9 37

9_36

9_38

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 37's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 37 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X7,12,8,13 X3,11,4,10 X11,3,12,2 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8
Gauss code	-1, 4, -3, 1, -5, 6, -2, 9, -8, 3, -4, 2, -6, 5, -7, 8, -9, 7
Dowker-Thistlethwaite code	4 10 14 12 16 2 6 18 8
Conway Notation	[3,21,21]

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

[edit Notes on presentations of 9 37]

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 37"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 14 12 16 2 6 18 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[3,21,21]

In[9]:=

br = BR[K]

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

Out[9]=

$\textrm{BR}(5,\{-1,-1,2,-1,-3,2,1,4,-3,2,-3,4\})$

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]=

{ 5, 12, 5 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index $\{4,7\}$ Nakanishi index 2 Maximal Thurston-Bennequin number [-6][-5] Hyperbolic Volume 10.9894 A-Polynomial See Data:9 37/A-polynomial

[edit Notes for 9 37's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 37's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2 t^2-11 t+19-11 t^{-1} +2 t^{-2}$
Conway polynomial	$2 z^4-3 z^2+1$
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 45, 0 }
Jones polynomial	$q^4-2 q^3+5 q^2-7 q+7-8 q^{-1} +7 q^{-2} -4 q^{-3} +3 q^{-4} - q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^2 a^4+z^4 a^2+z^2 a^2+2 a^2+z^4-z^2-2-2 z^2 a^{-2} + a^{-4}$
Kauffman polynomial (db, data sources)	$a^2 z^8+z^8+3 a^3 z^7+6 a z^7+3 z^7 a^{-1} +3 a^4 z^6+5 a^2 z^6+3 z^6 a^{-2} +5 z^6+a^5 z^5-6 a^3 z^5-13 a z^5-4 z^5 a^{-1} +2 z^5 a^{-3} -8 a^4 z^4-17 a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -13 z^4-2 a^5 z^3+3 a^3 z^3+13 a z^3+6 z^3 a^{-1} -2 z^3 a^{-3} +5 a^4 z^2+14 a^2 z^2+z^2 a^{-2} -2 z^2 a^{-4} +12 z^2-2 a^3 z-7 a z-5 z a^{-1} -2 a^2+ a^{-4} -2$
The A2 invariant	$-q^{16}+q^{14}+q^{12}-q^{10}+3 q^8+q^6-3-2 q^{-4} + q^{-6} +2 q^{-8} - q^{-10} + q^{-12} + q^{-14}$
The G2 invariant	$q^{80}-2 q^{78}+4 q^{76}-7 q^{74}+5 q^{72}-4 q^{70}-4 q^{68}+16 q^{66}-23 q^{64}+29 q^{62}-22 q^{60}+6 q^{58}+16 q^{56}-38 q^{54}+52 q^{52}-49 q^{50}+24 q^{48}+8 q^{46}-33 q^{44}+50 q^{42}-44 q^{40}+24 q^{38}+7 q^{36}-27 q^{34}+33 q^{32}-28 q^{30}-6 q^{28}+40 q^{26}-46 q^{24}+41 q^{22}-18 q^{20}-17 q^{18}+57 q^{16}-71 q^{14}+65 q^{12}-48 q^{10}+4 q^8+43 q^6-65 q^4+65 q^2-47+16 q^{-2} +18 q^{-4} -39 q^{-6} +32 q^{-8} -22 q^{-10} -7 q^{-12} +33 q^{-14} -38 q^{-16} +22 q^{-18} +6 q^{-20} -33 q^{-22} +50 q^{-24} -48 q^{-26} +29 q^{-28} -8 q^{-30} -20 q^{-32} +38 q^{-34} -40 q^{-36} +34 q^{-38} -14 q^{-40} +2 q^{-42} +9 q^{-44} -15 q^{-46} +15 q^{-48} -12 q^{-50} +8 q^{-52} -2 q^{-54} - q^{-56} +3 q^{-58} -3 q^{-60} +3 q^{-62} - q^{-64} + q^{-66}$

