9 27

9_26

9_28

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 27's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 27 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,1,12,18 X5,13,6,12 X13,17,14,16 X7,14,8,15 X15,6,16,7 X17,9,18,8 X9,2,10,3
Gauss code	-1, 9, -2, 1, -4, 7, -6, 8, -9, 2, -3, 4, -5, 6, -7, 5, -8, 3
Dowker-Thistlethwaite code	4 10 12 14 2 18 16 6 8
Conway Notation	[212112]

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

[edit Notes on presentations of 9 27]

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 27"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 12 14 2 18 16 6 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[212112]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 2 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-5] Hyperbolic Volume 11. A-Polynomial See Data:9 27/A-polynomial

[edit Notes for 9 27's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 0 }[/math] Topological 4 genus [math]\displaystyle{ 0 }[/math] Concordance genus [math]\displaystyle{ 0 }[/math] Rasmussen s-Invariant 0

[edit Notes for 9 27's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+5 t^2-11 t+15-11 t^{-1} +5 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6-z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 49, 0 }
Jones polynomial	[math]\displaystyle{ q^4-3 q^3+5 q^2-7 q+9-8 q^{-1} +7 q^{-2} -5 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6+2 a^2 z^4+z^4 a^{-2} -4 z^4-a^4 z^2+5 a^2 z^2+2 z^2 a^{-2} -6 z^2-a^4+3 a^2+ a^{-2} -2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^8+z^8+3 a^3 z^7+6 a z^7+3 z^7 a^{-1} +3 a^4 z^6+6 a^2 z^6+4 z^6 a^{-2} +7 z^6+a^5 z^5-4 a^3 z^5-8 a z^5+3 z^5 a^{-3} -7 a^4 z^4-17 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} -16 z^4-2 a^5 z^3-2 a^3 z^3-4 z^3 a^{-1} -4 z^3 a^{-3} +4 a^4 z^2+12 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +12 z^2+a^5 z+2 a^3 z+2 a z+2 z a^{-1} +z a^{-3} -a^4-3 a^2- a^{-2} -2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}+q^{12}-q^{10}+2 q^8+2 q^2-1+2 q^{-2} -2 q^{-4} + q^{-8} - q^{-10} + q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+7 q^{72}-4 q^{70}-6 q^{68}+19 q^{66}-29 q^{64}+33 q^{62}-29 q^{60}+6 q^{58}+22 q^{56}-50 q^{54}+65 q^{52}-56 q^{50}+32 q^{48}+6 q^{46}-42 q^{44}+61 q^{42}-58 q^{40}+33 q^{38}+3 q^{36}-32 q^{34}+41 q^{32}-25 q^{30}-q^{28}+36 q^{26}-53 q^{24}+50 q^{22}-24 q^{20}-20 q^{18}+64 q^{16}-89 q^{14}+88 q^{12}-53 q^{10}+8 q^8+45 q^6-81 q^4+87 q^2-67+26 q^{-2} +16 q^{-4} -47 q^{-6} +48 q^{-8} -25 q^{-10} -5 q^{-12} +31 q^{-14} -41 q^{-16} +24 q^{-18} + q^{-20} -35 q^{-22} +58 q^{-24} -59 q^{-26} +43 q^{-28} -9 q^{-30} -22 q^{-32} +45 q^{-34} -51 q^{-36} +45 q^{-38} -28 q^{-40} +7 q^{-42} +11 q^{-44} -23 q^{-46} +24 q^{-48} -18 q^{-50} +13 q^{-52} -4 q^{-54} -2 q^{-56} +4 q^{-58} -6 q^{-60} +4 q^{-62} -2 q^{-64} + q^{-66} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_27.