9 28: Difference between revisions

Revision as of 20:14, 28 August 2005

9_27

9_29

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9 28 Quick Notes

9 28 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_15,18,16,1 X_9,16,10,17 X_17,10,18,11 X₇₂₈₃
Gauss code	-1, 9, -2, 1, -4, 5, -9, 2, -7, 8, -3, 4, -5, 3, -6, 7, -8, 6
Dowker-Thistlethwaite code	4 8 12 2 16 14 6 18 10
Conway Notation	[21,21,2+]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	3
Bridge index	3
Super bridge index	$\{4,6\}$
Nakanishi index	1
Maximal Thurston-Bennequin number	[-9][-2]
Hyperbolic Volume	11.5632
A-Polynomial	See Data:9 28/A-polynomial

[edit Notes for 9 28's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 28's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-5t^{2}+12t-15+12t^{-1}-5t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 51, -2 }
Jones polynomial	$-q^{2}+3q-5+8q^{-1}-8q^{-2}+9q^{-3}-8q^{-4}+5q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}+a^{6}-2z^{4}a^{4}-5z^{2}a^{4}-4a^{4}+z^{6}a^{2}+4z^{4}a^{2}+7z^{2}a^{2}+5a^{2}-z^{4}-2z^{2}-1$
Kauffman polynomial (db, data sources)	$z^{4}a^{8}-z^{2}a^{8}+3z^{5}a^{7}-4z^{3}a^{7}+2za^{7}+4z^{6}a^{6}-4z^{4}a^{6}+2z^{2}a^{6}-a^{6}+3z^{7}a^{5}+2z^{5}a^{5}-9z^{3}a^{5}+6za^{5}+z^{8}a^{4}+8z^{6}a^{4}-17z^{4}a^{4}+12z^{2}a^{4}-4a^{4}+6z^{7}a^{3}-5z^{5}a^{3}-7z^{3}a^{3}+6za^{3}+z^{8}a^{2}+7z^{6}a^{2}-19z^{4}a^{2}+14z^{2}a^{2}-5a^{2}+3z^{7}a-3z^{5}a-4z^{3}a+3za+3z^{6}-7z^{4}+5z^{2}-1+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$
The A2 invariant	$q^{22}-q^{18}+q^{16}-3q^{14}-q^{12}+4q^{6}+3q^{2}-q^{-2}+q^{-4}-q^{-6}$
The G2 invariant	$q^{114}-2q^{112}+4q^{110}-6q^{108}+4q^{106}-3q^{104}-4q^{102}+13q^{100}-20q^{98}+27q^{96}-25q^{94}+15q^{92}+6q^{90}-29q^{88}+54q^{86}-62q^{84}+53q^{82}-29q^{80}-10q^{78}+49q^{76}-72q^{74}+73q^{72}-44q^{70}+3q^{68}+32q^{66}-53q^{64}+37q^{62}-7q^{60}-33q^{58}+56q^{56}-58q^{54}+23q^{52}+30q^{50}-81q^{48}+105q^{46}-99q^{44}+53q^{42}+8q^{40}-67q^{38}+103q^{36}-103q^{34}+79q^{32}-24q^{30}-28q^{28}+63q^{26}-64q^{24}+43q^{22}-34q^{18}+52q^{16}-35q^{14}+5q^{12}+41q^{10}-70q^{8}+77q^{6}-52q^{4}+5q^{2}+39-70q^{-2}+76q^{-4}-56q^{-6}+24q^{-8}+8q^{-10}-33q^{-12}+39q^{-14}-32q^{-16}+19q^{-18}-5q^{-20}-5q^{-22}+7q^{-24}-8q^{-26}+5q^{-28}-2q^{-30}+q^{-32}$

Further Quantum Invariants

Further quantum knot invariants for 9_28.

A1 Invariants.

