9 25

9_24

9_26

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 25's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 25 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3849 X5,12,6,13 X9,17,10,16 X13,18,14,1 X17,14,18,15 X15,11,16,10 X11,6,12,7 X7283
Gauss code	-1, 9, -2, 1, -3, 8, -9, 2, -4, 7, -8, 3, -5, 6, -7, 4, -6, 5
Dowker-Thistlethwaite code	4 8 12 2 16 6 18 10 14
Conway Notation	[22,21,2]

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

[edit Notes on presentations of 9 25]

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 25"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3849 X5,12,6,13 X9,17,10,16 X13,18,14,1 X17,14,18,15 X15,11,16,10 X11,6,12,7 X7283

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 12 2 16 6 18 10 14

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[22,21,2]

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]=

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

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

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 1 Maximal Thurston-Bennequin number [-10][-1] Hyperbolic Volume 11.3903 A-Polynomial See Data:9 25/A-polynomial

[edit Notes for 9 25's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 25's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3 t^2+12 t-17+12 t^{-1} -3 t^{-2}$
Conway polynomial	$1-3 z^4$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 47, -2 }
Jones polynomial	$q-2+5 q^{-1} -7 q^{-2} +8 q^{-3} -8 q^{-4} +7 q^{-5} -5 q^{-6} +3 q^{-7} - q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^8+3 z^2 a^6+3 a^6-2 z^4 a^4-4 z^2 a^4-3 a^4-z^4 a^2+a^2+z^2+1$
Kauffman polynomial (db, data sources)	$z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-7 z^4 a^8+4 z^2 a^8-a^8+3 z^7 a^7-4 z^5 a^7-2 z^3 a^7+z a^7+z^8 a^6+6 z^6 a^6-18 z^4 a^6+13 z^2 a^6-3 a^6+6 z^7 a^5-10 z^5 a^5+5 z^3 a^5-z a^5+z^8 a^4+6 z^6 a^4-15 z^4 a^4+13 z^2 a^4-3 a^4+3 z^7 a^3-3 z^5 a^3+3 z^3 a^3-z a^3+3 z^6 a^2-3 z^4 a^2+2 z^2 a^2-a^2+2 z^5 a-2 z^3 a+z^4-2 z^2+1$
The A2 invariant	$-q^{26}-q^{24}+2 q^{22}+q^{18}+2 q^{16}-2 q^{14}-2 q^{10}+q^6-q^4+3 q^2+ q^{-4}$
The G2 invariant	$q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+7 q^{120}-4 q^{118}-6 q^{116}+19 q^{114}-29 q^{112}+34 q^{110}-28 q^{108}+4 q^{106}+23 q^{104}-50 q^{102}+63 q^{100}-55 q^{98}+26 q^{96}+12 q^{94}-46 q^{92}+60 q^{90}-48 q^{88}+22 q^{86}+15 q^{84}-38 q^{82}+41 q^{80}-18 q^{78}-14 q^{76}+48 q^{74}-60 q^{72}+50 q^{70}-13 q^{68}-30 q^{66}+68 q^{64}-87 q^{62}+79 q^{60}-45 q^{58}-6 q^{56}+48 q^{54}-78 q^{52}+76 q^{50}-50 q^{48}+8 q^{46}+26 q^{44}-46 q^{42}+38 q^{40}-14 q^{38}-20 q^{36}+43 q^{34}-43 q^{32}+21 q^{30}+13 q^{28}-44 q^{26}+63 q^{24}-54 q^{22}+31 q^{20}-q^{18}-27 q^{16}+43 q^{14}-43 q^{12}+34 q^{10}-14 q^8+11 q^4-16 q^2+15-11 q^{-2} +8 q^{-4} -2 q^{-6} - q^{-8} +3 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 9_25.

A1 Invariants.