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+2 q^9-2 q^7+2 q^5-q^3+q+2 q^{-1} -2 q^{-3} +2 q^{-5} -2 q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{32}-2 q^{30}-2 q^{28}+7 q^{26}-2 q^{24}-9 q^{22}+11 q^{20}+2 q^{18}-15 q^{16}+9 q^{14}+7 q^{12}-13 q^{10}+3 q^8+8 q^6-4 q^4-4 q^2+5+9 q^{-2} -11 q^{-4} -3 q^{-6} +15 q^{-8} -10 q^{-10} -7 q^{-12} +13 q^{-14} -4 q^{-16} -5 q^{-18} +6 q^{-20} - q^{-22} -2 q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ -q^{63}+2 q^{61}+2 q^{59}-3 q^{57}-7 q^{55}+2 q^{53}+16 q^{51}+q^{49}-23 q^{47}-11 q^{45}+29 q^{43}+26 q^{41}-31 q^{39}-42 q^{37}+26 q^{35}+55 q^{33}-13 q^{31}-65 q^{29}-q^{27}+67 q^{25}+14 q^{23}-63 q^{21}-25 q^{19}+52 q^{17}+34 q^{15}-39 q^{13}-36 q^{11}+22 q^9+39 q^7-4 q^5-36 q^3-18 q+34 q^{-1} +42 q^{-3} -27 q^{-5} -55 q^{-7} +13 q^{-9} +68 q^{-11} -2 q^{-13} -69 q^{-15} -12 q^{-17} +62 q^{-19} +20 q^{-21} -48 q^{-23} -23 q^{-25} +34 q^{-27} +21 q^{-29} -21 q^{-31} -16 q^{-33} +11 q^{-35} +11 q^{-37} -7 q^{-39} -5 q^{-41} +3 q^{-43} +3 q^{-45} - q^{-47} -2 q^{-49} + q^{-51} }[/math]
4	[math]\displaystyle{ q^{104}-2 q^{102}-2 q^{100}+3 q^{98}+3 q^{96}+7 q^{94}-9 q^{92}-16 q^{90}-q^{88}+11 q^{86}+41 q^{84}-2 q^{82}-47 q^{80}-41 q^{78}-7 q^{76}+100 q^{74}+62 q^{72}-42 q^{70}-117 q^{68}-108 q^{66}+117 q^{64}+173 q^{62}+62 q^{60}-141 q^{58}-259 q^{56}+16 q^{54}+223 q^{52}+230 q^{50}-46 q^{48}-346 q^{46}-148 q^{44}+156 q^{42}+333 q^{40}+100 q^{38}-305 q^{36}-255 q^{34}+30 q^{32}+318 q^{30}+195 q^{28}-190 q^{26}-261 q^{24}-69 q^{22}+225 q^{20}+214 q^{18}-58 q^{16}-208 q^{14}-135 q^{12}+100 q^{10}+195 q^8+88 q^6-124 q^4-197 q^2-61+154 q^{-2} +254 q^{-4} -2 q^{-6} -232 q^{-8} -236 q^{-10} +52 q^{-12} +355 q^{-14} +151 q^{-16} -171 q^{-18} -343 q^{-20} -94 q^{-22} +321 q^{-24} +243 q^{-26} -34 q^{-28} -299 q^{-30} -185 q^{-32} +176 q^{-34} +204 q^{-36} +72 q^{-38} -160 q^{-40} -163 q^{-42} +51 q^{-44} +96 q^{-46} +80 q^{-48} -49 q^{-50} -85 q^{-52} +7 q^{-54} +23 q^{-56} +41 q^{-58} -9 q^{-60} -29 q^{-62} +4 q^{-64} +12 q^{-68} -2 q^{-70} -8 q^{-72} +3 q^{-74} +3 q^{-78} - q^{-80} -2 q^{-82} + q^{-84} }[/math]