Weight	Invariant
1	$q^{15}-2q^{13}+2q^{11}-3q^{9}+q^{7}+q^{5}+3q-2q^{-1}+2q^{-3}-q^{-5}$
2	$q^{42}-2q^{40}-q^{38}+6q^{36}-6q^{34}-4q^{32}+15q^{30}-8q^{28}-11q^{26}+18q^{24}-3q^{22}-13q^{20}+8q^{18}+4q^{16}-6q^{14}-5q^{12}+9q^{10}+3q^{8}-14q^{6}+9q^{4}+12q^{2}-16+3q^{-2}+13q^{-4}-10q^{-6}-3q^{-8}+7q^{-10}-2q^{-12}-2q^{-14}+q^{-16}$
3	$q^{81}-2q^{79}-q^{77}+3q^{75}+3q^{73}-6q^{71}-7q^{69}+14q^{67}+12q^{65}-20q^{63}-25q^{61}+27q^{59}+41q^{57}-31q^{55}-61q^{53}+28q^{51}+77q^{49}-13q^{47}-88q^{45}-2q^{43}+86q^{41}+18q^{39}-68q^{37}-36q^{35}+49q^{33}+41q^{31}-21q^{29}-49q^{27}-2q^{25}+46q^{23}+28q^{21}-47q^{19}-48q^{17}+42q^{15}+65q^{13}-31q^{11}-77q^{9}+18q^{7}+84q^{5}+3q^{3}-83q-17q^{-1}+68q^{-3}+35q^{-5}-51q^{-7}-39q^{-9}+31q^{-11}+37q^{-13}-12q^{-15}-28q^{-17}-q^{-19}+17q^{-21}+3q^{-23}-7q^{-25}-3q^{-27}+2q^{-29}+2q^{-31}-q^{-33}$
4	$q^{132}-2q^{130}-q^{128}+3q^{126}+3q^{122}-9q^{120}-2q^{118}+15q^{116}+2q^{114}+2q^{112}-37q^{110}-12q^{108}+52q^{106}+34q^{104}+6q^{102}-110q^{100}-67q^{98}+104q^{96}+136q^{94}+65q^{92}-218q^{90}-218q^{88}+96q^{86}+285q^{84}+234q^{82}-252q^{80}-413q^{78}-53q^{76}+339q^{74}+441q^{72}-118q^{70}-474q^{68}-249q^{66}+208q^{64}+488q^{62}+91q^{60}-318q^{58}-331q^{56}-12q^{54}+338q^{52}+226q^{50}-75q^{48}-277q^{46}-179q^{44}+117q^{42}+270q^{40}+139q^{38}-185q^{36}-288q^{34}-77q^{32}+285q^{30}+306q^{28}-87q^{26}-357q^{24}-255q^{22}+259q^{20}+433q^{18}+52q^{16}-347q^{14}-415q^{12}+125q^{10}+454q^{8}+228q^{6}-200q^{4}-477q^{2}-81+315q^{-2}+323q^{-4}+27q^{-6}-353q^{-8}-210q^{-10}+81q^{-12}+246q^{-14}+165q^{-16}-137q^{-18}-169q^{-20}-65q^{-22}+83q^{-24}+134q^{-26}+q^{-28}-53q^{-30}-63q^{-32}-7q^{-34}+46q^{-36}+16q^{-38}+q^{-40}-17q^{-42}-10q^{-44}+7q^{-46}+3q^{-48}+3q^{-50}-2q^{-52}-2q^{-54}+q^{-56}$
5	$q^{195}-2q^{193}-q^{191}+3q^{189}-4q^{181}-q^{179}+11q^{177}+5q^{175}-11q^{173}-19q^{171}-12q^{169}+18q^{167}+47q^{165}+39q^{163}-34q^{161}-107q^{159}-80q^{157}+47q^{155}+184q^{153}+187q^{151}-26q^{149}-320q^{147}-368q^{145}-32q^{143}+450q^{141}+644q^{139}+219q^{137}-573q^{135}-1018q^{133}-544q^{131}+601q^{129}+1424q^{127}+1036q^{125}-455q^{123}-1776q^{121}-1638q^{119}+85q^{117}+1977q^{115}+2238q^{113}+458q^{111}-1899q^{109}-2695q^{107}-1115q^{105}+1552q^{103}+2912q^{101}+1698q^{99}-994q^{97}-2790q^{95}-2114q^{93}+328q^{91}+2378q^{89}+2303q^{87}+276q^{85}-1794q^{83}-2190q^{81}-786q^{79}+1118q^{77}+1950q^{75}+1119q^{73}-514q^{71}-1563q^{69}-1322q^{67}-45q^{65}+1228q^{63}+1443q^{61}+470q^{59}-897q^{57}-1548q^{55}-862q^{53}+655q^{51}+1664q^{49}+1205q^{47}-426q^{45}-1801q^{43}-1574q^{41}+191q^{39}+1909q^{37}+1948q^{35}+121q^{33}-1950q^{31}-2308q^{29}-525q^{27}+1848q^{25}+2601q^{23}+1002q^{21}-1535q^{19}-2749q^{17}-1518q^{15}+1055q^{13}+2680q^{11}+1945q^{9}-409q^{7}-2337q^{5}-2241q^{3}-257q+1801q^{-1}+2235q^{-3}+841q^{-5}-1092q^{-7}-1992q^{-9}-1223q^{-11}+417q^{-13}+1515q^{-15}+1320q^{-17}+150q^{-19}-954q^{-21}-1167q^{-23}-488q^{-25}+436q^{-27}+862q^{-29}+581q^{-31}-62q^{-33}-507q^{-35}-498q^{-37}-146q^{-39}+229q^{-41}+337q^{-43}+181q^{-45}-51q^{-47}-172q^{-49}-146q^{-51}-28q^{-53}+74q^{-55}+83q^{-57}+33q^{-59}-18q^{-61}-34q^{-63}-25q^{-65}-q^{-67}+17q^{-69}+10q^{-71}-3q^{-75}-3q^{-77}-3q^{-79}+2q^{-81}+2q^{-83}-q^{-85}$

