10 71: Difference between revisions

Revision as of 21:03, 29 August 2005

10_70

10_72

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Visit 10 71's page at Knotilus!

Visit 10 71's page at the original Knot Atlas!

10 71 Quick Notes

10 71 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₄₂₅ X₃₈₄₉ X_11,15,12,14 X_5,13,6,12 X_13,7,14,6 X_9,19,10,18 X_15,20,16,1 X_19,16,20,17 X_17,11,18,10 X₇₂₈₃
Gauss code	-1, 10, -2, 1, -4, 5, -10, 2, -6, 9, -3, 4, -5, 3, -7, 8, -9, 6, -8, 7
Dowker-Thistlethwaite code	4 8 12 2 18 14 6 20 10 16
Conway Notation	[22,21,2+]

Minimum Braid Representative:

Length is 10, width is 5.

Braid index is 5.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-6][-6]
Hyperbolic Volume	13.3852
A-Polynomial	See Data:10 71/A-polynomial

[edit Notes for 10 71's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	$1$
Topological 4 genus	$1$
Concordance genus	$3$
Rasmussen s-Invariant	0

[edit Notes for 10 71's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+7t^{2}-18t+25-18t^{-1}+7t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 77, 0 }
Jones polynomial	$-q^{5}+3q^{4}-6q^{3}+10q^{2}-12q+13-12q^{-1}+10q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+2z^{4}a^{-2}-3z^{4}-a^{4}z^{2}+4a^{2}z^{2}+4z^{2}a^{-2}-z^{2}a^{-4}-5z^{2}-a^{4}+3a^{2}+3a^{-2}-a^{-4}-3$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+3a^{2}z^{8}+3z^{8}a^{-2}+6z^{8}+4a^{3}z^{7}+8az^{7}+8z^{7}a^{-1}+4z^{7}a^{-3}+3a^{4}z^{6}+2a^{2}z^{6}+2z^{6}a^{-2}+3z^{6}a^{-4}-2z^{6}+a^{5}z^{5}-5a^{3}z^{5}-15az^{5}-15z^{5}a^{-1}-5z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}-12a^{2}z^{4}-12z^{4}a^{-2}-6z^{4}a^{-4}-12z^{4}-2a^{5}z^{3}+7az^{3}+7z^{3}a^{-1}-2z^{3}a^{-5}+4a^{4}z^{2}+10a^{2}z^{2}+10z^{2}a^{-2}+4z^{2}a^{-4}+12z^{2}+a^{5}z+a^{3}z-az-za^{-1}+za^{-3}+za^{-5}-a^{4}-3a^{2}-3a^{-2}-a^{-4}-3$
The A2 invariant	$-q^{16}+q^{12}-2q^{10}+3q^{8}+q^{6}-q^{4}+2q^{2}-3+2q^{-2}-q^{-4}+q^{-6}+3q^{-8}-2q^{-10}+q^{-12}-q^{-16}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+8q^{72}-7q^{70}-2q^{68}+16q^{66}-32q^{64}+47q^{62}-52q^{60}+36q^{58}-3q^{56}-48q^{54}+102q^{52}-137q^{50}+132q^{48}-84q^{46}-6q^{44}+105q^{42}-179q^{40}+206q^{38}-155q^{36}+56q^{34}+61q^{32}-147q^{30}+164q^{28}-107q^{26}+10q^{24}+90q^{22}-135q^{20}+110q^{18}-14q^{16}-110q^{14}+208q^{12}-235q^{10}+166q^{8}-34q^{6}-128q^{4}+254q^{2}-299+253q^{-2}-128q^{-4}-32q^{-6}+166q^{-8}-233q^{-10}+206q^{-12}-108q^{-14}-12q^{-16}+109q^{-18}-135q^{-20}+90q^{-22}+10q^{-24}-107q^{-26}+164q^{-28}-150q^{-30}+63q^{-32}+55q^{-34}-156q^{-36}+206q^{-38}-180q^{-40}+106q^{-42}-6q^{-44}-84q^{-46}+132q^{-48}-136q^{-50}+102q^{-52}-47q^{-54}-4q^{-56}+36q^{-58}-51q^{-60}+46q^{-62}-32q^{-64}+16q^{-66}-2q^{-68}-7q^{-70}+8q^{-72}-8q^{-74}+5q^{-76}-2q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_71.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-3q^{7}+4q^{5}-2q^{3}+q+q^{-1}-2q^{-3}+4q^{-5}-3q^{-7}+2q^{-9}-q^{-11}$
2	$q^{32}-2q^{30}-q^{28}+7q^{26}-7q^{24}-7q^{22}+21q^{20}-9q^{18}-22q^{16}+30q^{14}-30q^{10}+21q^{8}+11q^{6}-20q^{4}+q^{2}+15+q^{-2}-20q^{-4}+11q^{-6}+21q^{-8}-30q^{-10}+30q^{-14}-22q^{-16}-9q^{-18}+21q^{-20}-7q^{-22}-7q^{-24}+7q^{-26}-q^{-28}-2q^{-30}+q^{-32}$