5	[math]\displaystyle{ -q^{155}+2 q^{153}+2 q^{151}-3 q^{149}-3 q^{147}-3 q^{145}+9 q^{141}+16 q^{139}+q^{137}-20 q^{135}-29 q^{133}-19 q^{131}+20 q^{129}+61 q^{127}+62 q^{125}-11 q^{123}-97 q^{121}-121 q^{119}-47 q^{117}+108 q^{115}+220 q^{113}+161 q^{111}-78 q^{109}-309 q^{107}-326 q^{105}-59 q^{103}+348 q^{101}+541 q^{99}+291 q^{97}-284 q^{95}-722 q^{93}-609 q^{91}+62 q^{89}+811 q^{87}+963 q^{85}+295 q^{83}-745 q^{81}-1264 q^{79}-732 q^{77}+497 q^{75}+1431 q^{73}+1192 q^{71}-123 q^{69}-1433 q^{67}-1558 q^{65}-315 q^{63}+1258 q^{61}+1785 q^{59}+742 q^{57}-978 q^{55}-1843 q^{53}-1070 q^{51}+644 q^{49}+1750 q^{47}+1273 q^{45}-324 q^{43}-1556 q^{41}-1345 q^{39}+50 q^{37}+1315 q^{35}+1314 q^{33}+149 q^{31}-1043 q^{29}-1228 q^{27}-316 q^{25}+799 q^{23}+1121 q^{21}+450 q^{19}-541 q^{17}-1013 q^{15}-623 q^{13}+280 q^{11}+929 q^9+805 q^7+27 q^5-818 q^3-1032 q-389 q^{-1} +672 q^{-3} +1265 q^{-5} +782 q^{-7} -438 q^{-9} -1410 q^{-11} -1218 q^{-13} +110 q^{-15} +1477 q^{-17} +1587 q^{-19} +274 q^{-21} -1350 q^{-23} -1852 q^{-25} -701 q^{-27} +1098 q^{-29} +1925 q^{-31} +1056 q^{-33} -717 q^{-35} -1801 q^{-37} -1283 q^{-39} +301 q^{-41} +1502 q^{-43} +1336 q^{-45} +71 q^{-47} -1114 q^{-49} -1217 q^{-51} -315 q^{-53} +702 q^{-55} +984 q^{-57} +430 q^{-59} -366 q^{-61} -710 q^{-63} -417 q^{-65} +137 q^{-67} +449 q^{-69} +335 q^{-71} -3 q^{-73} -256 q^{-75} -239 q^{-77} -36 q^{-79} +127 q^{-81} +138 q^{-83} +47 q^{-85} -51 q^{-87} -81 q^{-89} -32 q^{-91} +23 q^{-93} +34 q^{-95} +17 q^{-97} -5 q^{-99} -13 q^{-101} -10 q^{-103} +3 q^{-105} +8 q^{-107} - q^{-109} -3 q^{-111} +3 q^{-119} - q^{-121} -2 q^{-123} + q^{-125} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{16}+q^{12}-q^{10}+2 q^8+2 q^2-1+2 q^{-2} -2 q^{-4} + q^{-8} - q^{-10} + q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{44}-4 q^{42}+12 q^{40}-28 q^{38}+52 q^{36}-86 q^{34}+130 q^{32}-180 q^{30}+222 q^{28}-248 q^{26}+258 q^{24}-236 q^{22}+176 q^{20}-88 q^{18}-24 q^{16}+144 q^{14}-267 q^{12}+376 q^{10}-450 q^8+492 q^6-483 q^4+440 q^2-352+244 q^{-2} -120 q^{-4} -4 q^{-6} +102 q^{-8} -176 q^{-10} +222 q^{-12} -238 q^{-14} +228 q^{-16} -196 q^{-18} +162 q^{-20} -126 q^{-22} +88 q^{-24} -58 q^{-26} +36 q^{-28} -20 q^{-30} +10 q^{-32} -4 q^{-34} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{42}-2 q^{38}-2 q^{36}+2 q^{34}+4 q^{32}-3 q^{30}-3 q^{28}+4 q^{26}+4 q^{24}-5 q^{22}-5 q^{20}+5 q^{18}+2 q^{16}-6 q^{14}+6 q^{10}-2 q^8-q^6+6 q^4+q^2-2+3 q^{-2} +5 q^{-4} -7 q^{-6} -5 q^{-8} +7 q^{-10} +2 q^{-12} -7 q^{-14} +6 q^{-18} -3 q^{-22} +2 q^{-26} - q^{-28} - q^{-30} + q^{-32} }[/math]