A2 Invariants.

Weight	Invariant
1,0	$q^{22}-q^{18}+q^{16}-3q^{14}-q^{12}+4q^{6}+3q^{2}-q^{-2}+q^{-4}-q^{-6}$
1,1	$q^{60}-4q^{58}+10q^{56}-20q^{54}+38q^{52}-66q^{50}+100q^{48}-146q^{46}+198q^{44}-246q^{42}+284q^{40}-300q^{38}+289q^{36}-230q^{34}+132q^{32}-4q^{30}-149q^{28}+306q^{26}-448q^{24}+552q^{22}-611q^{20}+608q^{18}-560q^{16}+454q^{14}-320q^{12}+166q^{10}-2q^{8}-126q^{6}+239q^{4}-296q^{2}+326-308q^{-2}+262q^{-4}-208q^{-6}+148q^{-8}-96q^{-10}+54q^{-12}-28q^{-14}+12q^{-16}-4q^{-18}+q^{-20}$
2,0	$q^{56}-2q^{52}-q^{50}+2q^{48}+q^{46}-4q^{44}+7q^{40}-q^{38}-7q^{36}+2q^{34}+8q^{32}-6q^{28}+5q^{26}+q^{24}-9q^{22}-3q^{20}+q^{18}-5q^{16}-3q^{14}+8q^{12}+3q^{10}-2q^{8}+5q^{6}+11q^{4}-2q^{2}-6+5q^{-2}+3q^{-4}-5q^{-6}-3q^{-8}+2q^{-10}+2q^{-12}-2q^{-14}-q^{-16}+q^{-18}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{48}-2q^{46}+4q^{42}-7q^{40}-q^{38}+11q^{36}-8q^{34}-q^{32}+17q^{30}-8q^{28}-5q^{26}+7q^{24}-9q^{22}-10q^{20}-3q^{18}+3q^{16}-2q^{12}+14q^{10}+9q^{8}-10q^{6}+10q^{4}+6q^{2}-13+5q^{-2}+3q^{-4}-8q^{-6}+3q^{-8}+q^{-10}-2q^{-12}+q^{-14}$
1,0,0	$q^{29}+q^{25}-q^{23}+q^{21}-4q^{19}-q^{17}-3q^{15}+q^{11}+3q^{9}+4q^{7}+q^{5}+3q^{3}-q+q^{-1}-2q^{-3}+q^{-5}-q^{-7}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{62}-q^{58}+q^{54}-3q^{52}-7q^{50}+7q^{46}-q^{42}+16q^{40}+12q^{38}-4q^{36}-3q^{34}+3q^{32}-13q^{30}-22q^{28}-8q^{26}-4q^{24}-13q^{22}+19q^{18}+4q^{16}+4q^{14}+20q^{12}+13q^{10}-6q^{8}+q^{6}+8q^{4}-3q^{2}-10+3q^{-4}-5q^{-6}-2q^{-8}+3q^{-10}-q^{-14}+q^{-16}$
1,0,0,0	$q^{36}+q^{32}+q^{30}-q^{28}+q^{26}-4q^{24}-2q^{22}-3q^{20}-3q^{18}+q^{14}+4q^{12}+3q^{10}+5q^{8}+q^{6}+3q^{4}-q^{2}-2q^{-4}+q^{-6}-q^{-8}$

