9 20

9_19

9_21

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 20's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 20 at Knotilus!

From knotilus

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X5,14,6,15 X7,16,8,17 X11,1,12,18 X15,6,16,7 X17,13,18,12 X13,8,14,9 X9,2,10,3
Gauss code	-1, 9, -2, 1, -3, 6, -4, 8, -9, 2, -5, 7, -8, 3, -6, 4, -7, 5
Dowker-Thistlethwaite code	4 10 14 16 2 18 8 6 12
Conway Notation	[31212]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 9, width is 4, Braid index is 4		[{12, 2}, {1, 10}, {8, 11}, {10, 12}, {9, 3}, {2, 4}, {3, 5}, {4, 8}, {6, 9}, {5, 7}, {11, 6}, {7, 1}]

[edit Notes on presentations of 9 20]

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["9 20"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3,10,4,11 X5,14,6,15 X7,16,8,17 X11,1,12,18 X15,6,16,7 X17,13,18,12 X13,8,14,9 X9,2,10,3

In[5]:=

GaussCode[K]

Out[5]=

-1, 9, -2, 1, -3, 6, -4, 8, -9, 2, -5, 7, -8, 3, -6, 4, -7, 5

In[6]:=

DTCode[K]

Out[6]=

4 10 14 16 2 18 8 6 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[31212]

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]=

[math]\displaystyle{ \textrm{BR}(4,\{-1,-1,-1,2,-1,-3,2,-3,-3\}) }[/math]

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]=

{ 4, 9, 4 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-12][1] Hyperbolic Volume 9.6443 A-Polynomial See Data:9 20/A-polynomial

[edit Notes for 9 20's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 2 }[/math] Topological 4 genus [math]\displaystyle{ 2 }[/math] Concordance genus [math]\displaystyle{ 3 }[/math] Rasmussen s-Invariant -4