A3 Invariants.

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

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{44}-q^{40}+q^{38}+q^{36}-3 q^{34}-3 q^{32}+2 q^{30}-8 q^{26}-2 q^{24}+9 q^{22}+q^{20}-5 q^{18}+9 q^{16}+11 q^{14}-3 q^{12}-2 q^{10}+7 q^8-10 q^4+2 q^2+4-9 q^{-2} -4 q^{-4} +10 q^{-6} -2 q^{-8} -7 q^{-10} +7 q^{-12} +8 q^{-14} -4 q^{-16} -4 q^{-18} +6 q^{-20} +2 q^{-22} -5 q^{-24} - q^{-26} +3 q^{-28} - q^{-30} - q^{-32} + q^{-34} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{26}-q^{22}-q^{20}+q^{18}-q^{16}+3 q^{14}+q^{12}+2 q^{10}+2 q^8+q^4-q^2-2 q^{-2} + q^{-4} -2 q^{-6} + q^{-8} +2 q^{-14} - q^{-16} + q^{-18} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+2 q^{32}-5 q^{30}+7 q^{28}-9 q^{26}+11 q^{24}-12 q^{22}+12 q^{20}-9 q^{18}+6 q^{16}-5 q^{12}+12 q^{10}-17 q^8+21 q^6-22 q^4+23 q^2-20+16 q^{-2} -9 q^{-4} +3 q^{-6} +2 q^{-8} -7 q^{-10} +10 q^{-12} -12 q^{-14} +12 q^{-16} -10 q^{-18} +8 q^{-20} -6 q^{-22} +4 q^{-24} -2 q^{-26} + q^{-28} }[/math]
1,0	[math]\displaystyle{ q^{56}-2 q^{52}-2 q^{50}+3 q^{48}+5 q^{46}-2 q^{44}-8 q^{42}-3 q^{40}+9 q^{38}+8 q^{36}-7 q^{34}-13 q^{32}+13 q^{28}+7 q^{26}-9 q^{24}-9 q^{22}+5 q^{20}+11 q^{18}-8 q^{14}-q^{12}+8 q^{10}+3 q^8-7 q^6-5 q^4+7 q^2+7-5 q^{-2} -9 q^{-4} +4 q^{-6} +10 q^{-8} - q^{-10} -12 q^{-12} -4 q^{-14} +11 q^{-16} +9 q^{-18} -6 q^{-20} -12 q^{-22} +11 q^{-26} +6 q^{-28} -5 q^{-30} -7 q^{-32} +5 q^{-36} +2 q^{-38} -2 q^{-40} -2 q^{-42} + q^{-46} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{46}-2 q^{44}+3 q^{42}-4 q^{40}+6 q^{38}-8 q^{36}+7 q^{34}-10 q^{32}+9 q^{30}-11 q^{28}+6 q^{26}-6 q^{24}+6 q^{22}-q^{18}+9 q^{16}-4 q^{14}+15 q^{12}-14 q^{10}+16 q^8-16 q^6+18 q^4-19 q^2+12-15 q^{-2} +10 q^{-4} -6 q^{-6} + q^{-8} - q^{-10} -2 q^{-12} +9 q^{-14} -6 q^{-16} +8 q^{-18} -9 q^{-20} +11 q^{-22} -7 q^{-24} +6 q^{-26} -7 q^{-28} +5 q^{-30} -3 q^{-32} +2 q^{-34} -2 q^{-36} + q^{-38} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+7 q^{72}-4 q^{70}-6 q^{68}+19 q^{66}-29 q^{64}+33 q^{62}-29 q^{60}+6 q^{58}+22 q^{56}-50 q^{54}+65 q^{52}-56 q^{50}+32 q^{48}+6 q^{46}-42 q^{44}+61 q^{42}-58 q^{40}+33 q^{38}+3 q^{36}-32 q^{34}+41 q^{32}-25 q^{30}-q^{28}+36 q^{26}-53 q^{24}+50 q^{22}-24 q^{20}-20 q^{18}+64 q^{16}-89 q^{14}+88 q^{12}-53 q^{10}+8 q^8+45 q^6-81 q^4+87 q^2-67+26 q^{-2} +16 q^{-4} -47 q^{-6} +48 q^{-8} -25 q^{-10} -5 q^{-12} +31 q^{-14} -41 q^{-16} +24 q^{-18} + q^{-20} -35 q^{-22} +58 q^{-24} -59 q^{-26} +43 q^{-28} -9 q^{-30} -22 q^{-32} +45 q^{-34} -51 q^{-36} +45 q^{-38} -28 q^{-40} +7 q^{-42} +11 q^{-44} -23 q^{-46} +24 q^{-48} -18 q^{-50} +13 q^{-52} -4 q^{-54} -2 q^{-56} +4 q^{-58} -6 q^{-60} +4 q^{-62} -2 q^{-64} + q^{-66} }[/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 27"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+5 t^2-11 t+15-11 t^{-1} +5 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^6-z^4+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]=

{ 49, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^2 z^8+z^8+3 a^3 z^7+6 a z^7+3 z^7 a^{-1} +3 a^4 z^6+6 a^2 z^6+4 z^6 a^{-2} +7 z^6+a^5 z^5-4 a^3 z^5-8 a z^5+3 z^5 a^{-3} -7 a^4 z^4-17 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} -16 z^4-2 a^5 z^3-2 a^3 z^3-4 z^3 a^{-1} -4 z^3 a^{-3} +4 a^4 z^2+12 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +12 z^2+a^5 z+2 a^3 z+2 a z+2 z a^{-1} +z a^{-3} -a^4-3 a^2- a^{-2} -2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n4, K11n21, K11n172,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n83,}