3	$-q^{63}+2q^{61}+q^{59}-3q^{57}-4q^{55}+7q^{53}+10q^{51}-15q^{49}-21q^{47}+21q^{45}+44q^{43}-23q^{41}-77q^{39}+16q^{37}+115q^{35}+8q^{33}-148q^{31}-54q^{29}+171q^{27}+99q^{25}-165q^{23}-144q^{21}+138q^{19}+175q^{17}-96q^{15}-182q^{13}+46q^{11}+168q^{9}+7q^{7}-138q^{5}-56q^{3}+101q+101q^{-1}-56q^{-3}-138q^{-5}+7q^{-7}+167q^{-9}+46q^{-11}-180q^{-13}-95q^{-15}+174q^{-17}+137q^{-19}-144q^{-21}-166q^{-23}+99q^{-25}+172q^{-27}-54q^{-29}-150q^{-31}+8q^{-33}+119q^{-35}+17q^{-37}-79q^{-39}-25q^{-41}+44q^{-43}+22q^{-45}-21q^{-47}-15q^{-49}+10q^{-51}+7q^{-53}-4q^{-55}-3q^{-57}+q^{-59}+2q^{-61}-q^{-63}$
4	$q^{104}-2q^{102}-q^{100}+3q^{98}+4q^{94}-10q^{92}-5q^{90}+16q^{88}+6q^{86}+11q^{84}-45q^{82}-35q^{80}+50q^{78}+61q^{76}+63q^{74}-128q^{72}-170q^{70}+41q^{68}+205q^{66}+292q^{64}-168q^{62}-477q^{60}-208q^{58}+306q^{56}+779q^{54}+104q^{52}-753q^{50}-794q^{48}+37q^{46}+1242q^{44}+765q^{42}-599q^{40}-1349q^{38}-638q^{36}+1194q^{34}+1370q^{32}+24q^{30}-1377q^{28}-1238q^{26}+621q^{24}+1436q^{22}+642q^{20}-871q^{18}-1332q^{16}-66q^{14}+999q^{12}+932q^{10}-205q^{8}-1027q^{6}-598q^{4}+403q^{2}+976+409q^{-2}-588q^{-4}-1021q^{-6}-209q^{-8}+915q^{-10}+983q^{-12}-62q^{-14}-1311q^{-16}-851q^{-18}+637q^{-20}+1410q^{-22}+603q^{-24}-1234q^{-26}-1355q^{-28}+44q^{-30}+1375q^{-32}+1184q^{-34}-661q^{-36}-1371q^{-38}-596q^{-40}+798q^{-42}+1277q^{-44}+35q^{-46}-841q^{-48}-795q^{-50}+113q^{-52}+829q^{-54}+344q^{-56}-223q^{-58}-520q^{-60}-192q^{-62}+306q^{-64}+235q^{-66}+53q^{-68}-179q^{-70}-143q^{-72}+57q^{-74}+65q^{-76}+55q^{-78}-33q^{-80}-46q^{-82}+10q^{-84}+6q^{-86}+16q^{-88}-5q^{-90}-10q^{-92}+4q^{-94}+3q^{-98}-q^{-100}-2q^{-102}+q^{-104}$
5	$-q^{155}+2q^{153}+q^{151}-3q^{149}-q^{143}+5q^{141}+4q^{139}-12q^{137}-9q^{135}+7q^{133}+17q^{131}+24q^{129}+2q^{127}-49q^{125}-75q^{123}-12q^{121}+101q^{119}+158q^{117}+78q^{115}-145q^{113}-337q^{111}-254q^{109}+171q^{107}+604q^{105}+589q^{103}-39q^{101}-912q^{99}-1202q^{97}-373q^{95}+1165q^{93}+2062q^{91}+1214q^{89}-1099q^{87}-3064q^{85}-2613q^{83}+473q^{81}+3942q^{79}+4493q^{77}+902q^{75}-4300q^{73}-6533q^{71}-3114q^{69}+3789q^{67}+8352q^{65}+5868q^{63}-2305q^{61}-9333q^{59}-8700q^{57}-148q^{55}+9257q^{53}+11041q^{51}+3042q^{49}-8010q^{47}-12339q^{45}-5899q^{43}+5830q^{41}+12446q^{39}+8135q^{37}-3201q^{35}-11407q^{33}-9408q^{31}+579q^{29}+9517q^{27}+9738q^{25}+1643q^{23}-7232q^{21}-9264q^{19}-3313q^{17}+4893q^{15}+8320q^{13}+4522q^{11}-2747q^{9}-7229q^{7}-5426q^{5}+827q^{3}+6187q+6247q^{-1}+985q^{-3}-5254q^{-5}-7142q^{-7}-2811q^{-9}+4318q^{-11}+8093q^{-13}+4802q^{-15}-3177q^{-17}-8965q^{-19}-6968q^{-21}+1664q^{-23}+9472q^{-25}+9135q^{-27}+357q^{-29}-9294q^{-31}-11025q^{-33}-2813q^{-35}+8255q^{-37}+12209q^{-39}+5389q^{-41}-6250q^{-43}-12367q^{-45}-7686q^{-47}+3530q^{-49}+11371q^{-51}+9196q^{-53}-591q^{-55}-9250q^{-57}-9606q^{-59}-2079q^{-61}+6469q^{-63}+8899q^{-65}+3873q^{-67}-3574q^{-69}-7197q^{-71}-4670q^{-73}+1100q^{-75}+5092q^{-77}+4462q^{-79}+541q^{-81}-3006q^{-83}-3577q^{-85}-1341q^{-87}+1376q^{-89}+2443q^{-91}+1448q^{-93}-362q^{-95}-1411q^{-97}-1141q^{-99}-124q^{-101}+664q^{-103}+741q^{-105}+256q^{-107}-257q^{-109}-396q^{-111}-201q^{-113}+61q^{-115}+174q^{-117}+125q^{-119}+q^{-121}-75q^{-123}-56q^{-125}-3q^{-127}+23q^{-129}+18q^{-131}+8q^{-133}-9q^{-135}-12q^{-137}+4q^{-139}+5q^{-141}-q^{-143}-3q^{-149}+q^{-151}+2q^{-153}-q^{-155}$