[edit Notes for 9 20's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+5 t^2-9 t+11-9 t^{-1} +5 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6-z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 41, -4 }
Jones polynomial	[math]\displaystyle{ 1-2 q^{-1} +4 q^{-2} -5 q^{-3} +7 q^{-4} -7 q^{-5} +6 q^{-6} -5 q^{-7} +3 q^{-8} - q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^8-a^8+2 z^4 a^6+5 z^2 a^6+2 a^6-z^6 a^4-4 z^4 a^4-5 z^2 a^4-2 a^4+z^4 a^2+3 z^2 a^2+2 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^3 a^{11}+3 z^4 a^{10}-z^2 a^{10}+5 z^5 a^9-5 z^3 a^9+2 z a^9+5 z^6 a^8-6 z^4 a^8+3 z^2 a^8-a^8+3 z^7 a^7-7 z^3 a^7+2 z a^7+z^8 a^6+5 z^6 a^6-16 z^4 a^6+10 z^2 a^6-2 a^6+5 z^7 a^5-12 z^5 a^5+5 z^3 a^5+z^8 a^4+z^6 a^4-11 z^4 a^4+11 z^2 a^4-2 a^4+2 z^7 a^3-7 z^5 a^3+6 z^3 a^3+z^6 a^2-4 z^4 a^2+5 z^2 a^2-2 a^2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{28}+q^{24}-q^{22}+q^{20}-q^{18}+q^{14}-q^{12}+2 q^{10}-q^8+q^6+q^4+1 }[/math]
The G2 invariant	[math]\displaystyle{ q^{148}-2 q^{146}+3 q^{144}-4 q^{142}+2 q^{140}-q^{138}-2 q^{136}+9 q^{134}-12 q^{132}+15 q^{130}-13 q^{128}+5 q^{126}+2 q^{124}-13 q^{122}+23 q^{120}-27 q^{118}+23 q^{116}-13 q^{114}-q^{112}+15 q^{110}-23 q^{108}+25 q^{106}-21 q^{104}+5 q^{102}+7 q^{100}-17 q^{98}+16 q^{96}-5 q^{94}-9 q^{92}+23 q^{90}-24 q^{88}+14 q^{86}+3 q^{84}-26 q^{82}+44 q^{80}-44 q^{78}+30 q^{76}-4 q^{74}-21 q^{72}+43 q^{70}-45 q^{68}+33 q^{66}-17 q^{64}-6 q^{62}+22 q^{60}-28 q^{58}+22 q^{56}-6 q^{54}-9 q^{52}+19 q^{50}-19 q^{48}+7 q^{46}+8 q^{44}-23 q^{42}+31 q^{40}-28 q^{38}+12 q^{36}+10 q^{34}-26 q^{32}+36 q^{30}-30 q^{28}+17 q^{26}-q^{24}-13 q^{22}+21 q^{20}-19 q^{18}+14 q^{16}-3 q^{14}-2 q^{12}+5 q^{10}-5 q^8+4 q^6-q^4+q^2 }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_20.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{19}+2 q^{17}-2 q^{15}+q^{13}-q^{11}+2 q^7-q^5+2 q^3-q+ q^{-1} }[/math]
2	[math]\displaystyle{ q^{52}-2 q^{50}+5 q^{46}-6 q^{44}-q^{42}+8 q^{40}-8 q^{38}-q^{36}+9 q^{34}-4 q^{32}-4 q^{30}+4 q^{28}+2 q^{26}-5 q^{24}-3 q^{22}+6 q^{20}-q^{18}-7 q^{16}+8 q^{14}+3 q^{12}-8 q^{10}+5 q^8+5 q^6-6 q^4+q^2+4-2 q^{-2} - q^{-4} + q^{-6} }[/math]
3	[math]\displaystyle{ -q^{99}+2 q^{97}-3 q^{93}+5 q^{89}+2 q^{87}-10 q^{85}+13 q^{81}-2 q^{79}-18 q^{77}+2 q^{75}+25 q^{73}-2 q^{71}-28 q^{69}-2 q^{67}+30 q^{65}+7 q^{63}-26 q^{61}-13 q^{59}+16 q^{57}+19 q^{55}-6 q^{53}-20 q^{51}-5 q^{49}+21 q^{47}+13 q^{45}-16 q^{43}-22 q^{41}+13 q^{39}+23 q^{37}-9 q^{35}-27 q^{33}+2 q^{31}+28 q^{29}+4 q^{27}-27 q^{25}-11 q^{23}+26 q^{21}+17 q^{19}-19 q^{17}-20 q^{15}+13 q^{13}+23 q^{11}-3 q^9-20 q^7-2 q^5+14 q^3+7 q-8 q^{-1} -7 q^{-3} +3 q^{-5} +5 q^{-7} -2 q^{-11} - q^{-13} + q^{-15} }[/math]
4	[math]\displaystyle{ q^{160}-2 q^{158}+3 q^{154}-2 q^{152}+q^{150}-6 q^{148}+3 q^{146}+9 q^{144}-8 q^{142}+2 q^{140}-10 q^{138}+11 q^{136}+18 q^{134}-25 q^{132}-10 q^{130}-9 q^{128}+40 q^{126}+37 q^{124}-53 q^{122}-46 q^{120}-15 q^{118}+79 q^{116}+79 q^{114}-60 q^{112}-93 q^{110}-51 q^{108}+86 q^{106}+122 q^{104}-15 q^{102}-92 q^{100}-97 q^{98}+30 q^{96}+111 q^{94}+48 q^{92}-28 q^{90}-96 q^{88}-45 q^{86}+39 q^{84}+78 q^{82}+50 q^{80}-51 q^{78}-87 q^{76}-28 q^{74}+73 q^{72}+85 q^{70}-5 q^{68}-93 q^{66}-68 q^{64}+60 q^{62}+92 q^{60}+24 q^{58}-90 q^{56}-92 q^{54}+44 q^{52}+92 q^{50}+58 q^{48}-70 q^{46}-114 q^{44}+3 q^{42}+76 q^{40}+98 q^{38}-20 q^{36}-112 q^{34}-52 q^{32}+26 q^{30}+114 q^{28}+46 q^{26}-63 q^{24}-76 q^{22}-40 q^{20}+70 q^{18}+73 q^{16}+6 q^{14}-44 q^{12}-68 q^{10}+7 q^8+44 q^6+35 q^4+4 q^2-40-19 q^{-2} +3 q^{-4} +19 q^{-6} +18 q^{-8} -8 q^{-10} -9 q^{-12} -7 q^{-14} + q^{-16} +7 q^{-18} + q^{-20} -2 q^{-24} - q^{-26} + q^{-28} }[/math]
5	[math]\displaystyle{ -q^{235}+2 q^{233}-3 q^{229}+2 q^{227}+q^{225}+q^{221}-2 q^{219}-4 q^{217}+2 q^{215}+6 q^{213}-q^{211}-8 q^{209}-5 q^{207}+10 q^{205}+15 q^{203}+8 q^{201}-18 q^{199}-46 q^{197}-15 q^{195}+50 q^{193}+81 q^{191}+32 q^{189}-77 q^{187}-147 q^{185}-73 q^{183}+114 q^{181}+236 q^{179}+133 q^{177}-137 q^{175}-326 q^{173}-232 q^{171}+119 q^{169}+420 q^{167}+350 q^{165}-68 q^{163}-460 q^{161}-464 q^{159}-46 q^{157}+443 q^{155}+556 q^{153}+176 q^{151}-353 q^{149}-573 q^{147}-306 q^{145}+198 q^{143}+514 q^{141}+400 q^{139}-23 q^{137}-388 q^{135}-422 q^{133}-140 q^{131}+213 q^{129}+390 q^{127}+262 q^{125}-44 q^{123}-304 q^{121}-338 q^{119}-100 q^{117}+215 q^{115}+354 q^{113}+205 q^{111}-132 q^{109}-361 q^{107}-261 q^{105}+87 q^{103}+349 q^{101}+288 