Further Quantum Invariants

Further quantum knot invariants for 9_37.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2 q^9-q^7+3 q^5-q^3-q-2 q^{-3} +3 q^{-5} - q^{-7} + q^{-9}$
2	$q^{32}-2 q^{30}-2 q^{28}+6 q^{26}-2 q^{24}-7 q^{22}+10 q^{20}+3 q^{18}-12 q^{16}+8 q^{14}+6 q^{12}-13 q^{10}+6 q^6-3 q^4-4 q^2+5+9 q^{-2} -8 q^{-4} -2 q^{-6} +13 q^{-8} -9 q^{-10} -7 q^{-12} +11 q^{-14} -3 q^{-16} -5 q^{-18} +4 q^{-20} - q^{-24} + q^{-26}$
3	$-q^{63}+2 q^{61}+2 q^{59}-3 q^{57}-6 q^{55}+2 q^{53}+13 q^{51}-q^{49}-19 q^{47}-7 q^{45}+26 q^{43}+19 q^{41}-25 q^{39}-34 q^{37}+22 q^{35}+46 q^{33}-8 q^{31}-56 q^{29}-5 q^{27}+54 q^{25}+15 q^{23}-51 q^{21}-30 q^{19}+45 q^{17}+32 q^{15}-26 q^{13}-33 q^{11}+18 q^9+34 q^7+3 q^5-32 q^3-18 q+26 q^{-1} +32 q^{-3} -20 q^{-5} -51 q^{-7} +10 q^{-9} +56 q^{-11} +4 q^{-13} -58 q^{-15} -11 q^{-17} +49 q^{-19} +24 q^{-21} -38 q^{-23} -25 q^{-25} +25 q^{-27} +21 q^{-29} -10 q^{-31} -16 q^{-33} +4 q^{-35} +8 q^{-37} -2 q^{-39} -4 q^{-41} + q^{-43} + q^{-45} - q^{-49} + q^{-51}$
4	$q^{104}-2 q^{102}-2 q^{100}+3 q^{98}+3 q^{96}+6 q^{94}-9 q^{92}-13 q^{90}+2 q^{88}+10 q^{86}+31 q^{84}-8 q^{82}-41 q^{80}-26 q^{78}+5 q^{76}+82 q^{74}+38 q^{72}-48 q^{70}-91 q^{68}-69 q^{66}+104 q^{64}+137 q^{62}+39 q^{60}-122 q^{58}-208 q^{56}+11 q^{54}+186 q^{52}+197 q^{50}-30 q^{48}-287 q^{46}-149 q^{44}+110 q^{42}+288 q^{40}+120 q^{38}-235 q^{36}-241 q^{34}-11 q^{32}+262 q^{30}+200 q^{28}-122 q^{26}-225 q^{24}-87 q^{22}+164 q^{20}+187 q^{18}-17 q^{16}-167 q^{14}-127 q^{12}+61 q^{10}+153 q^8+85 q^6-91 q^4-160 q^2-60+115 q^{-2} +209 q^{-4} +16 q^{-6} -183 q^{-8} -204 q^{-10} +30 q^{-12} +292 q^{-14} +149 q^{-16} -128 q^{-18} -296 q^{-20} -105 q^{-22} +253 q^{-24} +229 q^{-26} +7 q^{-28} -247 q^{-30} -196 q^{-32} +108 q^{-34} +185 q^{-36} +105 q^{-38} -106 q^{-40} -159 q^{-42} -5 q^{-44} +68 q^{-46} +92 q^{-48} -3 q^{-50} -63 q^{-52} -22 q^{-54} +34 q^{-58} +9 q^{-60} -13 q^{-62} -2 q^{-64} -6 q^{-66} +6 q^{-68} + q^{-70} -3 q^{-72} +2 q^{-74} -2 q^{-76} + q^{-78} - q^{-82} + q^{-84}$