Weight	Invariant
1	$-q^{17}+2 q^{15}-2 q^{13}+2 q^{11}-q^9+q^5-2 q^3+3 q- q^{-1} + q^{-3}$
2	$q^{48}-2 q^{46}-2 q^{44}+7 q^{42}-2 q^{40}-9 q^{38}+11 q^{36}+3 q^{34}-15 q^{32}+7 q^{30}+8 q^{28}-12 q^{26}+q^{24}+8 q^{22}-3 q^{20}-6 q^{18}+3 q^{16}+9 q^{14}-10 q^{12}-3 q^{10}+16 q^8-8 q^6-8 q^4+12 q^2-2-5 q^{-2} +4 q^{-4} - q^{-8} + q^{-10}$
3	$-q^{93}+2 q^{91}+2 q^{89}-3 q^{87}-7 q^{85}+2 q^{83}+16 q^{81}+q^{79}-23 q^{77}-12 q^{75}+29 q^{73}+29 q^{71}-29 q^{69}-46 q^{67}+19 q^{65}+59 q^{63}-2 q^{61}-68 q^{59}-15 q^{57}+66 q^{55}+30 q^{53}-55 q^{51}-40 q^{49}+44 q^{47}+44 q^{45}-29 q^{43}-44 q^{41}+11 q^{39}+40 q^{37}+7 q^{35}-36 q^{33}-28 q^{31}+29 q^{29}+46 q^{27}-17 q^{25}-61 q^{23}+3 q^{21}+70 q^{19}+14 q^{17}-69 q^{15}-25 q^{13}+55 q^{11}+36 q^9-39 q^7-35 q^5+24 q^3+27 q-8 q^{-1} -18 q^{-3} +3 q^{-5} +9 q^{-7} - q^{-9} -4 q^{-11} + q^{-13} + q^{-15} - q^{-19} + q^{-21}$
4	$q^{152}-2 q^{150}-2 q^{148}+3 q^{146}+3 q^{144}+7 q^{142}-9 q^{140}-16 q^{138}-q^{136}+11 q^{134}+41 q^{132}-q^{130}-47 q^{128}-44 q^{126}-10 q^{124}+101 q^{122}+72 q^{120}-33 q^{118}-123 q^{116}-129 q^{114}+98 q^{112}+190 q^{110}+105 q^{108}-121 q^{106}-295 q^{104}-53 q^{102}+211 q^{100}+299 q^{98}+33 q^{96}-349 q^{94}-252 q^{92}+79 q^{90}+374 q^{88}+218 q^{86}-243 q^{84}-338 q^{82}-85 q^{80}+301 q^{78}+291 q^{76}-93 q^{74}-289 q^{72}-165 q^{70}+170 q^{68}+258 q^{66}+32 q^{64}-191 q^{62}-191 q^{60}+36 q^{58}+196 q^{56}+152 q^{54}-78 q^{52}-215 q^{50}-129 q^{48}+113 q^{46}+288 q^{44}+76 q^{42}-208 q^{40}-305 q^{38}-32 q^{36}+357 q^{34}+254 q^{32}-95 q^{30}-388 q^{28}-210 q^{26}+269 q^{24}+329 q^{22}+82 q^{20}-286 q^{18}-287 q^{16}+78 q^{14}+232 q^{12}+171 q^{10}-100 q^8-202 q^6-36 q^4+74 q^2+119+7 q^{-2} -73 q^{-4} -33 q^{-6} -2 q^{-8} +40 q^{-10} +13 q^{-12} -14 q^{-14} -4 q^{-16} -7 q^{-18} +7 q^{-20} +2 q^{-22} -3 q^{-24} +2 q^{-26} -2 q^{-28} + q^{-30} - q^{-34} + q^{-36}$
5	$-q^{225}+2 q^{223}+2 q^{221}-3 q^{219}-3 q^{217}-3 q^{215}+9 q^{211}+16 q^{209}+q^{207}-20 q^{205}-29 q^{203}-19 q^{201}+19 q^{199}+61 q^{197}+65 q^{195}-8 q^{193}-97 q^{191}-128 q^{189}-59 q^{187}+101 q^{185}+232 q^{183}+192 q^{181}-50 q^{179}-314 q^{177}-380 q^{175}-132 q^{173}+317 q^{171}+608 q^{169}+433 q^{167}-179 q^{165}-764 q^{163}-819 q^{161}-168 q^{159}+769 q^{157}+1205 q^{155}+673 q^{153}-549 q^{151}-1463 q^{149}-1236 q^{147}+92 q^{145}+1502 q^{143}+1753 q^{141}+499 q^{139}-1299 q^{137}-2079 q^{135}-1103 q^{133}+880 q^{131}+2172 q^{129}+1600 q^{127}-375 q^{125}-2039 q^{123}-1894 q^{121}-105 q^{119}+1723 q^{117}+1983 q^{115}+488 q^{113}-1347 q^{111}-1887 q^{109}-723 q^{107}+973 q^{105}+1668 q^{103}+846 q^{101}-645 q^{99}-1424 q^{97}-882 q^{95}+374 q^{93}+1185 q^{91}+903 q^{89}-124 q^{87}-979 q^{85}-964 q^{83}-132 q^{81}+798 q^{79}+1067 q^{77}+447 q^{75}-597 