} +4509 q^{-33} -2015 q^{-34} -5802 q^{-35} -4576 q^{-36} -259 q^{-37} +3308 q^{-38} +3688 q^{-39} +1360 q^{-40} -1367 q^{-41} -2477 q^{-42} -1556 q^{-43} +220 q^{-44} +1341 q^{-45} +1226 q^{-46} +304 q^{-47} -549 q^{-48} -768 q^{-49} -396 q^{-50} +123 q^{-51} +386 q^{-52} +290 q^{-53} +38 q^{-54} -129 q^{-55} -167 q^{-56} -81 q^{-57} +39 q^{-58} +71 q^{-59} +37 q^{-60} +15 q^{-61} -13 q^{-62} -35 q^{-63} -8 q^{-64} +10 q^{-65} -3 q^{-66} +7 q^{-67} +8 q^{-68} -5 q^{-69} -3 q^{-70} +3 q^{-71} -3 q^{-72} - q^{-73} +3 q^{-74} - q^{-75} </math> |
+coloured_jones_6 = <math>q^{105}-3 q^{104}+2 q^{103}+3 q^{102}-6 q^{101}+q^{100}+2 q^{99}+12 q^{98}-14 q^{97}-8 q^{96}+15 q^{95}-25 q^{94}+7 q^{93}+27 q^{92}+58 q^{91}-36 q^{90}-77 q^{89}-11 q^{88}-97 q^{87}+34 q^{86}+169 q^{85}+302 q^{84}+16 q^{83}-276 q^{82}-284 q^{81}-559 q^{80}-147 q^{79}+484 q^{78}+1234 q^{77}+849 q^{76}-80 q^{75}-793 q^{74}-2194 q^{73}-1887 q^{72}-313 q^{71}+2537 q^{70}+3551 q^{69}+2869 q^{68}+1050 q^{67}-3700 q^{66}-6342 q^{65}-5967 q^{64}-478 q^{63}+4858 q^{62}+9191 q^{61}+10410 q^{60}+2728 q^{59}-7034 q^{58}-15523 q^{57}-14134 q^{56}-6802 q^{55}+7861 q^{54}+23234 q^{53}+23849 q^{52}+11495 q^{51}-12269 q^{50}-29867 q^{49}-37265 q^{48}-19839 q^{47}+16729 q^{46}+45406 q^{45}+52441 q^{44}+24454 q^{43}-19536 q^{42}-65917 q^{41}-72831 q^{40}-29596 q^{39}+35411 q^{38}+89040 q^{37}+88092 q^{36}+34718 q^{35}-59002 q^{34}-119154 q^{33}-102892 q^{32}-19619 q^{31}+87988 q^{30}+143067 q^{29}+113841 q^{28}-8420 q^{27}-127717 q^{26}-165565 q^{25}-95838 q^{24}+46498 q^{23}+161601 q^{22}+180174 q^{21}+59278 q^{20}-99921 q^{19}-193612 q^{18}-157902 q^{17}-8568 q^{16}+147835 q^{15}+213957 q^{14}+113025 q^{13}-60656 q^{12}-192734 q^{11}-191017 q^{10}-51704 q^9+123426 q^8+222040 q^7+143230 q^6-29311 q^5-181179 q^4-203498 q^3-78262 q^2+102264 q+219914+159665 q^{-1} -6143 q^{-2} -168365 q^{-3} -209024 q^{-4} -99321 q^{-5} +81117 q^{-6} +213551 q^{-7} +174522 q^{-8} +21046 q^{-9} -148142 q^{-10} -210655 q^{-11} -125587 q^{-12} +46924 q^{-13} +194108 q^{-14} +187585 q^{-15} +61072 q^{-16} -106664 q^{-17} -196275 q^{-18} -152758 q^{-19} -6737 q^{-20} +146927 q^{-21} +182468 q^{-22} +104375 q^{-23} -40544 q^{-24} -149852 q^{-25} -159220 q^{-26} -64249 q^{-27} +72105 q^{-28} +141464 q^{-29} +123658 q^{-30} +28175 q^{-31} -74816 q^{-32} -125975 q^{-33} -93435 q^{-34} -2444 q^{-35} +71507 q^{-36} +99955 q^{-37} +64147 q^{-38} -3491 q^{-39} -64028 q^{-40} -77508 q^{-41} -40748 q^{-42} +7897 q^{-43} +48882 q^{-44} +54839 q^{-45} +30361 q^{-46} -10217 q^{-47} -36791 q^{-48} -36070 q^{-49} -19814 q^{-50} +7008 q^{-51} +24391 q^{-52} +26333 q^{-53} +11319 q^{-54} -5617 q^{-55} -14610 q^{-56} -16581 q^{-57} -7703 q^{-58} +2995 q^{-59} +10522 q^{-60} +9183 q^{-61} +4074 q^{-62} -1070 q^{-63} -5984 q^{-64} -5680 q^{-65} -2630 q^{-66} +1543 q^{-67} +2850 q^{-68} +2744 q^{-69} +1756 q^{-70} -741 q^{-71} -1724 q^{-72} -1606 q^{-73} -351 q^{-74} +186 q^{-75} +668 q^{-76} +952 q^{-77} +214 q^{-78} -212 q^{-79} -446 q^{-80} -185 q^{-81} -167 q^{-82} +8 q^{-83} +274 q^{-84} +117 q^{-85} +29 q^{-86} -75 q^{-87} -13 q^{-88} -72 q^{-89} -49 q^{-90} +53 q^{-91} +22 q^{-92} +18 q^{-93} -12 q^{-94} +14 q^{-95} -11 q^{-96} -18 q^{-97} +9 q^{-98} +3 q^{-100} -3 q^{-101} +3 q^{-102} + q^{-103} -3 q^{-104} + q^{-105} </math> |
+coloured_jones_7 = <math>-q^{140}+3 q^{139}-2 q^{138}-3 q^{137}+6 q^{136}-q^{135}-2 q^{134}-8 q^{133}-q^{132}+24 q^{131}-5 q^{130}-12 q^{129}+9 q^{128}-13 q^{127}-10 q^{126}-28 q^{125}+107 q^{123}+46 q^{122}-22 q^{121}-43 q^{120}-146 q^{119}-98 q^{118}-102 q^{117}+34 q^{116}+439 q^{115}+423 q^{114}+214 q^{113}-160 q^{112}-761 q^{111}-853 q^{110}-796 q^{109}-210 q^{108}+1276 q^{107}+2062 q^{106}+2114 q^{105}+966 q^{104}-1551 q^{103}-3331 q^{102}-4503 q^{101}-3647 q^{100}+359 q^{99}+4648 q^{98}+8226 q^{97}+8480 q^{96}+3476 q^{95}-3482 q^{94}-11393 q^{93}-15909 q^{92}-12223 q^{91}-2920 q^{90}+11191 q^{89}+23390 q^{88}+25430 q^{87}+17552 q^{86}-1768 q^{85}-24994 q^{84}-39645 q^{83}-41137 q^{82}-22049 q^{81}+12339 