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 27"];

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^2-11 t+15-11 t^{-1} +5 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ q^4-3 q^3+5 q^2-7 q+9-8 q^{-1} +7 q^{-2} -5 q^{-3} +3 q^{-4} - q^{-5} }[/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]=

{K11n4, K11n21, K11n172,}

In[6]:=

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

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

Out[6]=

{K11n83,}

Vassiliev invariants

V2 and V3:

(0, -1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{16}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 56 }[/math] [math]\displaystyle{ 56 }[/math] [math]\displaystyle{ -\frac{152}{3} }[/math] [math]\displaystyle{ \frac{40}{3} }[/math] [math]\displaystyle{ -24 }[/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]0 is the signature of 9 27. 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 χ 9                   1 1 7                 2   -2 5               3 1   2 3             4 2     -2 1           5 3       2 -1         4 5         1 -3       3 4           -1 -5     2 4             2 -7   1 3               -2 -9   2                 2 -11 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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^{12}-3 q^{11}+q^{10}+8 q^9-14 q^8+2 q^7+25 q^6-34 q^5-q^4+50 q^3-52 q^2-9 q+70-56 q^{-1} -18 q^{-2} +70 q^{-3} -44 q^{-4} -23 q^{-5} +54 q^{-6} -24 q^{-7} -21 q^{-8} +30 q^{-9} -7 q^{-10} -12 q^{-11} +10 q^{-12} -3 q^{-14} + q^{-15} }[/math]
3	[math]\displaystyle{ q^{24}-3 q^{23}+q^{22}+4 q^{21}+q^{20}-11 q^{19}-q^{18}+22 q^{17}+q^{16}-38 q^{15}-6 q^{14}+64 q^{13}+14 q^{12}-95 q^{11}-31 q^{10}+132 q^9+56 q^8-169 q^7-88 q^6+199 q^5+126 q^4-224 q^3-156 q^2+227 q+195-232 q^{-1} -208 q^{-2} +209 q^{-3} +227 q^{-4} -189 q^{-5} -225 q^{-6} +151 q^{-7} +224 q^{-8} -116 q^{-9} -207 q^{-10} +74 q^{-11} +186 q^{-12} -39 q^{-13} -154 q^{-14} +6 q^{-15} +122 q^{-16} +13 q^{-17} -86 q^{-18} -23 q^{-19} +54 q^{-20} +24 q^{-21} -29 q^{-22} -20 q^{-23} +14 q^{-24} +12 q^{-25} -5 q^{-26} -5 q^{-27} +3 q^{-29} - q^{-30} }[/math]
4	[math]\displaystyle{ q^{40}-3 q^{39}+q^{38}+4 q^{37}-3 q^{36}+4 q^{35}-14 q^{34}+7 q^{33}+18 q^{32}-15 q^{31}+8 q^{30}-47 q^{29}+27 q^{28}+68 q^{27}-33 q^{26}-8 q^{25}-139 q^{24}+63 q^{23}+197 q^{22}-17 q^{21}-53 q^{20}-353 q^{19}+66 q^{18}+429 q^{17}+115 q^{16}-81 q^{15}-714 q^{14}-48 q^{13}+694 q^{12}+392 q^{11}-3 q^{10}-1129 q^9-297 q^8+866 q^7+714 q^6+201 q^5-1432 q^4-585 q^3+870 q^2+944 q+457-1532 q^{-1} -800 q^{-2} +734 q^{-3} +1017 q^{-4} +669 q^{-5} -1425 q^{-6} -895 