5	$-q^{155}+2 q^{153}+2 q^{151}-3 q^{149}-3 q^{147}-3 q^{145}+q^{143}+9 q^{141}+13 q^{139}-2 q^{137}-20 q^{135}-22 q^{133}-7 q^{131}+24 q^{129}+49 q^{127}+36 q^{125}-30 q^{123}-87 q^{121}-77 q^{119}+q^{117}+113 q^{115}+163 q^{113}+70 q^{111}-122 q^{109}-251 q^{107}-195 q^{105}+42 q^{103}+322 q^{101}+387 q^{99}+126 q^{97}-304 q^{95}-569 q^{93}-409 q^{91}+149 q^{89}+686 q^{87}+733 q^{85}+168 q^{83}-652 q^{81}-1042 q^{79}-583 q^{77}+432 q^{75}+1201 q^{73}+1037 q^{71}-51 q^{69}-1202 q^{67}-1392 q^{65}-396 q^{63}+997 q^{61}+1595 q^{59}+828 q^{57}-676 q^{55}-1618 q^{53}-1126 q^{51}+330 q^{49}+1462 q^{47}+1286 q^{45}+2 q^{43}-1231 q^{41}-1298 q^{39}-210 q^{37}+938 q^{35}+1177 q^{33}+372 q^{31}-702 q^{29}-1041 q^{27}-424 q^{25}+476 q^{23}+875 q^{21}+485 q^{19}-293 q^{17}-756 q^{15}-533 q^{13}+112 q^{11}+662 q^9+652 q^7+95 q^5-573 q^3-803 q-357 q^{-1} +462 q^{-3} +977 q^{-5} +682 q^{-7} -284 q^{-9} -1125 q^{-11} -1033 q^{-13} +10 q^{-15} +1166 q^{-17} +1377 q^{-19} +345 q^{-21} -1089 q^{-23} -1605 q^{-25} -733 q^{-27} +812 q^{-29} +1696 q^{-31} +1087 q^{-33} -457 q^{-35} -1549 q^{-37} -1305 q^{-39} +15 q^{-41} +1246 q^{-43} +1358 q^{-45} +342 q^{-47} -825 q^{-49} -1191 q^{-51} -576 q^{-53} +399 q^{-55} +907 q^{-57} +640 q^{-59} -75 q^{-61} -579 q^{-63} -543 q^{-65} -127 q^{-67} +285 q^{-69} +390 q^{-71} +185 q^{-73} -99 q^{-75} -223 q^{-77} -155 q^{-79} -2 q^{-81} +105 q^{-83} +96 q^{-85} +29 q^{-87} -36 q^{-89} -49 q^{-91} -20 q^{-93} +9 q^{-95} +16 q^{-97} +11 q^{-99} +4 q^{-101} -8 q^{-103} -5 q^{-105} +2 q^{-107} - q^{-109} +3 q^{-113} - q^{-115} -2 q^{-117} + q^{-119} - q^{-123} + q^{-125}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{14}+q^{12}-q^{10}+3 q^8+q^6-3-2 q^{-4} + q^{-6} +2 q^{-8} - q^{-10} + q^{-12} + q^{-14}$
1,1	$q^{44}-4 q^{42}+10 q^{40}-22 q^{38}+42 q^{36}-70 q^{34}+100 q^{32}-136 q^{30}+169 q^{28}-186 q^{26}+186 q^{24}-156 q^{22}+117 q^{20}-40 q^{18}-44 q^{16}+138 q^{14}-229 q^{12}+284 q^{10}-348 q^8+352 q^6-343 q^4+300 q^2-214+146 q^{-2} -42 q^{-4} -30 q^{-6} +104 q^{-8} -154 q^{-10} +166 q^{-12} -170 q^{-14} +152 q^{-16} -132 q^{-18} +99 q^{-20} -72 q^{-22} +50 q^{-24} -32 q^{-26} +21 q^{-28} -10 q^{-30} +6 q^{-32} -2 q^{-34} + q^{-36}$
2,0	$q^{42}-q^{40}-2 q^{38}+3 q^{34}+q^{32}-6 q^{30}+8 q^{26}+5 q^{24}-6 q^{22}-2 q^{20}+8 q^{18}+4 q^{16}-9 q^{14}-5 q^{12}+2 q^{10}-5 q^8-4 q^6+2 q^2+2+8 q^{-2} +6 q^{-4} -3 q^{-6} - q^{-8} +8 q^{-10} -11 q^{-14} +7 q^{-18} -8 q^{-22} -3 q^{-24} +4 q^{-26} +2 q^{-28} - q^{-30} + q^{-34} + q^{-36}$