q^{99}-49 q^{97}-356 q^{95}-311 q^{93}+44 q^{91}+362 q^{89}+335 q^{87}-17 q^{85}-376 q^{83}-381 q^{81}-16 q^{79}+378 q^{77}+426 q^{75}+83 q^{73}-346 q^{71}-472 q^{69}-180 q^{67}+280 q^{65}+491 q^{63}+277 q^{61}-165 q^{59}-467 q^{57}-379 q^{55}+21 q^{53}+390 q^{51}+433 q^{49}+129 q^{47}-259 q^{45}-429 q^{43}-260 q^{41}+102 q^{39}+361 q^{37}+336 q^{35}+55 q^{33}-241 q^{31}-332 q^{29}-176 q^{27}+97 q^{25}+275 q^{23}+234 q^{21}+28 q^{19}-167 q^{17}-220 q^{15}-117 q^{13}+61 q^{11}+167 q^9+140 q^7+22 q^5-87 q^3-119 q-65 q^{-1} +23 q^{-3} +75 q^{-5} +67 q^{-7} +15 q^{-9} -32 q^{-11} -45 q^{-13} -29 q^{-15} +4 q^{-17} +25 q^{-19} +22 q^{-21} +5 q^{-23} -6 q^{-25} -11 q^{-27} -9 q^{-29} + q^{-31} +5 q^{-33} +3 q^{-35} + q^{-37} -2 q^{-41} - q^{-43} + q^{-45} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{28}+q^{24}-q^{22}+q^{20}-q^{18}+q^{14}-q^{12}+2 q^{10}-q^8+q^6+q^4+1 }[/math]
1,1	[math]\displaystyle{ q^{76}-4 q^{74}+8 q^{72}-12 q^{70}+22 q^{68}-36 q^{66}+46 q^{64}-56 q^{62}+72 q^{60}-84 q^{58}+84 q^{56}-82 q^{54}+77 q^{52}-62 q^{50}+32 q^{48}+4 q^{46}-43 q^{44}+90 q^{42}-132 q^{40}+168 q^{38}-192 q^{36}+198 q^{34}-190 q^{32}+158 q^{30}-126 q^{28}+82 q^{26}-30 q^{24}-18 q^{22}+57 q^{20}-82 q^{18}+106 q^{16}-110 q^{14}+104 q^{12}-84 q^{10}+70 q^8-48 q^6+29 q^4-16 q^2+8-2 q^{-2} + q^{-4} }[/math]
2,0	[math]\displaystyle{ q^{70}-q^{66}+q^{62}-3 q^{58}+q^{56}+2 q^{54}-3 q^{52}-q^{50}+4 q^{48}+3 q^{46}-4 q^{44}-q^{42}+3 q^{40}-2 q^{38}-4 q^{36}+2 q^{34}+q^{32}-4 q^{30}+q^{28}+2 q^{26}-2 q^{24}-2 q^{22}+4 q^{20}+3 q^{18}-2 q^{16}+q^{14}+5 q^{12}-2 q^8+q^6+2 q^4-1+ q^{-4} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{62}-2 q^{60}-q^{58}+5 q^{56}-3 q^{54}-3 q^{52}+8 q^{50}-3 q^{48}-6 q^{46}+6 q^{44}-2 q^{42}-6 q^{40}+4 q^{38}+q^{36}-2 q^{34}-q^{32}+3 q^{30}+3 q^{28}-6 q^{26}+3 q^{24}+5 q^{22}-7 q^{20}+5 q^{16}-5 q^{14}+2 q^{12}+5 q^{10}-2 q^8+2 q^6+2 q^4-q^2+1 }[/math]
1,0,0	[math]\displaystyle{ -q^{37}-q^{33}+q^{31}-q^{29}+2 q^{27}-q^{25}+q^{23}-q^{15}+q^{13}-q^{11}+2 q^9+2 q^5+q }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{80}-2 q^{76}-q^{74}+2 q^{72}+2 q^{70}-2 q^{68}+5 q^{64}-6 q^{60}+2 q^{56}-6 q^{54}-3 q^{52}+3 q^{50}+q^{48}-4 q^{46}+3 q^{44}+5 q^{42}-4 q^{40}-q^{38}+7 q^{36}-7 q^{32}+4 q^{30}+3 q^{28}-4 q^{26}-3 q^{24}+3 q^{22}+q^{20}-2 q^{18}+2 q^{16}+4 q^{14}+q^{12}+q^{10}+3 q^8+q^6+q^2 }[/math]
1,0,0,0	[math]\displaystyle{ -q^{46}-q^{42}-q^{40}+q^{38}-q^{36}+2 q^{34}+q^{30}+q^{28}-q^{22}-2 q^{18}+q^{16}-q^{14}+2 q^{12}+q^{10}+q^8+2 q^6+q^2 }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{62}+2 q^{60}-3 q^{58}+5 q^{56}-7 q^{54}+7 q^{52}-8 q^{50}+7 q^{48}-6 q^{46}+4 q^{44}-4 q^{40}+8 q^{38}-11 q^{36}+14 q^{34}-15 q^{32}+15 q^{30}-13 q^{28}+10 q^{26}-7 q^{24}+3 q^{22}+q^{20}-4 q^{18}+7 q^{16}-7 q^{14}+8 q^{12}-7 q^{10}+6 q^8-4 q^6+4 q^4-q^2+1 }[/math]
1,0	[math]\displaystyle{ q^{100}-2 q^{96}-2 q^{94}+q^{92}+5 q^{90}+2 q^{88}-5 q^{86}-5 q^{84}+2 q^{82}+8 q^{80}+2 q^{78}-7 q^{76}-6 q^{74}+3 q^{72}+7 q^{70}-q^{68}-7 q^{66}-2 q^{64}+6 q^{62}+4 q^{60}-5 q^{58}-5 q^{56}+3 q^{54}+5 q^{52}-2 q^{50}-5 q^{48}+q^{46}+5 q^{44}-6 q^{40}-q^{38}+7 q^{36}+5 q^{34}-5 q^{32}-8 q^{30}+q^{28}+9 q^{26}+3 q^{24}-6 q^{22}-6 q^{20}+4 q^{18}+7 q^{16}+q^{14}-4 q^{12}-2 q^{10}+3 q^8+3 q^6-q^4-q^2+ q^{-2} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{86}-2 q^{84}+q^{82}-2 q^{80}+5 q^{78}-5 q^{76}+4 q^{74}-5 q^{72}+8 q^{70}-5 q^{68}+4 q^{66}-6 q^{64}+3 q^{62}-2 q^{60}-2 q^{58}-6 q^{54}+7 q^{52}-8 q^{50}+11 q^{48}-11 q^{46}+13 q^{44}-9 q^{42}+12 q^{40}-9 q^{38}+8 q^{36}-5 q^{34}+3 q^{32}-3 q^{30}-2 q^{28}+2 q^{26}-5 q^{24}+4 q^{22}-6 q^{20}+8 q^{18}-4 q^{16}+7 q^{14}-3 q^{12}+6 q^{10}-2 q^8+3 q^6-q^4+q^2 }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{148}-2 q^{146}+3 q^{144}-4 q^{142}+2 q^{140}-q^{138}-2 q^{136}+9 q^{134}-12 q^{132}+15 q^{130}-13 q^{128}+5 q^{126}+2 q^{124}-13 q^{122}+23 q^{120}-27 q^{118}+23 q^{116}-13 q^{114}-q^{112}+15 q^{110}-23 q^{108}+25 q^{106}-21 q^{104}+5 q^{102}+7 q^{100}-17 q^{98}+16 q^{96}-5 q^{94}-9 q^{92}+23 q^{90}-24 q^{88}+14 q^{86}+3 q^{84}-26 q^{82}+44 q^{80}-44 q^{78}+30 q^{76}-4 q^{74}-21 q^{72}+43 q^{70}-45 q^{68}+33 q^{66}-17 q^{64}-6 q^{62}+22 q^{60}-28 q^{58}+22 q^{56}-6 q^{54}-9 q^{52}+19 q^{50}-19 q^{48}+7 q^{46}+8 q^{44}-23 q^{42}+31 q^{40}-28 q^{38}+12 q^{36}+10 q^{34}-26 q^{32}+36 q^{30}-30 q^{28}+17 q^{26}-q^{24}-13 q^{22}+21 q^{20}-19 q^{18}+14 q^{16}-3 q^{14}-2 q^{12}+5 q^{10}-5 q^8+4 q^6-q^4+q^2 }[/math]