A2 Invariants.

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

A3 Invariants.

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

B2 Invariants.

Weight

Invariant

0,1

-q^{34}+2q^{32}-5q^{30}+8q^{28}-13q^{26}+18q^{24}-23q^{22}+27q^{20}-27q^{18}+25q^{16}-17q^{14}+9q^{12}+5q^{10}-18q^{8}+32q^{6}-43q^{4}+50q^{2}-54+50q^{-2}-43q^{-4}+32q^{-6}-18q^{-8}+5q^{-10}+9q^{-12}-17q^{-14}+25q^{-16}-27q^{-18}+27q^{-20}-23q^{-22}+18q^{-24}-13q^{-26}+8q^{-28}-5q^{-30}+2q^{-32}-q^{-34}

1,0

q^{56}-2q^{52}-2q^{50}+3q^{48}+6q^{46}-q^{44}-11q^{42}-7q^{40}+11q^{38}+17q^{36}-4q^{34}-25q^{32}-10q^{30}+23q^{28}+24q^{26}-12q^{24}-29q^{22}-2q^{20}+28q^{18}+13q^{16}-19q^{14}-17q^{12}+12q^{10}+18q^{8}-6q^{6}-18q^{4}+2q^{2}+19+2q^{-2}-18q^{-4}-6q^{-6}+18q^{-8}+12q^{-10}-17q^{-12}-19q^{-14}+13q^{-16}+28q^{-18}-2q^{-20}-29q^{-22}-12q^{-24}+24q^{-26}+23q^{-28}-10q^{-30}-25q^{-32}-4q^{-34}+17q^{-36}+11q^{-38}-7q^{-40}-11q^{-42}-q^{-44}+6q^{-46}+3q^{-48}-2q^{-50}-2q^{-52}+q^{-56}

G2 Invariants.

Weight

Invariant

1,0

q^{80}-2q^{78}+5q^{76}-8q^{74}+8q^{72}-7q^{70}-2q^{68}+16q^{66}-32q^{64}+47q^{62}-52q^{60}+36q^{58}-3q^{56}-48q^{54}+102q^{52}-137q^{50}+132q^{48}-84q^{46}-6q^{44}+105q^{42}-179q^{40}+206q^{38}-155q^{36}+56q^{34}+61q^{32}-147q^{30}+164q^{28}-107q^{26}+10q^{24}+90q^{22}-135q^{20}+110q^{18}-14q^{16}-110q^{14}+208q^{12}-235q^{10}+166q^{8}-34q^{6}-128q^{4}+254q^{2}-299+253q^{-2}-128q^{-4}-32q^{-6}+166q^{-8}-233q^{-10}+206q^{-12}-108q^{-14}-12q^{-16}+109q^{-18}-135q^{-20}+90q^{-22}+10q^{-24}-107q^{-26}+164q^{-28}-150q^{-30}+63q^{-32}+55q^{-34}-156q^{-36}+206q^{-38}-180q^{-40}+106q^{-42}-6q^{-44}-84q^{-46}+132q^{-48}-136q^{-50}+102q^{-52}-47q^{-54}-4q^{-56}+36q^{-58}-51q^{-60}+46q^{-62}-32q^{-64}+16q^{-66}-2q^{-68}-7q^{-70}+8q^{-72}-8q^{-74}+5q^{-76}-2q^{-78}+q^{-80}

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["10 71"];

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

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

Out[4]= $-t^{3}+7t^{2}-18t+25-18t^{-1}+7t^{-2}-t^{-3}$

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

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

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= $\{1\}$

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

Out[7]= { 77, 0 }

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

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

Out[8]= $-q^{5}+3q^{4}-6q^{3}+10q^{2}-12q+13-12q^{-1}+10q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$

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

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

Out[9]= $-z^{6}+2a^{2}z^{4}+2z^{4}a^{-2}-3z^{4}-a^{4}z^{2}+4a^{2}z^{2}+4z^{2}a^{-2}-z^{2}a^{-4}-5z^{2}-a^{4}+3a^{2}+3a^{-2}-a^{-4}-3$