q^{80}+45867 q^{79}+67945 q^{78}+61275 q^{77}+23695 q^{76}-30157 q^{75}-85168 q^{74}-109777 q^{73}-86340 q^{72}-20184 q^{71}+73631 q^{70}+149589 q^{69}+167802 q^{68}+112693 q^{67}-13360 q^{66}-156006 q^{65}-248112 q^{64}-240273 q^{63}-106294 q^{62}+102642 q^{61}+295626 q^{60}+380223 q^{59}+282191 q^{58}+28014 q^{57}-278825 q^{56}-498532 q^{55}-491176 q^{54}-234937 q^{53}+174113 q^{52}+557006 q^{51}+697131 q^{50}+498890 q^{49}+23069 q^{48}-528379 q^{47}-859848 q^{46}-783000 q^{45}-295636 q^{44}+401905 q^{43}+946025 q^{42}+1045961 q^{41}+611059 q^{40}-189095 q^{39}-940980 q^{38}-1251817 q^{37}-927406 q^{36}-81393 q^{35}+847907 q^{34}+1379085 q^{33}+1208434 q^{32}+372926 q^{31}-688799 q^{30}-1424863 q^{29}-1428063 q^{28}-649734 q^{27}+493157 q^{26}+1401264 q^{25}+1577280 q^{24}+885628 q^{23}-292144 q^{22}-1329981 q^{21}-1660056 q^{20}-1067682 q^{19}+109185 q^{18}+1235684 q^{17}+1691414 q^{16}+1194554 q^{15}+40634 q^{14}-1138216 q^{13}-1688281 q^{12}-1275815 q^{11}-155218 q^{10}+1052033 q^9+1670034 q^8+1324510 q^7+237205 q^6-982038 q^5-1647416 q^4-1356453 q^3-300176 q^2+926937 q+1631196+1384848 q^{-1} +354411 q^{-2} -878768 q^{-3} -1619062 q^{-4} -1419481 q^{-5} -417387 q^{-6} +825039 q^{-7} +1609342 q^{-8} +1464942 q^{-9} +497790 q^{-10} -751578 q^{-11} -1587939 q^{-12} -1517934 q^{-13} -605726 q^{-14} +643232 q^{-15} +1541628 q^{-16} +1569085 q^{-17} +738971 q^{-18} -490362 q^{-19} -1451106 q^{-20} -1600154 q^{-21} -889095 q^{-22} +289369 q^{-23} +1302550 q^{-24} +1589981 q^{-25} +1035402 q^{-26} -49707 q^{-27} -1088279 q^{-28} -1516805 q^{-29} -1151186 q^{-30} -207003 q^{-31} +814744 q^{-32} +1366862 q^{-33} +1206888 q^{-34} +448642 q^{-35} -502674 q^{-36} -1140697 q^{-37} -1180870 q^{-38} -638782 q^{-39} +187587 q^{-40} +855626 q^{-41} +1064750 q^{-42} +747466 q^{-43} +90940 q^{-44} -545478 q^{-45} -871645 q^{-46} -759540 q^{-47} -295978 q^{-48} +252223 q^{-49} +630712 q^{-50} +680531 q^{-51} +407382 q^{-52} -14127 q^{-53} -382931 q^{-54} -536608 q^{-55} -424989 q^{-56} -143221 q^{-57} +167515 q^{-58} +364088 q^{-59} +368655 q^{-60} +215994 q^{-61} -10817 q^{-62} -201155 q^{-63} -271514 q^{-64} -217968 q^{-65} -77732 q^{-66} +74444 q^{-67} +166205 q^{-68} +175493 q^{-69} +107793 q^{-70} +5076 q^{-71} -78580 q^{-72} -116915 q^{-73} -98125 q^{-74} -40710 q^{-75} +19953 q^{-76} +62796 q^{-77} +70393 q^{-78} +46335 q^{-79} +10071 q^{-80} -24744 q^{-81} -41254 q^{-82} -36242 q^{-83} -19084 q^{-84} +3547 q^{-85} +19079 q^{-86} +22466 q^{-87} +17185 q^{-88} +4852 q^{-89} -6219 q^{-90} -11397 q^{-91} -11531 q^{-92} -5868 q^{-93} +431 q^{-94} +4384 q^{-95} +6340 q^{-96} +4485 q^{-97} +1356 q^{-98} -1230 q^{-99} -3135 q^{-100} -2534 q^{-101} -1169 q^{-102} -75 q^{-103} +1210 q^{-104} +1319 q^{-105} +912 q^{-106} +321 q^{-107} -593 q^{-108} -618 q^{-109} -388 q^{-110} -264 q^{-111} +134 q^{-112} +228 q^{-113} +277 q^{-114} +241 q^{-115} -99 q^{-116} -138 q^{-117} -84 q^{-118} -96 q^{-119} +5 q^{-121} +52 q^{-122} +97 q^{-123} -7 q^{-124} -24 q^{-125} -12 q^{-126} -22 q^{-127} +2 q^{-128} -12 q^{-129} +22 q^{-131} + q^{-132} -4 q^{-133} -3 q^{-135} +3 q^{-136} -3 q^{-137} - q^{-138} +3 q^{-139} - q^{-140} </math> |
+computer_talk =
+         <table>
+         <tr valign=top>
+         <td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
+         <td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
+         </tr>
+         <tr valign=top><td colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></td></tr>
+         </table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[2]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[Knot[10, 91]]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[2]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[X[6, 2, 7, 1], X[20, 6, 1, 5], X[16, 9, 17, 10], X[10, 3, 11, 4],
+  X[2, 18, 3, 17], X[14, 7, 15, 8], X[8, 15, 9, 16], X[12, 20, 13, 19],