.

Computer Talk

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

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

In[1]:=

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

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 20"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+5 t^2-9 t+11-9 t^{-1} +5 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6-z^4+2 z^2+1 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 41, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ 1-2 q^{-1} +4 q^{-2} -5 q^{-3} +7 q^{-4} -7 q^{-5} +6 q^{-6} -5 q^{-7} +3 q^{-8} - q^{-9} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^2 a^8-a^8+2 z^4 a^6+5 z^2 a^6+2 a^6-z^6 a^4-4 z^4 a^4-5 z^2 a^4-2 a^4+z^4 a^2+3 z^2 a^2+2 a^2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^3 a^{11}+3 z^4 a^{10}-z^2 a^{10}+5 z^5 a^9-5 z^3 a^9+2 z a^9+5 z^6 a^8-6 z^4 a^8+3 z^2 a^8-a^8+3 z^7 a^7-7 z^3 a^7+2 z a^7+z^8 a^6+5 z^6 a^6-16 z^4 a^6+10 z^2 a^6-2 a^6+5 z^7 a^5-12 z^5 a^5+5 z^3 a^5+z^8 a^4+z^6 a^4-11 z^4 a^4+11 z^2 a^4-2 a^4+2 z^7 a^3-7 z^5 a^3+6 z^3 a^3+z^6 a^2-4 z^4 a^2+5 z^2 a^2-2 a^2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_149, K11n26,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n90,}