A3 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2 q^{32}-4 q^{30}+6 q^{28}-8 q^{26}+10 q^{24}-10 q^{22}+10 q^{20}-7 q^{18}+6 q^{16}+q^{14}-4 q^{12}+12 q^{10}-15 q^8+18 q^6-20 q^4+18 q^2-19+11 q^{-2} -8 q^{-4} + q^{-6} +2 q^{-8} -6 q^{-10} +10 q^{-12} -10 q^{-14} +11 q^{-16} -8 q^{-18} +7 q^{-20} -4 q^{-22} +3 q^{-24} - q^{-26} + q^{-28}$
1,0	$q^{56}-2 q^{52}-2 q^{50}+2 q^{48}+4 q^{46}-2 q^{44}-7 q^{42}-2 q^{40}+9 q^{38}+8 q^{36}-4 q^{34}-9 q^{32}+2 q^{30}+12 q^{28}+7 q^{26}-9 q^{24}-9 q^{22}+2 q^{20}+7 q^{18}-4 q^{16}-10 q^{14}-2 q^{12}+6 q^{10}+2 q^8-6 q^6-q^4+8 q^2+9-3 q^{-2} -5 q^{-4} +4 q^{-6} +9 q^{-8} -2 q^{-10} -11 q^{-12} -5 q^{-14} +8 q^{-16} +7 q^{-18} -7 q^{-20} -11 q^{-22} +10 q^{-26} +4 q^{-28} -4 q^{-30} -5 q^{-32} + q^{-34} +4 q^{-36} +2 q^{-38} - q^{-40} - q^{-42} + q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2 q^{78}+4 q^{76}-7 q^{74}+5 q^{72}-4 q^{70}-4 q^{68}+16 q^{66}-23 q^{64}+29 q^{62}-22 q^{60}+6 q^{58}+16 q^{56}-38 q^{54}+52 q^{52}-49 q^{50}+24 q^{48}+8 q^{46}-33 q^{44}+50 q^{42}-44 q^{40}+24 q^{38}+7 q^{36}-27 q^{34}+33 q^{32}-28 q^{30}-6 q^{28}+40 q^{26}-46 q^{24}+41 q^{22}-18 q^{20}-17 q^{18}+57 q^{16}-71 q^{14}+65 q^{12}-48 q^{10}+4 q^8+43 q^6-65 q^4+65 q^2-47+16 q^{-2} +18 q^{-4} -39 q^{-6} +32 q^{-8} -22 q^{-10} -7 q^{-12} +33 q^{-14} -38 q^{-16} +22 q^{-18} +6 q^{-20} -33 q^{-22} +50 q^{-24} -48 q^{-26} +29 q^{-28} -8 q^{-30} -20 q^{-32} +38 q^{-34} -40 q^{-36} +34 q^{-38} -14 q^{-40} +2 q^{-42} +9 q^{-44} -15 q^{-46} +15 q^{-48} -12 q^{-50} +8 q^{-52} -2 q^{-54} - q^{-56} +3 q^{-58} -3 q^{-60} +3 q^{-62} - q^{-64} + q^{-66}$

. </div></div>

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 37"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2 t^2-11 t+19-11 t^{-1} +2 t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$2 z^4-3 z^2+1$