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

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

Out[10]= $az^{9}+z^{9}a^{-1}+3a^{2}z^{8}+3z^{8}a^{-2}+6z^{8}+4a^{3}z^{7}+8az^{7}+8z^{7}a^{-1}+4z^{7}a^{-3}+3a^{4}z^{6}+2a^{2}z^{6}+2z^{6}a^{-2}+3z^{6}a^{-4}-2z^{6}+a^{5}z^{5}-5a^{3}z^{5}-15az^{5}-15z^{5}a^{-1}-5z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}-12a^{2}z^{4}-12z^{4}a^{-2}-6z^{4}a^{-4}-12z^{4}-2a^{5}z^{3}+7az^{3}+7z^{3}a^{-1}-2z^{3}a^{-5}+4a^{4}z^{2}+10a^{2}z^{2}+10z^{2}a^{-2}+4z^{2}a^{-4}+12z^{2}+a^{5}z+a^{3}z-az-za^{-1}+za^{-3}+za^{-5}-a^{4}-3a^{2}-3a^{-2}-a^{-4}-3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n156, K11n179, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {10_104, ...}

Vassiliev invariants

V₂ and V₃:

(1, 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

4

0

8

{\frac {62}{3}}

-{\frac {14}{3}}

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 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 \frac{32}{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 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 \frac{248}{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{56}{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{3631}{30}}

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{326}{5}}

-{\frac {2578}{45}}

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{209}{18}}

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{1169}{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 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 \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

r

). The squares with yellow highlighting are those on the "critical diagonals", where 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-2r=s+1} or 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-2r=s-1} , where 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 s=} 0 is the signature of 10 71. 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

6

2

4

3

6

4

-2

1

7

6

1

-1

6

7

1

-3

4

6

-2

-5

2

6

4

-7

1

4

-3

-9

2

-11

1

-1

Integral Khovanov Homology

(db, data source)