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.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 20"];

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]=

{ [math]\displaystyle{ -t^3+5 t^2-9 t+11-9 t^{-1} +5 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ 1-2 q^{-1} +4 q^{-2} -5 q^{-3} +7 q^{-4} -7 q^{-5} +6 q^{-6} -5 q^{-7} +3 q^{-8} - q^{-9} }[/math] }

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]=

{10_149, K11n26,}

In[6]:=

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

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

Out[6]=

{K11n90,}

Vassiliev invariants

V2 and V3:

(2, -4)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{412}{3} }[/math] [math]\displaystyle{ \frac{68}{3} }[/math] [math]\displaystyle{ -256 }[/math] [math]\displaystyle{ -\frac{1856}{3} }[/math] [math]\displaystyle{ -\frac{224}{3} }[/math] [math]\displaystyle{ -128 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ \frac{3296}{3} }[/math] [math]\displaystyle{ \frac{544}{3} }[/math] [math]\displaystyle{ \frac{42751}{15} }[/math] [math]\displaystyle{ -\frac{4124}{15} }[/math] [math]\displaystyle{ \frac{64924}{45} }[/math] [math]\displaystyle{ \frac{833}{9} }[/math] [math]\displaystyle{ \frac{2911}{15} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-4 is the signature of 9 20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                   1 1 -1                 1   -1 -3               3 1   2 -5             3 2     -1 -7           4 2       2 -9         3 3         0 -11       3 4           -1 -13     2 3             1 -15   1 3               -2 -17   2                 2 -19 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