+  X[18, 12, 19, 11], X[4, 13, 5, 14]]</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[3]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[Knot[10, 91]]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[3]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[1, -5, 4, -10, 2, -1, 6, -7, 3, -4, 9, -8, 10, -6, 7, -3, 5,
+  -9, 8, -2]</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[4]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[Knot[10, 91]]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[4]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[6, 10, 20, 14, 16, 18, 4, 8, 2, 12]</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[5]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>br = BR[Knot[10, 91]]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[5]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BR[3, {-1, -1, -1, 2, -1, 2, 2, -1, 2, 2}]</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[6]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{First[br], Crossings[br]}</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[6]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{3, 10}</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[7]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BraidIndex[Knot[10, 91]]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[7]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>3</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[8]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Show[DrawMorseLink[Knot[10, 91]]]</nowiki></code></td></tr>
+<tr align=left><td></td><td>[[Image:10_91_ML.gif]]</td></tr><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[8]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>-Graphics-</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[9]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> (#[Knot[10, 91]]&) /@ {
+                   SymmetryType, UnknottingNumber, ThreeGenus,
+                   BridgeIndex, SuperBridgeIndex, NakanishiIndex
+                  }</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[9]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Chiral, 1, 4, 3, NotAvailable, 1}</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[10]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>alex = Alexander[Knot[10, 91]][t]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[10]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>      -4   4    9    14             2      3    4
++ t   - -- + -- - -- - 14 t + 9 t  - 4 t  + t
+2   t
+           t    t</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[11]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Conway[Knot[10, 91]][z]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[11]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>       2      4      6    8
++ 2 z  + 5 z  + 4 z  + z</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[12]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[12]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[10, 91]}</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[13]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{KnotDet[Knot[10, 91]], KnotSignature[Knot[10, 91]]}</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[13]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{73, 0}</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[14]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Jones[Knot[10, 91]][q]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[14]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>      -5   3    6    9    11             2      3      4    5
+- q   + -- - -- + -- - -- - 11 q + 9 q  - 6 q  + 3 q  - q
+3    2   q
+           q    q    q</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[15]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[15]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[10, 43], Knot[10, 91]}</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[16]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>A2Invariant[Knot[10, 91]][q]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[16]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>      -14    -12    2     -8    -4   4       2    4    8      10