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 \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	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 i=-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 i=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 r=-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 {\mathbb Z}}
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=-4}	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
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=-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 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=-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 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=-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 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{7}}
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=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 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}\oplus{\mathbb Z}_2^{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 {\mathbb Z}^{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 r=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 {\mathbb Z}_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 {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the 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 n+1} -dimensional representation of 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 sl(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 n}	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_n}
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^{15}-3 q^{14}+q^{13}+9 q^{12}-17 q^{11}+q^{10}+37 q^9-47 q^8-12 q^7+89 q^6-77 q^5-42 q^4+140 q^3-87 q^2-73 q+161-73 q^{-1} -87 q^{-2} +140 q^{-3} -42 q^{-4} -77 q^{-5} +89 q^{-6} -12 q^{-7} -47 q^{-8} +37 q^{-9} + q^{-10} -17 q^{-11} +9 q^{-12} + q^{-13} -3 q^{-14} + q^{-15} }
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 -q^{30}+3 q^{29}-q^{28}-4 q^{27}-2 q^{26}+14 q^{25}+2 q^{24}-29 q^{23}-8 q^{22}+57 q^{21}+24 q^{20}-98 q^{19}-62 q^{18}+153 q^{17}+126 q^{16}-209 q^{15}-220 q^{14}+249 q^{13}+352 q^{12}-282 q^{11}-485 q^{10}+271 q^9+633 q^8-245 q^7-754 q^6+186 q^5+859 q^4-124 q^3-914 q^2+41 q+941+33 q^{-1} -914 q^{-2} -116 q^{-3} +859 q^{-4} +178 q^{-5} -753 q^{-6} -238 q^{-7} +631 q^{-8} +264 q^{-9} -482 q^{-10} -275 q^{-11} +349 q^{-12} +243 q^{-13} -218 q^{-14} -203 q^{-15} +124 q^{-16} +149 q^{-17} -62 q^{-18} -96 q^{-19} +25 q^{-20} +56 q^{-21} -8 q^{-22} -29 q^{-23} +2 q^{-24} +14 q^{-25} -2 q^{-26} -4 q^{-27} - q^{-28} +3 q^{-29} - q^{-30} }
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 q^{50}-3 q^{49}+q^{48}+4 q^{47}-3 q^{46}+5 q^{45}-17 q^{44}+6 q^{43}+25 q^{42}-13 q^{41}+9 q^{40}-73 q^{39}+19 q^{38}+113 q^{37}-3 q^{36}+q^{35}-273 q^{34}-17 q^{33}+345 q^{32}+179 q^{31}+72 q^{30}-771 q^{29}-345 q^{28}+642 q^{27}+746 q^{26}+557 q^{25}-1487 q^{24}-1253 q^{23}+596 q^{22}+1622 q^{21}+1799 q^{20}-1966 q^{19}-2647 q^{18}-179 q^{17}+2332 q^{16}+3644 q^{15}-1775 q^{14}-3978 q^{13}-1578 q^{12}+2453 q^{11}+5481 q^{10}-968 q^9-4751 q^8-3066 q^7+1993 q^6+6730 q^5+77 q^4-4819 q^3-4190 q^2+1181 q+7163+1074 q^{-1} -4252 q^{-2} -4763 q^{-3} +180 q^{-4} +6734 q^{-5} +1896 q^{-6} -3115 q^{-7} -4696 q^{-8} -885 q^{-9} +5468 q^{-10} +2357 q^{-11} -1602 q^{-12} -3902 q^{-13} -1700 q^{-14} +3609 q^{-15} +2218 q^{-16} -201 q^{-17} -2556 q^{-18} -1876 q^{-19} +1777 q^{-20} +1507 q^{-21} +549 q^{-22} -1192 q^{-23} -1399 q^{-24} +572 q^{-25} +676 q^{-26} +590 q^{-27} -335 q^{-28} -724 q^{-29} +99 q^{-30} +162 q^{-31} +321 q^{-32} -26 q^{-33} -264 q^{-34} +12 q^{-35} -2 q^{-36} +110 q^{-37} +16 q^{-38} -73 q^{-39} +10 q^{-40} -13 q^{-41} +25 q^{-42} +6 q^{-43} -17 q^{-44} +5 q^{-45} -3 q^{-46} +4 q^{-47} + q^{-48} -3 q^{-49} + q^{-50} }