q^{-7} +499 q^{-8} +944 q^{-9} +819 q^{-10} -1153 q^{-11} -884 q^{-12} +205 q^{-13} +752 q^{-14} +890 q^{-15} -768 q^{-16} -761 q^{-17} -83 q^{-18} +467 q^{-19} +840 q^{-20} -363 q^{-21} -528 q^{-22} -260 q^{-23} +163 q^{-24} +642 q^{-25} -63 q^{-26} -252 q^{-27} -267 q^{-28} -44 q^{-29} +367 q^{-30} +55 q^{-31} -49 q^{-32} -156 q^{-33} -100 q^{-34} +142 q^{-35} +46 q^{-36} +26 q^{-37} -52 q^{-38} -62 q^{-39} +35 q^{-40} +12 q^{-41} +20 q^{-42} -7 q^{-43} -19 q^{-44} +5 q^{-45} +5 q^{-47} -3 q^{-49} + q^{-50} }[/math]
5	[math]\displaystyle{ q^{60}-3 q^{59}+q^{58}+4 q^{57}-3 q^{56}+q^{54}-6 q^{53}+3 q^{52}+13 q^{51}-8 q^{50}-13 q^{49}-2 q^{48}+2 q^{47}+25 q^{46}+30 q^{45}-19 q^{44}-68 q^{43}-51 q^{42}+32 q^{41}+123 q^{40}+121 q^{39}-30 q^{38}-231 q^{37}-254 q^{36}+15 q^{35}+376 q^{34}+459 q^{33}+84 q^{32}-543 q^{31}-808 q^{30}-278 q^{29}+720 q^{28}+1255 q^{27}+638 q^{26}-825 q^{25}-1825 q^{24}-1180 q^{23}+823 q^{22}+2440 q^{21}+1903 q^{20}-659 q^{19}-3026 q^{18}-2764 q^{17}+305 q^{16}+3524 q^{15}+3676 q^{14}+210 q^{13}-3853 q^{12}-4563 q^{11}-846 q^{10}+4026 q^9+5300 q^8+1523 q^7-3963 q^6-5930 q^5-2174 q^4+3834 q^3+6272 q^2+2743 q-3480-6523 q^{-1} -3235 q^{-2} +3191 q^{-3} +6486 q^{-4} +3588 q^{-5} -2702 q^{-6} -6399 q^{-7} -3884 q^{-8} +2288 q^{-9} +6096 q^{-10} +4060 q^{-11} -1711 q^{-12} -5728 q^{-13} -4206 q^{-14} +1173 q^{-15} +5184 q^{-16} +4245 q^{-17} -519 q^{-18} -4563 q^{-19} -4205 q^{-20} -92 q^{-21} +3789 q^{-22} +4034 q^{-23} +713 q^{-24} -2966 q^{-25} -3728 q^{-26} -1198 q^{-27} +2075 q^{-28} +3261 q^{-29} +1578 q^{-30} -1246 q^{-31} -2685 q^{-32} -1725 q^{-33} +502 q^{-34} +2018 q^{-35} +1703 q^{-36} +64 q^{-37} -1370 q^{-38} -1486 q^{-39} -432 q^{-40} +789 q^{-41} +1171 q^{-42} +583 q^{-43} -330 q^{-44} -818 q^{-45} -584 q^{-46} +40 q^{-47} +500 q^{-48} +470 q^{-49} +108 q^{-50} -243 q^{-51} -334 q^{-52} -153 q^{-53} +93 q^{-54} +203 q^{-55} +125 q^{-56} -12 q^{-57} -95 q^{-58} -94 q^{-59} -19 q^{-60} +48 q^{-61} +51 q^{-62} +12 q^{-63} -9 q^{-64} -21 q^{-65} -20 q^{-66} +7 q^{-67} +12 q^{-68} +2 q^{-69} -5 q^{-72} +3 q^{-74} - q^{-75} }[/math]
6	[math]\displaystyle{ q^{84}-3 q^{83}+q^{82}+4 q^{81}-3 q^{80}-3 q^{78}+9 q^{77}-10 q^{76}-2 q^{75}+20 q^{74}-18 q^{73}-7 q^{72}-7 q^{71}+40 q^{70}-12 q^{69}-5 q^{68}+48 q^{67}-73 q^{66}-57 q^{65}-29 q^{64}+139 q^{63}+38 q^{62}+43 q^{61}+114 q^{60}-242 q^{59}-269 q^{58}-176 q^{57}+330 q^{56}+284 q^{55}+350 q^{54}+381 q^{53}-586 q^{52}-932 q^{51}-826 q^{50}+414 q^{49}+873 q^{48}+1420 q^{47}+1430 q^{46}-839 q^{45}-2346 q^{44}-2782 q^{43}-428 q^{42}+1474 q^{41}+3757 q^{40}+4381 q^{39}+103 q^{38}-4115 