+-1 - q    + q    - --- + q   - q   + -- + 4 q  - q  + q  - 2 q   +
+2
+                   q                 q
+14
+  q   - q</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[17]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>HOMFLYPT[Knot[10, 91]][a, z]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[17]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>                           2                        4
+2       2   5 z       2  2       4   4 z       2  4
+- -- - 2 a  + 12 z  - ---- - 5 a  z  + 13 z  - ---- - 4 a  z  +
+2                        2
+    a                    a                        a
+z     2  6    8
+z  - -- - a  z  + z
+         a</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[18]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kauffman[Knot[10, 91]][a, z]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[18]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>                                                    2      2
+2   3 z   6 z            5         2   z    9 z
++ -- + 2 a  - --- - --- - 4 a z + a  z - 19 z  + -- - ---- -
+3     a                            4     2
+    a           a                                  a     a
+3       3
+2      4  2   2 z    9 z    18 z         3      5  3       4
+a  z  + 2 a  z  - ---- + ---- + ----- + 9 a z  - 2 a  z  + 35 z  -
+3      a
+                       a      a
+4                        5       5       5
+z    16 z       2  4      4  4   z    12 z    13 z         5
+  ---- + ----- + 7 a  z  - 6 a  z  + -- - ----- - ----- - 7 a z  -
+2                          5     3       a
+   a      a                          a     a
+6                          7
+5    5  5       6   3 z    13 z       2  6      4  6   5 z
+a  z  + a  z  - 26 z  + ---- - ----- - 7 a  z  + 3 a  z  + ---- +
+2                           3
+                             a      a                           a
+8                9
+z       7      3  7      8   5 z       2  8   2 z         9
+  ---- + a z  + 4 a  z  + 9 z  + ---- + 4 a  z  + ---- + 2 a z
+   a                               2               a
+                                  a</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[19]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Vassiliev[2][Knot[10, 91]], Vassiliev[3][Knot[10, 91]]}</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[19]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{2, 0}</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[20]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kh[Knot[10, 91]][q, t]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[20]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>7           1        2       1       4       2       5       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
+5               3        3  2      5  2      5  3      7  3
+  ---- + --- + 5 q t + 6 q  t + 4 q  t  + 5 q  t  + 2 q  t  + 4 q  t  +
+q t
+  q  t
+4      9  4    11  5
+  q  t  + 2 q  t  + q   t</nowiki></code></td></tr>
+</table>
+         <table><tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[21]:=</code></td>
+<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>ColouredJones[Knot[10, 91], 2][q]</nowiki></code></td></tr>
+<tr align=left>
+<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[21]:=</code></td>
+<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>       -15    3     -13    8    15     2    30   40   7    71   64
++ q    - --- + q    + --- - --- + --- + -- - -- - -- + -- - -- -
+12    11    10    9    8    7    6    5
+             q            q     q     q     q    q    q    q    q
+113   71   60              2        3       4       5       6
+  -- + --- - -- - -- - 58 q - 73 q  + 114 q  - 31 q  - 66 q  + 72 q  -
+3     2   q
+  q    q     q
+8       9      10       11      12      13      14    15
+q  - 43 q  + 30 q  + 5 q   - 17 q   + 7 q   + 2 q   - 3 q   + q</nowiki></code></td></tr>
+</table>  }}