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]=

$\{3,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 45, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^4-2 q^3+5 q^2-7 q+7-8 q^{-1} +7 q^{-2} -4 q^{-3} +3 q^{-4} - q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^2 a^4+z^4 a^2+z^2 a^2+2 a^2+z^4-z^2-2-2 z^2 a^{-2} + a^{-4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^2 z^8+z^8+3 a^3 z^7+6 a z^7+3 z^7 a^{-1} +3 a^4 z^6+5 a^2 z^6+3 z^6 a^{-2} +5 z^6+a^5 z^5-6 a^3 z^5-13 a z^5-4 z^5 a^{-1} +2 z^5 a^{-3} -8 a^4 z^4-17 a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -13 z^4-2 a^5 z^3+3 a^3 z^3+13 a z^3+6 z^3 a^{-1} -2 z^3 a^{-3} +5 a^4 z^2+14 a^2 z^2+z^2 a^{-2} -2 z^2 a^{-4} +12 z^2-2 a^3 z-7 a z-5 z a^{-1} -2 a^2+ a^{-4} -2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n100,}

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

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 37"];

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]=

{ $2 t^2-11 t+19-11 t^{-1} +2 t^{-2}$, $q^4-2 q^3+5 q^2-7 q+7-8 q^{-1} +7 q^{-2} -4 q^{-3} +3 q^{-4} - q^{-5}$ }

In[5]:=

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

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

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

Out[5]=

{K11n100,}

In[6]:=

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

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

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-3, -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 $-12$ $-8$ $72$ $82$ $22$ $96$ $\frac{496}{3}$ $\frac{160}{3}$ $24$ $-288$ $32$ $-984$ $-264$ $-\frac{8191}{10}$ $-\frac{974}{15}$ $-\frac{1954}{5}$ $\frac{85}{2}$ $-\frac{671}{10}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of 9 37. 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                 1   -1 5               4 1   3 3             3 1     -2 1           4 4       0 -1         5 4         -1 -3       2 3           -1 -5     2 5             3 -7   1 2               -1 -9   2                 2 -11 1                   -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