[math]\displaystyle{ n }[/math]	[math]\displaystyle{ J_n }[/math]
2	[math]\displaystyle{ q^2-2 q-1+7 q^{-1} -5 q^{-2} -8 q^{-3} +18 q^{-4} -5 q^{-5} -21 q^{-6} +29 q^{-7} -36 q^{-9} +35 q^{-10} +7 q^{-11} -45 q^{-12} +33 q^{-13} +14 q^{-14} -43 q^{-15} +25 q^{-16} +14 q^{-17} -30 q^{-18} +15 q^{-19} +7 q^{-20} -14 q^{-21} +6 q^{-22} +2 q^{-23} -3 q^{-24} + q^{-25} }[/math]
3	[math]\displaystyle{ q^6-2 q^5-q^4+2 q^3+6 q^2-4 q-11+ q^{-1} +21 q^{-2} +3 q^{-3} -27 q^{-4} -17 q^{-5} +38 q^{-6} +29 q^{-7} -37 q^{-8} -50 q^{-9} +39 q^{-10} +65 q^{-11} -28 q^{-12} -87 q^{-13} +23 q^{-14} +96 q^{-15} -4 q^{-16} -113 q^{-17} -6 q^{-18} +114 q^{-19} +28 q^{-20} -123 q^{-21} -41 q^{-22} +120 q^{-23} +57 q^{-24} -115 q^{-25} -67 q^{-26} +105 q^{-27} +71 q^{-28} -90 q^{-29} -70 q^{-30} +76 q^{-31} +58 q^{-32} -57 q^{-33} -47 q^{-34} +44 q^{-35} +32 q^{-36} -31 q^{-37} -20 q^{-38} +21 q^{-39} +12 q^{-40} -15 q^{-41} -5 q^{-42} +8 q^{-43} +2 q^{-44} -3 q^{-45} -2 q^{-46} +3 q^{-47} - q^{-48} }[/math]
4	[math]\displaystyle{ q^{12}-2 q^{11}-q^{10}+2 q^9+q^8+7 q^7-8 q^6-9 q^5+2 q^3+33 q^2-7 q-25-22 q^{-1} -19 q^{-2} +77 q^{-3} +24 q^{-4} -16 q^{-5} -59 q^{-6} -94 q^{-7} +101 q^{-8} +74 q^{-9} +51 q^{-10} -62 q^{-11} -204 q^{-12} +65 q^{-13} +87 q^{-14} +160 q^{-15} +6 q^{-16} -292 q^{-17} -13 q^{-18} +27 q^{-19} +252 q^{-20} +124 q^{-21} -314 q^{-22} -86 q^{-23} -90 q^{-24} +296 q^{-25} +252 q^{-26} -280 q^{-27} -134 q^{-28} -226 q^{-29} +298 q^{-30} +366 q^{-31} -212 q^{-32} -166 q^{-33} -354 q^{-34} +273 q^{-35} +454 q^{-36} -122 q^{-37} -178 q^{-38} -455 q^{-39} +214 q^{-40} +490 q^{-41} -21 q^{-42} -150 q^{-43} -494 q^{-44} +130 q^{-45} +439 q^{-46} +47 q^{-47} -74 q^{-48} -431 q^{-49} +49 q^{-50} +312 q^{-51} +52 q^{-52} +3 q^{-53} -294 q^{-54} +13 q^{-55} +175 q^{-56} +10 q^{-57} +36 q^{-58} -155 q^{-59} +13 q^{-60} +81 q^{-61} -21 q^{-62} +29 q^{-63} -65 q^{-64} +16 q^{-65} +32 q^{-66} -22 q^{-67} +14 q^{-68} -22 q^{-69} +9 q^{-70} +11 q^{-71} -10 q^{-72} +4 q^{-73} -5 q^{-74} +3 q^{-75} +2 q^{-76} -3 q^{-77} + q^{-78} }[/math]
5	[math]\displaystyle{ q^{20}-2 q^{19}-q^{18}+2 q^{17}+q^{16}+2 q^{15}+3 q^{14}-6 q^{13}-11 q^{12}+6 q^{10}+13 q^9+20 q^8-3 q^7-32 q^6-33 q^5-10 q^4+26 q^3+67 q^2+49 q-24-85 q^{-1} -98 q^{-2} -28 q^{-3} +99 q^{-4} +158 q^{-5} +94 q^{-6} -58 q^{-7} -204 q^{-8} -206 q^{-9} -4 q^{-10} +211 q^{-11} +289 q^{-12} +148 q^{-13} -163 q^{-14} -384 q^{-15} -277 q^{-16} +55 q^{-17} +380 q^{-18} +444 q^{-19} +118 q^{-20} -359 q^{-21} -536 q^{-22} -307 q^{-23} +211 q^{-24} +614 q^{-25} +506 q^{-26} -55 q^{-27} -579 q^{-28} -676 q^{-29} -189 q^{-30} +526 q^{-31} +808 q^{-32} +387 q^{-33} -365 q^{-34} -887 q^{-35} -649 q^{-36} +234 q^{-37} +934 q^{-38} +816 q^{-39} -22 q^{-40} -935 q^{-41} -1043 q^{-42} -131 q^{-43} +939 q^{-44} +1175 q^{-45} +330 q^{-46} -908 q^{-47} -1361 q^{-48} -486 q^{-49} +894 q^{-50} +1482 q^{-51} +667 q^{-52} -847 q^{-53} -1623 q^{-54} -834 q^{-55} +794 q^{-56} +1711 q^{-57} +1004 q^{-58} -698 q^{-59} -1762 q^{-60} -1149 q^{-61} +556 q^{-62} +1745 q^{-63} +1264 q^{-64} -392 q^{-65} -1634 q^{-66} -1326 q^{-67} +203 q^{-68} +1463 q^{-69} +1298 q^{-70} -27 q^{-71} -1211 q^{-72} -1212 q^{-73} -113 q^{-74} +959 q^{-75} +1031 