q^{73}-1215 q^{71}-837 q^{69}+336 q^{67}+1349 q^{65}+1278 q^{63}+33 q^{61}-1400 q^{59}-1722 q^{57}-511 q^{55}+1301 q^{53}+2085 q^{51}+1038 q^{49}-1011 q^{47}-2255 q^{45}-1542 q^{43}+537 q^{41}+2191 q^{39}+1913 q^{37}-10 q^{35}-1840 q^{33}-2044 q^{31}-521 q^{29}+1324 q^{27}+1924 q^{25}+878 q^{23}-749 q^{21}-1559 q^{19}-1024 q^{17}+235 q^{15}+1098 q^{13}+964 q^{11}+102 q^9-649 q^7-735 q^5-264 q^3+292 q+489 q^{-1} +271 q^{-3} -86 q^{-5} -266 q^{-7} -199 q^{-9} -12 q^{-11} +122 q^{-13} +116 q^{-15} +34 q^{-17} -43 q^{-19} -59 q^{-21} -23 q^{-23} +13 q^{-25} +20 q^{-27} +12 q^{-29} +3 q^{-31} -10 q^{-33} -6 q^{-35} +3 q^{-37} +3 q^{-43} - q^{-45} -2 q^{-47} + q^{-49} - q^{-53} + q^{-55}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{26}-q^{24}+2 q^{22}+q^{18}+2 q^{16}-2 q^{14}-2 q^{10}+q^6-q^4+3 q^2+ q^{-4}$
1,1	$q^{68}-4 q^{66}+12 q^{64}-28 q^{62}+52 q^{60}-86 q^{58}+130 q^{56}-176 q^{54}+212 q^{52}-234 q^{50}+232 q^{48}-194 q^{46}+126 q^{44}-30 q^{42}-80 q^{40}+196 q^{38}-303 q^{36}+380 q^{34}-432 q^{32}+438 q^{30}-410 q^{28}+342 q^{26}-244 q^{24}+138 q^{22}-19 q^{20}-76 q^{18}+154 q^{16}-198 q^{14}+214 q^{12}-206 q^{10}+174 q^8-142 q^6+107 q^4-74 q^2+50-30 q^{-2} +21 q^{-4} -10 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12}$
2,0	$q^{66}+q^{64}-q^{62}-4 q^{60}-2 q^{58}+4 q^{56}+2 q^{54}-3 q^{52}+8 q^{48}+5 q^{46}-8 q^{44}-5 q^{42}+4 q^{40}-q^{38}-8 q^{36}-3 q^{34}+6 q^{32}+q^{30}-q^{28}+4 q^{26}+q^{24}-2 q^{22}+5 q^{20}+4 q^{18}-7 q^{16}-2 q^{14}+7 q^{12}+q^{10}-9 q^8-2 q^6+9 q^4-5+ q^{-2} +4 q^{-4} + q^{-6} - q^{-8} + q^{-12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{54}-2 q^{52}+q^{50}+3 q^{48}-7 q^{46}+3 q^{44}+4 q^{42}-11 q^{40}+6 q^{38}+7 q^{36}-10 q^{34}+4 q^{32}+7 q^{30}-5 q^{28}-q^{26}+3 q^{24}+2 q^{22}-4 q^{20}-4 q^{18}+8 q^{16}-5 q^{14}-8 q^{12}+12 q^{10}-3 q^8-7 q^6+10 q^4-3+4 q^{-2} + q^{-4} - q^{-6} + q^{-8}$
1,0,0	$-q^{35}-q^{33}-q^{31}+2 q^{29}+3 q^{25}+q^{23}+2 q^{21}-2 q^{19}-q^{17}-2 q^{15}-2 q^{13}+2 q^7-q^5+3 q^3+ q^{-1} + q^{-5}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{54}+2 q^{52}-5 q^{50}+7 q^{48}-9 q^{46}+11 q^{44}-12 q^{42}+11 q^{40}-8 q^{38}+5 q^{36}+2 q^{34}-6 q^{32}+13 q^{30}-17 q^{28}+21 q^{26}-23 q^{24}+20 q^{22}-18 q^{20}+12 q^{18}-8 q^{16}+q^{14}+4 q^{12}-8 q^{10}+11 q^8-11 q^6+12 q^4-8 q^2+7-4 q^{-2} +3 q^{-4} - q^{-6} + q^{-8}$
1,0	$q^{88}-2 q^{84}-2 q^{82}+3 q^{80}+5 q^{78}-2 q^{76}-8 q^{74}-3 q^{72}+9 q^{70}+8 q^{68}-7 q^{66}-12 q^{64}+q^{62}+13 q^{60}+6 q^{58}-10 q^{56}-8 q^{54}+5 q^{52}+9 q^{50}-2 q^{48}-8 q^{46}+8 q^{42}+2 q^{40}-8 q^{38}-3 q^{36}+7 q^{34}+6 q^{32}-6 q^{30}-8 q^{28}+4 q^{26}+10 q^{24}-2 q^{22}-12 q^{20}-3 q^{18}+11 q^{16}+8 q^{14}-7 q^{12}-11 q^{10}+q^8+11 q^6+4 q^4-4 q^2-5+ q^{-2} +4 q^{-4} +2 q^{-6} - q^{-8} - q^{-10} + q^{-14}$