B2 Invariants.

Weight

Invariant

0,1

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{48}-2 q^{46}+4 q^{44}-6 q^{42}+9 q^{40}-11 q^{38}+13 q^{36}-14 q^{34}+11 q^{32}-9 q^{30}+2 q^{28}+3 q^{26}-11 q^{24}+17 q^{22}-22 q^{20}+25 q^{18}-25 q^{16}+24 q^{14}-18 q^{12}+14 q^{10}-5 q^8+8 q^4-10 q^2+13-13 q^{-2} +13 q^{-4} -10 q^{-6} +7 q^{-8} -5 q^{-10} +2 q^{-12} - q^{-14} }

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{78}-2 q^{74}-2 q^{72}+2 q^{70}+5 q^{68}-8 q^{64}-6 q^{62}+6 q^{60}+12 q^{58}+q^{56}-13 q^{54}-6 q^{52}+11 q^{50}+14 q^{48}-4 q^{46}-13 q^{44}-q^{42}+11 q^{40}+2 q^{38}-12 q^{36}-8 q^{34}+5 q^{32}+5 q^{30}-7 q^{28}-9 q^{26}+3 q^{24}+10 q^{22}-7 q^{18}+q^{16}+14 q^{14}+7 q^{12}-9 q^{10}-10 q^8+9 q^6+15 q^4-14-7 q^{-2} +9 q^{-4} +10 q^{-6} -4 q^{-8} -9 q^{-10} -2 q^{-12} +5 q^{-14} +3 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-24} }

D4 Invariants.

Weight

Invariant

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{66}-2 q^{64}+2 q^{62}-3 q^{60}+5 q^{58}-8 q^{56}+6 q^{54}-8 q^{52}+12 q^{50}-10 q^{48}+10 q^{46}-6 q^{44}+12 q^{42}-q^{40}+q^{36}-9 q^{34}+8 q^{32}-21 q^{30}+8 q^{28}-25 q^{26}+17 q^{24}-19 q^{22}+19 q^{20}-13 q^{18}+20 q^{16}-q^{14}+13 q^{12}+q^{10}+q^8+8 q^6-7 q^4+7 q^2-12+10 q^{-2} -11 q^{-4} +8 q^{-6} -9 q^{-8} +6 q^{-10} -4 q^{-12} +3 q^{-14} -2 q^{-16} + q^{-18} }