q^{-76} +193 q^{-77} -682 q^{-78} -832 q^{-79} -226 q^{-80} +470 q^{-81} +613 q^{-82} +197 q^{-83} -290 q^{-84} -414 q^{-85} -156 q^{-86} +169 q^{-87} +262 q^{-88} +103 q^{-89} -101 q^{-90} -144 q^{-91} -53 q^{-92} +47 q^{-93} +75 q^{-94} +29 q^{-95} -31 q^{-96} -35 q^{-97} -4 q^{-98} +16 q^{-99} +10 q^{-100} -2 q^{-101} -3 q^{-102} -9 q^{-103} +3 q^{-104} +11 q^{-105} -5 q^{-106} -5 q^{-107} +4 q^{-108} -2 q^{-109} - q^{-110} +5 q^{-111} -3 q^{-112} -2 q^{-113} +3 q^{-114} - q^{-115} }[/math]
6	[math]\displaystyle{ q^{30}-2 q^{29}-q^{28}+2 q^{27}+q^{26}+2 q^{25}-2 q^{24}+5 q^{23}-8 q^{22}-11 q^{21}+3 q^{20}+5 q^{19}+14 q^{18}+3 q^{17}+25 q^{16}-17 q^{15}-41 q^{14}-24 q^{13}-12 q^{12}+27 q^{11}+23 q^{10}+109 q^9+22 q^8-57 q^7-90 q^6-109 q^5-49 q^4-22 q^3+243 q^2+184 q+88-62 q^{-1} -217 q^{-2} -287 q^{-3} -325 q^{-4} +204 q^{-5} +320 q^{-6} +435 q^{-7} +283 q^{-8} -9 q^{-9} -430 q^{-10} -849 q^{-11} -249 q^{-12} +18 q^{-13} +593 q^{-14} +793 q^{-15} +731 q^{-16} +5 q^{-17} -1053 q^{-18} -838 q^{-19} -865 q^{-20} +33 q^{-21} +803 q^{-22} +1568 q^{-23} +1062 q^{-24} -397 q^{-25} -804 q^{-26} -1744 q^{-27} -1183 q^{-28} -186 q^{-29} +1676 q^{-30} +2041 q^{-31} +949 q^{-32} +289 q^{-33} -1778 q^{-34} -2289 q^{-35} -1890 q^{-36} +688 q^{-37} +2133 q^{-38} +2187 q^{-39} +2082 q^{-40} -690 q^{-41} -2542 q^{-42} -3509 q^{-43} -996 q^{-44} +1158 q^{-45} +2661 q^{-46} +3815 q^{-47} +1083 q^{-48} -1848 q^{-49} -4466 q^{-50} -2685 q^{-51} -431 q^{-52} +2345 q^{-53} +5003 q^{-54} +2900 q^{-55} -651 q^{-56} -4768 q^{-57} -3983 q^{-58} -2053 q^{-59} +1658 q^{-60} +5676 q^{-61} +4418 q^{-62} +550 q^{-63} -4773 q^{-64} -4945 q^{-65} -3433 q^{-66} +1016 q^{-67} +6136 q^{-68} +5677 q^{-69} +1565 q^{-70} -4761 q^{-71} -5802 q^{-72} -4645 q^{-73} +475 q^{-74} +6536 q^{-75} +6838 q^{-76} +2552 q^{-77} -4613 q^{-78} -6563 q^{-79} -5846 q^{-80} -262 q^{-81} +6595 q^{-82} +7795 q^{-83} +3705 q^{-84} -3898 q^{-85} -6802 q^{-86} -6858 q^{-87} -1426 q^{-88} +5798 q^{-89} +8017 q^{-90} +4789 q^{-91} -2421 q^{-92} -5975 q^{-93} -7076 q^{-94} -2684 q^{-95} +4049 q^{-96} +6995 q^{-97} +5132 q^{-98} -696 q^{-99} -4123 q^{-100} -6036 q^{-101} -3274 q^{-102} +2024 q^{-103} +4932 q^{-104} +4330 q^{-105} +434 q^{-106} -2061 q^{-107} -4086 q^{-108} -2838 q^{-109} +604 q^{-110} +2741 q^{-111} +2797 q^{-112} +655 q^{-113} -632 q^{-114} -2152 q^{-115} -1811 q^{-116} +37 q^{-117} +1200 q^{-118} +1372 q^{-119} +365 q^{-120} -12 q^{-121} -878 q^{-122} -878 q^{-123} -27 q^{-124} +421 q^{-125} +509 q^{-126} +74 q^{-127} +118 q^{-128} -275 q^{-129} -337 q^{-130} +20 q^{-131} +123 q^{-132} +139 q^{-133} -41 q^{-134} +86 q^{-135} -63 q^{-136} -108 q^{-137} +32 q^{-138} +32 q^{-139} +25 q^{-140} -45 q^{-141} +42 q^{-142} -9 q^{-143} -30 q^{-144} +19 q^{-145} +5 q^{-146} +4 q^{-147} -22 q^{-148} +16 q^{-149} + q^{-150} -10 q^{-151} +8 q^{-152} - q^{-153} + q^{-154} -5 q^{-155} +3 q^{-156} +2 q^{-157} -3 q^{-158} + q^{-159} }[/math]