G2 Invariants.

Weight

Invariant

1,0

$q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+7 q^{120}-4 q^{118}-6 q^{116}+19 q^{114}-29 q^{112}+34 q^{110}-28 q^{108}+4 q^{106}+23 q^{104}-50 q^{102}+63 q^{100}-55 q^{98}+26 q^{96}+12 q^{94}-46 q^{92}+60 q^{90}-48 q^{88}+22 q^{86}+15 q^{84}-38 q^{82}+41 q^{80}-18 q^{78}-14 q^{76}+48 q^{74}-60 q^{72}+50 q^{70}-13 q^{68}-30 q^{66}+68 q^{64}-87 q^{62}+79 q^{60}-45 q^{58}-6 q^{56}+48 q^{54}-78 q^{52}+76 q^{50}-50 q^{48}+8 q^{46}+26 q^{44}-46 q^{42}+38 q^{40}-14 q^{38}-20 q^{36}+43 q^{34}-43 q^{32}+21 q^{30}+13 q^{28}-44 q^{26}+63 q^{24}-54 q^{22}+31 q^{20}-q^{18}-27 q^{16}+43 q^{14}-43 q^{12}+34 q^{10}-14 q^8+11 q^4-16 q^2+15-11 q^{-2} +8 q^{-4} -2 q^{-6} - q^{-8} +3 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18}$

. </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 25"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3 t^2+12 t-17+12 t^{-1} -3 t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$1-3 z^4$

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]=

{ 47, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q-2+5 q^{-1} -7 q^{-2} +8 q^{-3} -8 q^{-4} +7 q^{-5} -5 q^{-6} +3 q^{-7} - q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^8+3 z^2 a^6+3 a^6-2 z^4 a^4-4 z^2 a^4-3 a^4-z^4 a^2+a^2+z^2+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-7 z^4 a^8+4 z^2 a^8-a^8+3 z^7 a^7-4 z^5 a^7-2 z^3 a^7+z a^7+z^8 a^6+6 z^6 a^6-18 z^4 a^6+13 z^2 a^6-3 a^6+6 z^7 a^5-10 z^5 a^5+5 z^3 a^5-z a^5+z^8 a^4+6 z^6 a^4-15 z^4 a^4+13 z^2 a^4-3 a^4+3 z^7 a^3-3 z^5 a^3+3 z^3 a^3-z a^3+3 z^6 a^2-3 z^4 a^2+2 z^2 a^2-a^2+2 z^5 a-2 z^3 a+z^4-2 z^2+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n134,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n25,}

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 25"];