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^{75}+3 q^{74}-q^{73}-4 q^{72}+3 q^{71}-2 q^{69}+9 q^{68}-2 q^{67}-20 q^{66}+6 q^{65}+17 q^{64}+8 q^{63}+14 q^{62}-28 q^{61}-73 q^{60}-13 q^{59}+93 q^{58}+132 q^{57}+63 q^{56}-141 q^{55}-335 q^{54}-208 q^{53}+232 q^{52}+645 q^{51}+548 q^{50}-218 q^{49}-1123 q^{48}-1225 q^{47}-38 q^{46}+1694 q^{45}+2358 q^{44}+777 q^{43}-2190 q^{42}-3942 q^{41}-2274 q^{40}+2265 q^{39}+5905 q^{38}+4698 q^{37}-1560 q^{36}-7934 q^{35}-8044 q^{34}-262 q^{33}+9528 q^{32}+12145 q^{31}+3466 q^{30}-10364 q^{29}-16592 q^{28}-7789 q^{27}+9884 q^{26}+20804 q^{25}+13253 q^{24}-8189 q^{23}-24433 q^{22}-19005 q^{21}+5203 q^{20}+26921 q^{19}+24892 q^{18}-1369 q^{17}-28387 q^{16}-30073 q^{15}-3009 q^{14}+28652 q^{13}+34543 q^{12}+7409 q^{11}-28050 q^{10}-37881 q^9-11641 q^8+26655 q^7+40331 q^6+15388 q^5-24759 q^4-41656 q^3-18770 q^2+22324 q+42219+21627 q^{-1} -19497 q^{-2} -41716 q^{-3} -24130 q^{-4} +16071 q^{-5} +40416 q^{-6} +26109 q^{-7} -12228 q^{-8} -37918 q^{-9} -27557 q^{-10} +7865 q^{-11} +34465 q^{-12} +28141 q^{-13} -3353 q^{-14} -29823 q^{-15} -27778 q^{-16} -1073 q^{-17} +24478 q^{-18} +26142 q^{-19} +4853 q^{-20} -18487 q^{-21} -23467 q^{-22} -7689 q^{-23} +12749 q^{-24} +19702 q^{-25} +9182 q^{-26} -7435 q^{-27} -15468 q^{-28} -9473 q^{-29} +3344 q^{-30} +11150 q^{-31} +8549 q^{-32} -407 q^{-33} -7295 q^{-34} -6989 q^{-35} -1219 q^{-36} +4247 q^{-37} +5130 q^{-38} +1826 q^{-39} -2093 q^{-40} -3398 q^{-41} -1770 q^{-42} +778 q^{-43} +2044 q^{-44} +1375 q^{-45} -128 q^{-46} -1085 q^{-47} -922 q^{-48} -119 q^{-49} +506 q^{-50} +546 q^{-51} +162 q^{-52} -212 q^{-53} -294 q^{-54} -104 q^{-55} +73 q^{-56} +121 q^{-57} +79 q^{-58} -20 q^{-59} -71 q^{-60} -24 q^{-61} +16 q^{-62} +8 q^{-63} +16 q^{-64} +6 q^{-65} -20 q^{-66} -2 q^{-67} +9 q^{-68} -2 q^{-69} +3 q^{-71} -4 q^{-72} - q^{-73} +3 q^{-74} - q^{-75} }
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 q^{105}-3 q^{104}+q^{103}+4 q^{102}-3 q^{101}-3 q^{99}+10 q^{98}-13 q^{97}-3 q^{96}+27 q^{95}-16 q^{94}-11 q^{93}-18 q^{92}+44 q^{91}-17 q^{90}-4 q^{89}+93 q^{88}-59 q^{87}-94 q^{86}-120 q^{85}+122 q^{84}+43 q^{83}+129 q^{82}+374 q^{81}-127 q^{80}-445 q^{79}-704 q^{78}-23 q^{77}+163 q^{76}+820 q^{75}+1679 q^{74}+417 q^{73}-1100 q^{72}-2754 q^{71}-1860 q^{70}-890 q^{69}+2140 q^{68}+5766 q^{67}+4320 q^{66}+105 q^{65}-6429 q^{64}-8228 q^{63}-7743 q^{62}+486 q^{61}+12671 q^{60}+16071 q^{59}+10267 q^{58}-6146 q^{57}-18829 q^{56}-26781 q^{55}-14288 q^{54}+14126 q^{53}+34903 q^{52}+37649 q^{51}+11689 q^{50}-22346 q^{49}-56318 q^{48}-52069 q^{47}-6912 q^{46}+46334 q^{45}+79069 q^{44}+58082 q^{43}+1062 q^{42}-78635 q^{41}-107682 q^{40}-61642 q^{39}+28514 q^{38}+113841 q^{37}+125521 q^{36}+61506 q^{35}-70706 q^{34}-157830 q^{33}-139732 q^{32}-27071 q^{31}+118633 q^{30}+188775 q^{29}+145658 q^{28}-25843 q^{27}-179362 q^{26}-214439 q^{25}-105180 q^{24}+88395 q^{23}+225359 q^{22}+226146 q^{21}+40067 q^{20}-168064 q^{19}-264145 q^{18}-179823 q^{17}+38263 q^{16}+231417 q^{15}+283013 q^{14}+103990 q^{13}-136883 q^{12}-285072 q^{11}-234321 q^{10}-12878 q^9+217193 q^8+313167 q^7+153420 q^6-99859 q^5-284929 q^4-267400 q^3-56989 q^2+192187 q+322667+189245 q^{-1} -60743 q^{-2} -269454 q^{-3} -284321 q^{-4} -96924 q^{-5} +156541 q^{-6} +314357 q^{-7} +215957 q^{-8} -15153 q^{-9} -235908 q^{-10} -285368 q^{-11} -135952 q^{-12} +104928 q^{-13} +283046 q^{-14} +230811 q^{-15} +38316 q^{-16} -178507 q^{-17} -262541 q^{-18} -167495 q^{-19} +38345 q^{-20} +222561 q^{-21} +222168 q^{-22} +88504 q^{-23} -100992 q^{-24} -208570 q^{-25} -175561 q^{-26} -27261 q^{-27} +139141 q^{-28} +180916 q^{-29} +114899 q^{-30} -23730 q^{-31} -131093 q^{-32} -149140 q^{-33} -67729 q^{-34} +56486 q^{-35} +115371 q^{-36} +105182 q^{-37} +26906 q^{-38} -55236 q^{-39} -97296 q^{-40} -70861 q^{-41} +1388 q^{-42} +50932 q^{-43} +69344 q^{-44} +40128 q^{-45} -6076 q^{-46} -45190 q^{-47} -48017 q^{-48} -17786 q^{-49} +10095 q^{-50} +31626 q^{-51} +28508 