<p>

$n$	$J_n$
2	$q^{12}-2 q^{11}+q^{10}+5 q^9-11 q^8+3 q^7+19 q^6-29 q^5+q^4+41 q^3-44 q^2-5 q+58-48 q^{-1} -14 q^{-2} +59 q^{-3} -39 q^{-4} -20 q^{-5} +46 q^{-6} -20 q^{-7} -18 q^{-8} +26 q^{-9} -5 q^{-10} -11 q^{-11} +9 q^{-12} -3 q^{-14} + q^{-15}$
3	$q^{24}-2 q^{23}+q^{22}+q^{21}+q^{20}-7 q^{19}+3 q^{18}+11 q^{17}-3 q^{16}-27 q^{15}+9 q^{14}+42 q^{13}+q^{12}-77 q^{11}-4 q^{10}+104 q^9+26 q^8-137 q^7-51 q^6+166 q^5+78 q^4-183 q^3-112 q^2+197 q+130-189 q^{-1} -156 q^{-2} +183 q^{-3} +165 q^{-4} -158 q^{-5} -172 q^{-6} +132 q^{-7} +172 q^{-8} -100 q^{-9} -159 q^{-10} +57 q^{-11} +151 q^{-12} -34 q^{-13} -120 q^{-14} -2 q^{-15} +100 q^{-16} +14 q^{-17} -66 q^{-18} -26 q^{-19} +44 q^{-20} +23 q^{-21} -22 q^{-22} -19 q^{-23} +11 q^{-24} +11 q^{-25} -4 q^{-26} -5 q^{-27} +3 q^{-29} - q^{-30}$
4	$q^{40}-2 q^{39}+q^{38}+q^{37}-3 q^{36}+5 q^{35}-7 q^{34}+5 q^{33}+6 q^{32}-15 q^{31}+9 q^{30}-18 q^{29}+27 q^{28}+31 q^{27}-49 q^{26}-13 q^{25}-59 q^{24}+87 q^{23}+126 q^{22}-73 q^{21}-86 q^{20}-213 q^{19}+140 q^{18}+337 q^{17}+7 q^{16}-163 q^{15}-517 q^{14}+89 q^{13}+591 q^{12}+229 q^{11}-139 q^{10}-875 q^9-102 q^8+759 q^7+506 q^6+4 q^5-1137 q^4-336 q^3+780 q^2+705 q+197-1231 q^{-1} -511 q^{-2} +680 q^{-3} +774 q^{-4} +373 q^{-5} -1163 q^{-6} -603 q^{-7} +492 q^{-8} +734 q^{-9} +523 q^{-10} -959 q^{-11} -626 q^{-12} +241 q^{-13} +596 q^{-14} +626 q^{-15} -637 q^{-16} -564 q^{-17} -32 q^{-18} +366 q^{-19} +632 q^{-20} -282 q^{-21} -396 q^{-22} -210 q^{-23} +107 q^{-24} +494 q^{-25} -25 q^{-26} -169 q^{-27} -221 q^{-28} -68 q^{-29} +275 q^{-30} +61 q^{-31} -8 q^{-32} -123 q^{-33} -101 q^{-34} +102 q^{-35} +39 q^{-36} +35 q^{-37} -37 q^{-38} -57 q^{-39} +25 q^{-40} +8 q^{-41} +20 q^{-42} -4 q^{-43} -18 q^{-44} +4 q^{-45} +5 q^{-47} -3 q^{-49} + q^{-50}$
5	$q^{60}-2 q^{59}+q^{58}+q^{57}-3 q^{56}+q^{55}+5 q^{54}-5 q^{53}+4 q^{51}-10 q^{50}-2 q^{49}+17 q^{48}+2 q^{47}+5 q^{46}-3 q^{45}-39 q^{44}-31 q^{43}+30 q^{42}+67 q^{41}+72 q^{40}+6 q^{39}-146 q^{38}-184 q^{37}-38 q^{36}+191 q^{35}+356 q^{34}+211 q^{33}-251 q^{32}-596 q^{31}-454 q^{30}+155 q^{29}+860 q^{28}+926 q^{27}+16 q^{26}-1104 q^{25}-1429 q^{24}-460 q^{23}+1226 q^{22}+2093 q^{21}+1032 q^{20}-1216 q^{19}-2660 q^{18}-1780 q^{17}+982 q^{16}+3185 q^{15}+2576 q^{14}-607 q^{13}-3544 q^{12}-3325 q^{11}+110 q^{10}+3701 q^9+4010 q^8+425 q^7-3755 q^6-4481 q^5-933 q^4+3609 q^3+4851 q^2+1391 q-3460-4996 q^{-1} -1752 q^{-2} +3163 q^{-3} +5081 q^{-4} +2059 q^{-5} -2903 q^{-6} -4986 q^{-7} -2302 q^{-8} +2518 q^{-9} +4858 q^{-10} +2522 q^{-11} -2125 q^{-12} -4596 q^{-13} -2701 q^{-14} +1618 q^{-15} +4241 q^{-16} +2861 q^{-17} -1051 q^{-18} -3791 q^{-19} -2940 q^{-20} +470 q^{-21} +3153 q^{-22} +2928 q^{-23} +182 q^{-24} -2507 q^{-25} -2764 q^{-26} -662 q^{-27} +1697 q^{-28} +2436 q^{-29} +1124 q^{-30} -1003 q^{-31} -1997 q^{-32} -1260 q^{-33} +304 q^{-34} +1440 q^{-35} +1314 q^{-36} +148 q^{-37} -909 q^{-38} -1096 q^{-39} -465 q^{-40} +425 q^{-41} +855 q^{-42} +538 q^{-43} -89 q^{-44} -531 q^{-45} -512 q^{-46} -112 q^{-47} +297 q^{-48} +378 q^{-49} +176 q^{-50} -101 q^{-51} -251 q^{-52} -177 q^{-53} +17 q^{-54} +141 q^{-55} +120 q^{-56} +28 q^{-57} -59 q^{-58} -84 q^{-59} -33 q^{-60} +29 q^{-61} +42 q^{-62} +18 q^{-63} -2 q^{-64} -18 q^{-65} -20 q^{-66} +4 q^{-67} +11 q^{-68} +3 q^{-69} -5 q^{-72} +3 q^{-74} - q^{-75}$

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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