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]=

{ $-3 t^2+12 t-17+12 t^{-1} -3 t^{-2}$, $q-2+5 q^{-1} -7 q^{-2} +8 q^{-3} -8 q^{-4} +7 q^{-5} -5 q^{-6} +3 q^{-7} - q^{-8}$ }

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]=

{K11n134,}

In[6]:=

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

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

Out[6]=

{K11n25,}

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 $0$ $-8$ $0$ $64$ $24$ $0$ $-\frac{656}{3}$ $-\frac{128}{3}$ $-72$ $0$ $32$ $0$ $0$ $528$ $-\frac{248}{3}$ $344$ $48$ $32$

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 $s=$-2 is the signature of 9 25. 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 3                   1 1 1                 1   -1 -1               4 1   3 -3             4 2     -2 -5           4 3       1 -7         4 4         0 -9       3 4           -1 -11     2 4             2 -13   1 3               -2 -15   2                 2 -17 1                   -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

<p>

$n$	$J_n$
2	$q^4-2 q^3+q^2+5 q-11+4 q^{-1} +19 q^{-2} -31 q^{-3} +4 q^{-4} +43 q^{-5} -50 q^{-6} -3 q^{-7} +62 q^{-8} -56 q^{-9} -12 q^{-10} +65 q^{-11} -45 q^{-12} -19 q^{-13} +52 q^{-14} -25 q^{-15} -20 q^{-16} +30 q^{-17} -7 q^{-18} -12 q^{-19} +10 q^{-20} -3 q^{-22} + q^{-23}$
3	$q^9-2 q^8+q^7+q^6+q^5-7 q^4+4 q^3+11 q^2-5 q-28+14 q^{-1} +46 q^{-2} -8 q^{-3} -87 q^{-4} +10 q^{-5} +121 q^{-6} +11 q^{-7} -167 q^{-8} -34 q^{-9} +204 q^{-10} +67 q^{-11} -234 q^{-12} -98 q^{-13} +248 q^{-14} +130 q^{-15} -251 q^{-16} -155 q^{-17} +240 q^{-18} +173 q^{-19} -218 q^{-20} -184 q^{-21} +185 q^{-22} +188 q^{-23} -145 q^{-24} -184 q^{-25} +101 q^{-26} +173 q^{-27} -60 q^{-28} -148 q^{-29} +20 q^{-30} +120 q^{-31} +6 q^{-32} -87 q^{-33} -20 q^{-34} +55 q^{-35} +23 q^{-36} -29 q^{-37} -20 q^{-38} +14 q^{-39} +12 q^{-40} -5 q^{-41} -5 q^{-42} +3 q^{-44} - q^{-45}$
4	$q^{16}-2 q^{15}+q^{14}+q^{13}-3 q^{12}+5 q^{11}-7 q^{10}+6 q^9+6 q^8-17 q^7+8 q^6-17 q^5+33 q^4+33 q^3-59 q^2-23 q-57+113 q^{-1} +145 q^{-2} -104 q^{-3} -133 q^{-4} -223 q^{-5} +215 q^{-6} +416 q^{-7} -43 q^{-8} -287 q^{-9} -588 q^{-10} +216 q^{-11} +784 q^{-12} +204 q^{-13} -347 q^{-14} -1067 q^{-15} +38 q^{-16} +1077 q^{-17} +553 q^{-18} -244 q^{-19} -1456 q^{-20} -235 q^{-21} +1174 q^{-22} +837 q^{-23} -32 q^{-24} -1631 q^{-25} -477 q^{-26} +1088 q^{-27} +974 q^{-28} +198 q^{-29} -1587 q^{-30} -637 q^{-31} +861 q^{-32} +974 q^{-33} +421 q^{-34} -1361 q^{-35} -725 q^{-36} +526 q^{-37} +850 q^{-38} +617 