G2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-3 q^{104}-4 q^{102}+13 q^{100}-20 q^{98}+27 q^{96}-25 q^{94}+15 q^{92}+6 q^{90}-29 q^{88}+54 q^{86}-62 q^{84}+53 q^{82}-29 q^{80}-10 q^{78}+49 q^{76}-72 q^{74}+73 q^{72}-44 q^{70}+3 q^{68}+32 q^{66}-53 q^{64}+37 q^{62}-7 q^{60}-33 q^{58}+56 q^{56}-58 q^{54}+23 q^{52}+30 q^{50}-81 q^{48}+105 q^{46}-99 q^{44}+53 q^{42}+8 q^{40}-67 q^{38}+103 q^{36}-103 q^{34}+79 q^{32}-24 q^{30}-28 q^{28}+63 q^{26}-64 q^{24}+43 q^{22}-34 q^{18}+52 q^{16}-35 q^{14}+5 q^{12}+41 q^{10}-70 q^8+77 q^6-52 q^4+5 q^2+39-70 q^{-2} +76 q^{-4} -56 q^{-6} +24 q^{-8} +8 q^{-10} -33 q^{-12} +39 q^{-14} -32 q^{-16} +19 q^{-18} -5 q^{-20} -5 q^{-22} +7 q^{-24} -8 q^{-26} +5 q^{-28} -2 q^{-30} + q^{-32} }

.

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["9 28"];

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

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

Out[4]= $t^{3}-5t^{2}+12t-15+12t^{-1}-5t^{-2}+t^{-3}$

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]= { 51, -2 }

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

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

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

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

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

Out[9]= $z^{2}a^{6}+a^{6}-2z^{4}a^{4}-5z^{2}a^{4}-4a^{4}+z^{6}a^{2}+4z^{4}a^{2}+7z^{2}a^{2}+5a^{2}-z^{4}-2z^{2}-1$

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

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

Out[10]= $z^{4}a^{8}-z^{2}a^{8}+3z^{5}a^{7}-4z^{3}a^{7}+2za^{7}+4z^{6}a^{6}-4z^{4}a^{6}+2z^{2}a^{6}-a^{6}+3z^{7}a^{5}+2z^{5}a^{5}-9z^{3}a^{5}+6za^{5}+z^{8}a^{4}+8z^{6}a^{4}-17z^{4}a^{4}+12z^{2}a^{4}-4a^{4}+6z^{7}a^{3}-5z^{5}a^{3}-7z^{3}a^{3}+6za^{3}+z^{8}a^{2}+7z^{6}a^{2}-19z^{4}a^{2}+14z^{2}a^{2}-5a^{2}+3z^{7}a-3z^{5}a-4z^{3}a+3za+3z^{6}-7z^{4}+5z^{2}-1+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$

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

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 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 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 8}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{34}{3}}

-{\frac {14}{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 32}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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{136}{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}}

-{\frac {2609}{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{622}{15}}

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{1742}{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{79}{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{271}{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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where 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=} -2 is the signature of 9 28. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

2

1

3

1

-2

-1

5

2

3

-3

4

0

-5

5

4

1

-7

3

4

1

-9

2

5

-3

-11

1

3

2

-13

2

-2

-15

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=-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 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=-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}}
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}\oplus{\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}}
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}^{3}\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=-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}^{5}\oplus{\mathbb Z}_2^{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}^{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 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^{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}^{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 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}^{4}\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=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}^{4}\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}^{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 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}^{2}\oplus{\mathbb Z}_2^{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}^{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 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}\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=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}	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}}

Computer Talk

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

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 \textrm{Include}(\textrm{ColouredJonesM.mhtml})}