q^{37}-6737 q^{36}-3643 q^{35}+708 q^{34}+6888 q^{33}+10003 q^{32}+3935 q^{31}-4578 q^{30}-11996 q^{29}-9990 q^{28}-3275 q^{27}+8867 q^{26}+17250 q^{25}+11236 q^{24}-1796 q^{23}-16228 q^{22}-18033 q^{21}-10774 q^{20}+7676 q^{19}+23398 q^{18}+20147 q^{17}+4318 q^{16}-17280 q^{15}-24815 q^{14}-19617 q^{13}+3325 q^{12}+26221 q^{11}+27600 q^{10}+11576 q^9-15114 q^8-28245 q^7-26869 q^6-2200 q^5+25655 q^4+31734 q^3+17494 q^2-11382 q-28381-31040 q^{-1} -6955 q^{-2} +23051 q^{-3} +32709 q^{-4} +21116 q^{-5} -7491 q^{-6} -26411 q^{-7} -32451 q^{-8} -10430 q^{-9} +19418 q^{-10} +31516 q^{-11} +22950 q^{-12} -3607 q^{-13} -23043 q^{-14} -31967 q^{-15} -13222 q^{-16} +14740 q^{-17} +28621 q^{-18} +23696 q^{-19} +801 q^{-20} -18128 q^{-21} -29822 q^{-22} -15749 q^{-23} +8657 q^{-24} +23742 q^{-25} +23215 q^{-26} +5727 q^{-27} -11405 q^{-28} -25504 q^{-29} -17374 q^{-30} +1624 q^{-31} +16648 q^{-32} +20570 q^{-33} +9923 q^{-34} -3615 q^{-35} -18663 q^{-36} -16662 q^{-37} -4583 q^{-38} +8272 q^{-39} +15172 q^{-40} +11451 q^{-41} +3148 q^{-42} -10332 q^{-43} -12819 q^{-44} -7710 q^{-45} +946 q^{-46} +8115 q^{-47} +9410 q^{-48} +6546 q^{-49} -3018 q^{-50} -7051 q^{-51} -6939 q^{-52} -2991 q^{-53} +1982 q^{-54} +5182 q^{-55} +6032 q^{-56} +1013 q^{-57} -2007 q^{-58} -3888 q^{-59} -3236 q^{-60} -1163 q^{-61} +1418 q^{-62} +3455 q^{-63} +1696 q^{-64} +487 q^{-65} -1130 q^{-66} -1693 q^{-67} -1520 q^{-68} -326 q^{-69} +1202 q^{-70} +870 q^{-71} +799 q^{-72} +80 q^{-73} -392 q^{-74} -797 q^{-75} -496 q^{-76} +203 q^{-77} +167 q^{-78} +380 q^{-79} +208 q^{-80} +65 q^{-81} -238 q^{-82} -227 q^{-83} -37 q^{-85} +89 q^{-86} +80 q^{-87} +79 q^{-88} -44 q^{-89} -58 q^{-90} - q^{-91} -29 q^{-92} +9 q^{-93} +12 q^{-94} +29 q^{-95} -7 q^{-96} -12 q^{-97} +5 q^{-98} -7 q^{-99} +5 q^{-102} -3 q^{-104} + q^{-105} }[/math]
7	[math]\displaystyle{ q^{112}-3 q^{111}+q^{110}+4 q^{109}-3 q^{108}-3 q^{106}+5 q^{105}+5 q^{104}-15 q^{103}+5 q^{102}+10 q^{101}-12 q^{100}-q^{99}-8 q^{98}+23 q^{97}+34 q^{96}-39 q^{95}-2 q^{94}-2 q^{93}-52 q^{92}-4 q^{91}-17 q^{90}+92 q^{89}+152 q^{88}-35 q^{87}-30 q^{86}-116 q^{85}-239 q^{84}-75 q^{83}-27 q^{82}+301 q^{81}+560 q^{80}+197 q^{79}+q^{78}-497 q^{77}-943 q^{76}-585 q^{75}-243 q^{74}+792 q^{73}+1766 q^{72}+1377 q^{71}+671 q^{70}-1120 q^{69}-2892 q^{68}-2758 q^{67}-1804 q^{66}+1140 q^{65}+4477 q^{64}+5209 q^{63}+4045 q^{62}-629 q^{61}-6374 q^{60}-8732 q^{59}-7897 q^{58}-1353 q^{57}+7965 q^{56}+13612 q^{55}+14172 q^{54}+5467 q^{53}-8715 q^{52}-19328 q^{51}-22895 q^{50}-12804 q^{49}+7179 q^{48}+25168 q^{47}+34266 q^{46}+24009 q^{45}-2397 q^{44}-29861 q^{43}-47360 