q^{-39} -977 q^{-40} -715 q^{-41} +140 q^{-42} +597 q^{-43} +712 q^{-44} -516 q^{-45} -559 q^{-46} -155 q^{-47} +266 q^{-48} +615 q^{-49} -134 q^{-50} -293 q^{-51} -243 q^{-52} +2 q^{-53} +373 q^{-54} +40 q^{-55} -67 q^{-56} -158 q^{-57} -90 q^{-58} +146 q^{-59} +46 q^{-60} +23 q^{-61} -53 q^{-62} -61 q^{-63} +35 q^{-64} +12 q^{-65} +20 q^{-66} -7 q^{-67} -19 q^{-68} +5 q^{-69} +5 q^{-71} -3 q^{-73} + q^{-74}$
5	$q^{25}-2 q^{24}+q^{23}+q^{22}-3 q^{21}+q^{20}+5 q^{19}-5 q^{18}+q^{17}+4 q^{16}-12 q^{15}-3 q^{14}+18 q^{13}+4 q^{12}+9 q^{11}-3 q^{10}-48 q^9-39 q^8+34 q^7+81 q^6+91 q^5+3 q^4-182 q^3-226 q^2-33 q+261+448 q^{-1} +221 q^{-2} -379 q^{-3} -782 q^{-4} -504 q^{-5} +347 q^{-6} +1199 q^{-7} +1083 q^{-8} -245 q^{-9} -1645 q^{-10} -1763 q^{-11} -188 q^{-12} +2009 q^{-13} +2710 q^{-14} +801 q^{-15} -2245 q^{-16} -3608 q^{-17} -1711 q^{-18} +2213 q^{-19} +4540 q^{-20} +2724 q^{-21} -1967 q^{-22} -5262 q^{-23} -3790 q^{-24} +1500 q^{-25} +5784 q^{-26} +4773 q^{-27} -920 q^{-28} -6046 q^{-29} -5602 q^{-30} +289 q^{-31} +6106 q^{-32} +6206 q^{-33} +325 q^{-34} -5975 q^{-35} -6615 q^{-36} -884 q^{-37} +5728 q^{-38} +6824 q^{-39} +1369 q^{-40} -5355 q^{-41} -6884 q^{-42} -1814 q^{-43} +4896 q^{-44} +6809 q^{-45} +2224 q^{-46} -4328 q^{-47} -6602 q^{-48} -2625 q^{-49} +3640 q^{-50} +6267 q^{-51} +3003 q^{-52} -2837 q^{-53} -5780 q^{-54} -3320 q^{-55} +1944 q^{-56} +5103 q^{-57} +3543 q^{-58} -1002 q^{-59} -4285 q^{-60} -3580 q^{-61} +116 q^{-62} +3314 q^{-63} +3398 q^{-64} +662 q^{-65} -2310 q^{-66} -3008 q^{-67} -1176 q^{-68} +1331 q^{-69} +2422 q^{-70} +1442 q^{-71} -512 q^{-72} -1754 q^{-73} -1427 q^{-74} -79 q^{-75} +1094 q^{-76} +1215 q^{-77} +402 q^{-78} -532 q^{-79} -895 q^{-80} -515 q^{-81} +157 q^{-82} +564 q^{-83} +457 q^{-84} +53 q^{-85} -283 q^{-86} -340 q^{-87} -134 q^{-88} +115 q^{-89} +209 q^{-90} +119 q^{-91} -19 q^{-92} -98 q^{-93} -94 q^{-94} -16 q^{-95} +49 q^{-96} +50 q^{-97} +12 q^{-98} -9 q^{-99} -21 q^{-100} -20 q^{-101} +7 q^{-102} +12 q^{-103} +2 q^{-104} -5 q^{-107} +3 q^{-109} - q^{-110}$
6	$q^{36}-2 q^{35}+q^{34}+q^{33}-3 q^{32}+q^{31}+q^{30}+7 q^{29}-10 q^{28}-q^{27}+9 q^{26}-13 q^{25}+q^{24}+9 q^{23}+28 q^{22}-24 q^{21}-19 q^{20}+13 q^{19}-46 q^{18}-4 q^{17}+51 q^{16}+119 q^{15}-15 q^{14}-69 q^{13}-52 q^{12}-220 q^{11}-82 q^{10}+166 q^9+468 q^8+248 q^7-36 q^6-281 q^5-904 q^4-671 q^3+129 q^2+1296 q+1406+843 q^{-1} -286 q^{-2} -2463 q^{-3} -2814 q^{-4} -1280 q^{-5} +2022 q^{-6} +3948 q^{-7} +4080 q^{-8} +1692 q^{-9} -3997 q^{-10} -7111 q^{-11} -6048 q^{-12} +341 