q^{-52} +11165 q^{-53} -12553 q^{-54} -22005 q^{-55} -14284 q^{-56} -4461 q^{-57} +8682 q^{-58} +12688 q^{-59} +9763 q^{-60} -403 q^{-61} -6499 q^{-62} -6038 q^{-63} -4750 q^{-64} +488 q^{-65} +3493 q^{-66} +4503 q^{-67} +1237 q^{-68} -1008 q^{-69} -1368 q^{-70} -2102 q^{-71} -677 q^{-72} +440 q^{-73} +1427 q^{-74} +509 q^{-75} -6 q^{-76} -17 q^{-77} -589 q^{-78} -320 q^{-79} -67 q^{-80} +366 q^{-81} +81 q^{-82} +3 q^{-83} +104 q^{-84} -114 q^{-85} -79 q^{-86} -49 q^{-87} +97 q^{-88} -7 q^{-89} -20 q^{-90} +42 q^{-91} -18 q^{-92} -10 q^{-93} -16 q^{-94} +27 q^{-95} -3 q^{-96} -13 q^{-97} +10 q^{-98} -3 q^{-99} -3 q^{-101} +4 q^{-102} + q^{-103} -3 q^{-104} + q^{-105} }
7	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^{140}+3 q^{139}-q^{138}-4 q^{137}+3 q^{136}+3 q^{134}-5 q^{133}-6 q^{132}+18 q^{131}-4 q^{130}-17 q^{129}+10 q^{128}+5 q^{127}+19 q^{126}-22 q^{125}-49 q^{124}+46 q^{123}-q^{122}-25 q^{121}+50 q^{120}+35 q^{119}+95 q^{118}-67 q^{117}-237 q^{116}-40 q^{115}-72 q^{114}+13 q^{113}+305 q^{112}+327 q^{111}+499 q^{110}-6 q^{109}-834 q^{108}-835 q^{107}-981 q^{106}-329 q^{105}+1035 q^{104}+1848 q^{103}+2739 q^{102}+1559 q^{101}-1414 q^{100}-3589 q^{99}-5583 q^{98}-4419 q^{97}+249 q^{96}+5324 q^{95}+10843 q^{94}+10858 q^{93}+3756 q^{92}-6224 q^{91}-18002 q^{90}-22048 q^{89}-13892 q^{88}+2682 q^{87}+25611 q^{86}+39508 q^{85}+33822 q^{84}+10089 q^{83}-29584 q^{82}-61860 q^{81}-66529 q^{80}-38785 q^{79}+22063 q^{78}+84042 q^{77}+112851 q^{76}+90294 q^{75}+7196 q^{74}-96800 q^{73}-168206 q^{72}-167942 q^{71}-69313 q^{70}+85394 q^{69}+221631 q^{68}+270081 q^{67}+172974 q^{66}-34028 q^{65}-256579 q^{64}-386292 q^{63}-319674 q^{62}-71005 q^{61}+252161 q^{60}+498162 q^{59}+502190 q^{58}+237245 q^{57}-189446 q^{56}-583385 q^{55}-703245 q^{54}-460322 q^{53}+56550 q^{52}+617416 q^{51}+897896 q^{50}+726787 q^{49}+149273 q^{48}-583820 q^{47}-1061619 q^{46}-1012012 q^{45}-415748 q^{44}+474654 q^{43}+1169976 q^{42}+1289213 q^{41}+724519 q^{40}-294998 q^{39}-1212370 q^{38}-1533060 q^{37}-1046266 q^{36}+60784 q^{35}+1184229 q^{34}+1724267 q^{33}+1357122 q^{32}+206118 q^{31}-1096577 q^{30}-1854829 q^{29}-1633207 q^{28}-480434 q^{27}+963299 q^{26}+1925041 q^{25}+1862512 q^{24}+741680 q^{23}-804916 q^{22}-1944029 q^{21}-2038995 q^{20}-974677 q^{19}+638143 q^{18}+1924202 q^{17}+2166570 q^{16}+1172133 q^{15}-477017 q^{14}-1878611 q^{13}-2252020 q^{12}-1334234 q^{11}+327516 q^{10}+1818177 q^9+2306060 q^8+1465819 q^7-191775 q^6-1748494 q^5-2335919 q^4-1575157 q^3+63986 q^2+1670691 q+2348802+1669944 q^{-1} +61475 q^{-2} -1580758 q^{-3} -2343857 q^{-4} -1755851 q^{-5} -194546 q^{-6} +1471175 q^{-7} +2319234 q^{-8} +1834551 q^{-9} +339718 q^{-10} -1333529 q^{-11} -2265344 q^{-12} -1901929 q^{-13} -500698 q^{-14} +1159908 q^{-15} +2173423 q^{-16} +1949453 q^{-17} +672252 q^{-18} -947289 q^{-19} -2032365 q^{-20} -1963836 q^{-21} -844916 q^{-22} +698683 q^{-23} +1836854 q^{-24} +1930889 q^{-25} +1001339 q^{-26} -425443 q^{-27} -1586172 q^{-28} -1839147 q^{-29} -1122928 q^{-30} +146973 q^{-31} +1291084 q^{-32} +1682678 q^{-33} +1189349 q^{-34} +113341 q^{-35} -968596 q^{-36} -1466308 q^{-37} -1189743 q^{-38} -329709 q^{-39} +646595 q^{-40} +1203823 q^{-41} +1118353 q^{-42} +482769 q^{-43} -350263 q^{-44} -919307 q^{-45} -986055 q^{-46} -561821 q^{-47} +106912 q^{-48} +640217 q^{-49} +808886 q^{-50} +568347 q^{-51} +70019 q^{-52} -391940 q^{-53} -613950 q^{-54} -516053 q^{-55} -174862 q^{-56} +194438 q^{-57} +425070 q^{-58} +424638 q^{-59} +216201 q^{-60} -55025 q^{-61} -263003 q^{-62} -318303 q^{-63} -209546 q^{-64} -27811 q^{-65} +139438 q^{-66} +216128 q^{-67} +173949 q^{-68} +65113 q^{-69} -56221 q^{-70} -131497 q^{-71} -127792 q^{-72} -71267 q^{-73} +8490 q^{-74} +70271 q^{-75} +83625 q^{-76} +60143 q^{-77} +13260 q^{-78} -31096 q^{-79} -48559 q^{-80} -43446 q^{-81} -18945 q^{-82} +9772 q^{-83} +24774 q^{-84} +27496 q^{-85} +16498 