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 28]]
Out[2]=	9
In[3]:=	PD[Knot[9, 28]]
Out[3]=	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[15, 18, 16, 1], X[9, 16, 10, 17], X[17, 10, 18, 11], X[7, 2, 8, 3]]
In[4]:=	GaussCode[Knot[9, 28]]
Out[4]=	GaussCode[-1, 9, -2, 1, -4, 5, -9, 2, -7, 8, -3, 4, -5, 3, -6, 7, -8, 6]
In[5]:=	BR[Knot[9, 28]]
Out[5]=	BR[4, {-1, -1, 2, -1, -3, 2, 2, -3, -3}]
In[6]:=	alex = Alexander[Knot[9, 28]][t]
Out[6]=	-3 5 12 2 3 -15 + t - -- + -- + 12 t - 5 t + t 2 t t
In[7]:=	Conway[Knot[9, 28]][z]
Out[7]=	2 4 6 1 + z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 28], Knot[9, 29], Knot[10, 163], Knot[11, NonAlternating, 87]}
In[9]:=	{KnotDet[Knot[9, 28]], KnotSignature[Knot[9, 28]]}
Out[9]=	{51, -2}
In[10]:=	J=Jones[Knot[9, 28]][q]
Out[10]=	-7 3 5 8 9 8 8 2 -5 + q - -- + -- - -- + -- - -- + - + 3 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 28]}
In[12]:=	A2Invariant[Knot[9, 28]][q]
Out[12]=	-22 -18 -16 3 -12 4 3 2 4 6 q - q + q - --- - q + -- + -- - q + q - q 14 6 2 q q q
In[13]:=	Kauffman[Knot[9, 28]][a, z]
Out[13]=	2 4 6 z 3 5 7 2 -1 - 5 a - 4 a - a + - + 3 a z + 6 a z + 6 a z + 2 a z + 5 z + a 3 2 2 4 2 6 2 8 2 2 z 3 3 3 14 a z + 12 a z + 2 a z - a z - ---- - 4 a z - 7 a z - a 5 3 7 3 4 2 4 4 4 6 4 8 4 9 a z - 4 a z - 7 z - 19 a z - 17 a z - 4 a z + a z + 5 z 5 3 5 5 5 7 5 6 2 6 -- - 3 a z - 5 a z + 2 a z + 3 a z + 3 z + 7 a z + a 4 6 6 6 7 3 7 5 7 2 8 4 8 8 a z + 4 a z + 3 a z + 6 a z + 3 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[9, 28]], Vassiliev[3][Knot[9, 28]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[9, 28]][q, t]
Out[15]=	4 5 1 2 1 3 2 5 3 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t 4 5 4 4 2 t 2 3 2 5 3 ----- + ----- + ---- + ---- + --- + 3 q t + q t + 2 q t + q t 7 2 5 2 5 3 q q t q t q t q t

@@ Line 1: / Line 1: @@
 <!--  -->
+<!-- -->
+<!--  -->
+<!-- -->
 <!-- provide an anchor so we can return to the top of the page -->
 <span id="top"></span>
+<!-- -->
 <!-- this relies on transclusion for next and previous links -->
 {{Knot Navigation Links|ext=gif}}
+{{Rolfsen Knot Page Header|n=9|k=28|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,9,-2,1,-4,5,-9,2,-7,8,-3,4,-5,3,-6,7,-8,6/goTop.html}}
-{| align=left
-|- valign=top
-|[[Image:{{PAGENAME}}.gif]]
-|{{Rolfsen Knot Site Links|n=9|k=28|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,9,-2,1,-4,5,-9,2,-7,8,-3,4,-5,3,-6,7,-8,6/goTop.html}}
-|{{:{{PAGENAME}} Quick Notes}}
-|}
 <br style="clear:both" />
@@ Line 24: / Line 21: @@
 {{Vassiliev Invariants}}
-===[[Khovanov Homology]]===
+{{Khovanov Homology|table=<table border=1>
-The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
-<center><table border=1>
 <tr align=center>
 <td width=14.2857%><table cellpadding=0 cellspacing=0>
@@ Line 47: / Line 40: @@
 <tr align=center><td>-13</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>-2</td></tr>
 <tr align=center><td>-15</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>1</td></tr>
-</table></center>
+</table>}}
 {{Computer Talk Header}}
@@ Line 127: / Line 119: @@
   q  t    q  t    q  t   q  t</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

9 28: Difference between revisions

Revision as of 20:14, 28 August 2005

Contents

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Three dimensional invariants

Four dimensional invariants

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