q^{-13} +6591 q^{-14} +10268 q^{-15} +7942 q^{-16} -2671 q^{-17} -11812 q^{-18} -14534 q^{-19} -6166 q^{-20} +6068 q^{-21} +17110 q^{-22} +18345 q^{-23} +3858 q^{-24} -13313 q^{-25} -23788 q^{-26} -16818 q^{-27} +302 q^{-28} +20740 q^{-29} +29289 q^{-30} +14325 q^{-31} -9763 q^{-32} -29726 q^{-33} -27651 q^{-34} -8996 q^{-35} +19526 q^{-36} +36660 q^{-37} +24647 q^{-38} -2982 q^{-39} -30850 q^{-40} -34893 q^{-41} -17980 q^{-42} +15180 q^{-43} +39209 q^{-44} +31596 q^{-45} +3769 q^{-46} -28659 q^{-47} -37739 q^{-48} -24145 q^{-49} +10297 q^{-50} +38309 q^{-51} +34815 q^{-52} +8783 q^{-53} -25138 q^{-54} -37536 q^{-55} -27577 q^{-56} +5901 q^{-57} +35503 q^{-58} +35603 q^{-59} +12527 q^{-60} -20802 q^{-61} -35488 q^{-62} -29521 q^{-63} +1281 q^{-64} +30969 q^{-65} +34848 q^{-66} +16129 q^{-67} -14860 q^{-68} -31473 q^{-69} -30505 q^{-70} -4448 q^{-71} +23891 q^{-72} +32060 q^{-73} +19627 q^{-74} -6779 q^{-75} -24549 q^{-76} -29514 q^{-77} -10684 q^{-78} +14058 q^{-79} +25993 q^{-80} +21279 q^{-81} +2141 q^{-82} -14620 q^{-83} -24835 q^{-84} -14902 q^{-85} +3352 q^{-86} +16539 q^{-87} +18845 q^{-88} +8669 q^{-89} -3934 q^{-90} -16341 q^{-91} -14471 q^{-92} -4504 q^{-93} +6245 q^{-94} +12251 q^{-95} +10001 q^{-96} +3585 q^{-97} -6897 q^{-98} -9567 q^{-99} -6822 q^{-100} -880 q^{-101} +4656 q^{-102} +6712 q^{-103} +5644 q^{-104} -498 q^{-105} -3620 q^{-106} -4667 q^{-107} -2986 q^{-108} -165 q^{-109} +2389 q^{-110} +3804 q^{-111} +1480 q^{-112} -58 q^{-113} -1619 q^{-114} -1905 q^{-115} -1366 q^{-116} -17 q^{-117} +1418 q^{-118} +950 q^{-119} +732 q^{-120} -54 q^{-121} -510 q^{-122} -820 q^{-123} -452 q^{-124} +252 q^{-125} +207 q^{-126} +391 q^{-127} +194 q^{-128} +39 q^{-129} -248 q^{-130} -225 q^{-131} +3 q^{-132} -31 q^{-133} +92 q^{-134} +80 q^{-135} +76 q^{-136} -45 q^{-137} -57 q^{-138} - q^{-139} -29 q^{-140} +9 q^{-141} +12 q^{-142} +29 q^{-143} -7 q^{-144} -12 q^{-145} +5 q^{-146} -7 q^{-147} +5 q^{-150} -3 q^{-152} + q^{-153}$
7	$q^{49}-2 q^{48}+q^{47}+q^{46}-3 q^{45}+q^{44}+q^{43}+3 q^{42}+2 q^{41}-12 q^{40}+4 q^{39}+8 q^{38}-9 q^{37}+2 q^{36}+2 q^{35}+15 q^{34}+8 q^{33}-47 q^{32}-4 q^{31}+20 q^{30}-8 q^{29}+23 q^{28}+17 q^{27}+58 q^{26}+23 q^{25}-143 q^{24}-96 q^{23}-36 q^{22}-9 q^{21}+152 q^{20}+191 q^{19}+277 q^{18}+144 q^{17}-377 q^{16}-540 q^{15}-567 q^{14}-327 q^{13}+414 q^{12}+954 q^{11}+1430 q^{10}+1109 q^9-402 q^8-1739 q^7-2768 q^6-2573 q^5-380 q^4+2316 q^3+5012 q^2+5548 q+2388-2370 q^{-1} -7721 q^{-2} -10214 q^{-3} -6877 q^{-4} +623 q^{-5} +10484 q^{-6} +16861 q^{-7} +14375 