q^{-86} -164 q^{-87} -10549 q^{-88} -15397 q^{-89} -11692 q^{-90} -2922 q^{-91} +3329 q^{-92} +7818 q^{-93} +7170 q^{-94} +2757 q^{-95} -364 q^{-96} -3350 q^{-97} -3768 q^{-98} -2015 q^{-99} -706 q^{-100} +1365 q^{-101} +1990 q^{-102} +1035 q^{-103} +541 q^{-104} -395 q^{-105} -735 q^{-106} -483 q^{-107} -569 q^{-108} +88 q^{-109} +442 q^{-110} +196 q^{-111} +192 q^{-112} -44 q^{-113} -70 q^{-114} - q^{-115} -195 q^{-116} -42 q^{-117} +98 q^{-118} +25 q^{-119} +39 q^{-120} -32 q^{-121} -4 q^{-122} +48 q^{-123} -46 q^{-124} -20 q^{-125} +19 q^{-126} +4 q^{-127} +10 q^{-128} -17 q^{-129} -4 q^{-130} +18 q^{-131} -6 q^{-132} -5 q^{-133} +3 q^{-134} +3 q^{-136} -4 q^{-137} - q^{-138} +3 q^{-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, 71]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[11, 15, 12, 14], X[5, 13, 6, 12], X[13, 7, 14, 6], X[9, 19, 10, 18], X[15, 20, 16, 1], X[19, 16, 20, 17], X[17, 11, 18, 10], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 71]]
Out[3]=	GaussCode[-1, 10, -2, 1, -4, 5, -10, 2, -6, 9, -3, 4, -5, 3, -7, 8, -9, 6, -8, 7]
In[4]:=	DTCode[Knot[10, 71]]
Out[4]=	DTCode[4, 8, 12, 2, 18, 14, 6, 20, 10, 16]
In[5]:=	br = BR[Knot[10, 71]]
Out[5]=	BR[5, {-1, -1, 2, -1, -3, 2, 2, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 10}
In[7]:=	BraidIndex[Knot[10, 71]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 71]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[10, 71]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 71]][t]
Out[10]=	-3 7 18 2 3 25 - t + -- - -- - 18 t + 7 t - t 2 t t
In[11]:=	Conway[Knot[10, 71]][z]
Out[11]=	2 4 6 1 + z + z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 71], Knot[11, NonAlternating, 156], Knot[11, NonAlternating, 179]}
In[13]:=	{KnotDet[Knot[10, 71]], KnotSignature[Knot[10, 71]]}
Out[13]=	{77, 0}
In[14]:=	Jones[Knot[10, 71]][q]
Out[14]=	-5 3 6 10 12 2 3 4 5 13 - q + -- - -- + -- - -- - 12 q + 10 q - 6 q + 3 q - q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 71], Knot[10, 104]}
In[16]:=	A2Invariant[Knot[10, 71]][q]
Out[16]=	-16 -12 2 3 -6 -4 2 2 4 6 8 -3 - q + q - --- + -- + q - q + -- + 2 q - q + q + 3 q - 10 8 2 q q q 10 12 16 2 q + q - q
In[17]:=	HOMFLYPT[Knot[10, 71]][a, z]
Out[17]=	2 2 -4 3 2 4 2 z 4 z 2 2 4 2 4 -3 - a + -- + 3 a - a - 5 z - -- + ---- + 4 a z - a z - 3 z + 2 4 2 a a a 4 2 z 2 4 6 ---- + 2 a z - z 2 a
In[18]:=	Kauffman[Knot[10, 71]][a, z]
Out[18]=	-4 3 2 4 z z z 3 5 2 -3 - a - -- - 3 a - a + -- + -- - - - a z + a z + a z + 12 z + 2 5 3 a a a a 2 2 3 3 4 z 10 z 2 2 4 2 2 z 7 z 3 5 3 ---- + ----- + 10 a z + 4 a z - ---- + ---- + 7 a z - 2 a z - 4 2 5 a a a a 4 4 5 5 5 4 6 z 12 z 2 4 4 4 z 5 z 15 z 12 z - ---- - ----- - 12 a z - 6 a z + -- - ---- - ----- - 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 3 z 2 z 2 6 4 6 15 a z - 5 a z + a z - 2 z + ---- + ---- + 2 a z + 3 a z + 4 2 a a 7 7 8 9 4 z 8 z 7 3 7 8 3 z 2 8 z 9 ---- + ---- + 8 a z + 4 a z + 6 z + ---- + 3 a z + -- + a z 3 a 2 a a a
In[19]:=	{Vassiliev[2][Knot[10, 71]], Vassiliev[3][Knot[10, 71]]}
Out[19]=	{1, 0}
In[20]:=	Kh[Knot[10, 71]][q, t]
Out[20]=	7 1 2 1 4 2 6 4 - + 7 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 6 6 3 3 2 5 2 5 3 7 3 ---- + --- + 6 q t + 6 q t + 4 q t + 6 q t + 2 q t + 4 q t + 3 q t q t 7 4 9 4 11 5 q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 71], 2][q]
Out[21]=	-15 3 -13 9 17 -10 37 47 12 89 77 161 + q - --- + q + --- - --- + q + -- - -- - -- + -- - -- - 14 12 11 9 8 7 6 5 q q q q q q q q 42 140 87 73 2 3 4 5 6 -- + --- - -- - -- - 73 q - 87 q + 140 q - 42 q - 77 q + 89 q - 4 3 2 q q q q 7 8 9 10 11 12 13 14 15 12 q - 47 q + 37 q + q - 17 q + 9 q + q - 3 q + q

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10 71: Difference between revisions

Revision as of 21:03, 29 August 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

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