7	[math]\displaystyle{ q^{42}-2 q^{41}-q^{40}+2 q^{39}+q^{38}+2 q^{37}-2 q^{36}+3 q^{34}-8 q^{33}-8 q^{32}+2 q^{31}+5 q^{30}+16 q^{29}+5 q^{28}+q^{27}+14 q^{26}-23 q^{25}-36 q^{24}-27 q^{23}-14 q^{22}+40 q^{21}+41 q^{20}+42 q^{19}+78 q^{18}-69 q^{16}-112 q^{15}-153 q^{14}-32 q^{13}+31 q^{12}+104 q^{11}+267 q^{10}+202 q^9+88 q^8-98 q^7-371 q^6-348 q^5-296 q^4-127 q^3+347 q^2+536 q+633+471 q^{-1} -158 q^{-2} -528 q^{-3} -909 q^{-4} -1003 q^{-5} -343 q^{-6} +262 q^{-7} +1027 q^{-8} +1527 q^{-9} +1042 q^{-10} +407 q^{-11} -740 q^{-12} -1855 q^{-13} -1795 q^{-14} -1400 q^{-15} -67 q^{-16} +1661 q^{-17} +2346 q^{-18} +2576 q^{-19} +1350 q^{-20} -880 q^{-21} -2302 q^{-22} -3469 q^{-23} -2945 q^{-24} -678 q^{-25} +1477 q^{-26} +3901 q^{-27} +4430 q^{-28} +2558 q^{-29} +225 q^{-30} -3270 q^{-31} -5354 q^{-32} -4701 q^{-33} -2643 q^{-34} +1750 q^{-35} +5395 q^{-36} +6267 q^{-37} +5374 q^{-38} +877 q^{-39} -4237 q^{-40} -7151 q^{-41} -8021 q^{-42} -4017 q^{-43} +1983 q^{-44} +6786 q^{-45} +10027 q^{-46} +7477 q^{-47} +1237 q^{-48} -5317 q^{-49} -11107 q^{-50} -10603 q^{-51} -4985 q^{-52} +2639 q^{-53} +11066 q^{-54} +13216 q^{-55} +8885 q^{-56} +726 q^{-57} -9928 q^{-58} -14829 q^{-59} -12558 q^{-60} -4718 q^{-61} +7866 q^{-62} +15709 q^{-63} +15733 q^{-64} +8624 q^{-65} -5173 q^{-66} -15529 q^{-67} -18237 q^{-68} -12589 q^{-69} +2095 q^{-70} +14934 q^{-71} +20144 q^{-72} +15948 q^{-73} +969 q^{-74} -13680 q^{-75} -21411 q^{-76} -19067 q^{-77} -3977 q^{-78} +12473 q^{-79} +22377 q^{-80} +21542 q^{-81} +6560 q^{-82} -11132 q^{-83} -23000 q^{-84} -23780 q^{-85} -8926 q^{-86} +10107 q^{-87} +23703 q^{-88} +25666 q^{-89} +10835 q^{-90} -9234 q^{-91} -24359 q^{-92} -27500 q^{-93} -12663 q^{-94} +8611 q^{-95} +25234 q^{-96} +29289 q^{-97} +14367 q^{-98} -8002 q^{-99} -26076 q^{-100} -31165 q^{-101} -16233 q^{-102} +7241 q^{-103} +26777 q^{-104} +33006 q^{-105} +18333 q^{-106} -6020 q^{-107} -27075 q^{-108} -34654 q^{-109} -20609 q^{-110} +4190 q^{-111} +26566 q^{-112} +35774 q^{-113} +23007 q^{-114} -1698 q^{-115} -25097 q^{-116} -36023 q^{-117} -25085 q^{-118} -1340 q^{-119} +22486 q^{-120} +35116 q^{-121} +26516 q^{-122} +4549 q^{-123} -18913 q^{-124} -32827 q^{-125} -26913 q^{-126} -7552 q^{-127} +14704 q^{-128} +29362 q^{-129} +26036 q^{-130} +9756 q^{-131} -10333 q^{-132} -24876 q^{-133} -23885 q^{-134} -11045 q^{-135} +6306 q^{-136} +20052 q^{-137} +20748 q^{-138} +11104 q^{-139} -3086 q^{-140} -15200 q^{-141} -16953 q^{-142} -10261 q^{-143} +730 q^{-144} +10941 q^{-145} +13142 q^{-146} +8685 q^{-147} +584 q^{-148} -7446 q^{-149} -9521 q^{-150} -6816 q^{-151} -1223 q^{-152} +4803 q^{-153} +6555 q^{-154} +5001 q^{-155} +1293 q^{-156} -3000 q^{-157} -4253 q^{-158} -3373 q^{-159} -1084 q^{-160} +1769 q^{-161} +2570 q^{-162} +2169 q^{-163} +843 q^{-164} -1064 q^{-165} -1513 q^{-166} -1265 q^{-167} -526 q^{-168} +610 q^{-169} +775 q^{-170} +701 q^{-171} +358 q^{-172} -354 q^{-173} -407 q^{-174} -363 q^{-175} -191 q^{-176} +225 q^{-177} +170 q^{-178} +147 q^{-179} +109 q^{-180} -110 q^{-181} -56 q^{-182} -82 q^{-183} -68 q^{-184} +92 q^{-185} +22 q^{-186} +2 q^{-187} +24 q^{-188} -29 q^{-189} +17 q^{-190} -17 q^{-191} -29 q^{-192} +38 q^{-193} - q^{-194} -12 q^{-195} +3 q^{-196} -6 q^{-197} +15 q^{-198} -5 q^{-199} -12 q^{-200} +14 q^{-201} -2 q^{-202} -5 q^{-203} + q^{-204} - q^{-205} +5 q^{-206} -3 q^{-207} -2 q^{-208} +3 q^{-209} - q^{-210} }[/math]

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.

9_19 9_21