q^{42}-39195 q^{41}-6809 q^{40}+31862 q^{39}+60924 q^{38}+57869 q^{37}+20902 q^{36}-29940 q^{35}-73286 q^{34}-78632 q^{33}-39407 q^{32}+23183 q^{31}+82637 q^{30}+99660 q^{29}+61251 q^{28}-11558 q^{27}-87846 q^{26}-118947 q^{25}-84539 q^{24}-4031 q^{23}+88237 q^{22}+134783 q^{21}+107278 q^{20}+22113 q^{19}-84114 q^{18}-146119 q^{17}-127548 q^{16}-41030 q^{15}+76399 q^{14}+152853 q^{13}+144158 q^{12}+58779 q^{11}-66536 q^{10}-154976 q^9-156376 q^8-74618 q^7+55602 q^6+154039 q^5+164612 q^4+87139 q^3-45095 q^2-150246 q-168875-97203 q^{-1} +35059 q^{-2} +145379 q^{-3} +170719 q^{-4} +104201 q^{-5} -26401 q^{-6} -139098 q^{-7} -170054 q^{-8} -109677 q^{-9} +17979 q^{-10} +132477 q^{-11} +168325 q^{-12} +113539 q^{-13} -10181 q^{-14} -124683 q^{-15} -164926 q^{-16} -116950 q^{-17} +1602 q^{-18} +115820 q^{-19} +160641 q^{-20} +119763 q^{-21} +7502 q^{-22} -105075 q^{-23} -154443 q^{-24} -122222 q^{-25} -18015 q^{-26} +92245 q^{-27} +146427 q^{-28} +123742 q^{-29} +29301 q^{-30} -76916 q^{-31} -135600 q^{-32} -123865 q^{-33} -41201 q^{-34} +59434 q^{-35} +121883 q^{-36} +121506 q^{-37} +52501 q^{-38} -40112 q^{-39} -104949 q^{-40} -116130 q^{-41} -62182 q^{-42} +20326 q^{-43} +85363 q^{-44} +106842 q^{-45} +68749 q^{-46} -1236 q^{-47} -63810 q^{-48} -93870 q^{-49} -71343 q^{-50} -15247 q^{-51} +42006 q^{-52} +77551 q^{-53} +69026 q^{-54} +27826 q^{-55} -21387 q^{-56} -59219 q^{-57} -62302 q^{-58} -35356 q^{-59} +4051 q^{-60} +40538 q^{-61} +51752 q^{-62} +37531 q^{-63} +8960 q^{-64} -23359 q^{-65} -39145 q^{-66} -34988 q^{-67} -16789 q^{-68} +9320 q^{-69} +26200 q^{-70} +29011 q^{-71} +19680 q^{-72} +628 q^{-73} -14663 q^{-74} -21232 q^{-75} -18685 q^{-76} -6407 q^{-77} +5811 q^{-78} +13504 q^{-79} +15164 q^{-80} +8382 q^{-81} +7 q^{-82} -6901 q^{-83} -10683 q^{-84} -7935 q^{-85} -2965 q^{-86} +2343 q^{-87} +6480 q^{-88} +6056 q^{-89} +3688 q^{-90} +353 q^{-91} -3173 q^{-92} -3898 q^{-93} -3265 q^{-94} -1477 q^{-95} +1172 q^{-96} +2100 q^{-97} +2246 q^{-98} +1532 q^{-99} -61 q^{-100} -810 q^{-101} -1330 q^{-102} -1265 q^{-103} -306 q^{-104} +231 q^{-105} +640 q^{-106} +754 q^{-107} +306 q^{-108} +120 q^{-109} -213 q^{-110} -468 q^{-111} -246 q^{-112} -120 q^{-113} +78 q^{-114} +196 q^{-115} +93 q^{-116} +117 q^{-117} +43 q^{-118} -97 q^{-119} -75 q^{-120} -65 q^{-121} -5 q^{-122} +43 q^{-123} -3 q^{-124} +29 q^{-125} +29 q^{-126} -9 q^{-127} -12 q^{-128} -20 q^{-129} -2 q^{-130} +12 q^{-131} -5 q^{-132} +7 q^{-134} -5 q^{-137} +3 q^{-139} - q^{-140} }[/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.

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