q^{-8} +4244 q^{-9} -11540 q^{-10} -24697 q^{-11} -25884 q^{-12} -13868 q^{-13} +9592 q^{-14} +32412 q^{-15} +40381 q^{-16} +28983 q^{-17} -1848 q^{-18} -37526 q^{-19} -57242 q^{-20} -49960 q^{-21} -12323 q^{-22} +38102 q^{-23} +73166 q^{-24} +74995 q^{-25} +34149 q^{-26} -31469 q^{-27} -86279 q^{-28} -102321 q^{-29} -61840 q^{-30} +17521 q^{-31} +93437 q^{-32} +128109 q^{-33} +93677 q^{-34} +3851 q^{-35} -93581 q^{-36} -150173 q^{-37} -126202 q^{-38} -30131 q^{-39} +86429 q^{-40} +165930 q^{-41} +156317 q^{-42} +58993 q^{-43} -73228 q^{-44} -174769 q^{-45} -181702 q^{-46} -87249 q^{-47} +56185 q^{-48} +176989 q^{-49} +200831 q^{-50} +112483 q^{-51} -37527 q^{-52} -173938 q^{-53} -213557 q^{-54} -133252 q^{-55} +19431 q^{-56} +167410 q^{-57} +220610 q^{-58} +149010 q^{-59} -3390 q^{-60} -159021 q^{-61} -223169 q^{-62} -160099 q^{-63} -10179 q^{-64} +150009 q^{-65} +222740 q^{-66} +167515 q^{-67} +21258 q^{-68} -141101 q^{-69} -220251 q^{-70} -172291 q^{-71} -30665 q^{-72} +132140 q^{-73} +216507 q^{-74} +175603 q^{-75} +39315 q^{-76} -122795 q^{-77} -211642 q^{-78} -178042 q^{-79} -48156 q^{-80} +112108 q^{-81} +205276 q^{-82} +180019 q^{-83} +58062 q^{-84} -99274 q^{-85} -196848 q^{-86} -181260 q^{-87} -69252 q^{-88} +83491 q^{-89} +185321 q^{-90} +181081 q^{-91} +81626 q^{-92} -64344 q^{-93} -169860 q^{-94} -178470 q^{-95} -94293 q^{-96} +42096 q^{-97} +149911 q^{-98} +172010 q^{-99} +105726 q^{-100} -17597 q^{-101} -125246 q^{-102} -160638 q^{-103} -114329 q^{-104} -7281 q^{-105} +96865 q^{-106} +143639 q^{-107} +117833 q^{-108} +30299 q^{-109} -66111 q^{-110} -121194 q^{-111} -115123 q^{-112} -49018 q^{-113} +35809 q^{-114} +94658 q^{-115} +105401 q^{-116} +61050 q^{-117} -8534 q^{-118} -66080 q^{-119} -89559 q^{-120} -65447 q^{-121} -12985 q^{-122} +38731 q^{-123} +69413 q^{-124} +62059 q^{-125} +26964 q^{-126} -15169 q^{-127} -47756 q^{-128} -52635 q^{-129} -33026 q^{-130} -2300 q^{-131} +27660 q^{-132} +39557 q^{-133} +32200 q^{-134} +12759 q^{-135} -11470 q^{-136} -25676 q^{-137} -26572 q^{-138} -16839 q^{-139} +373 q^{-140} +13644 q^{-141} +18824 q^{-142} +16002 q^{-143} +5406 q^{-144} -4743 q^{-145} -11038 q^{-146} -12491 q^{-147} -7209 q^{-148} -564 q^{-149} +5046 q^{-150} +8191 q^{-151} +6327 q^{-152} +2770 q^{-153} -1113 q^{-154} -4371 q^{-155} -4446 q^{-156} -3090 q^{-157} -820 q^{-158} +1866 q^{-159} +2551 q^{-160} +2323 q^{-161} +1311 q^{-162} -374 q^{-163} -1079 q^{-164} -1465 q^{-165} -1226 q^{-166} -177 q^{-167} +368 q^{-168} +717 q^{-169} +757 q^{-170} +279 q^{-171} +71 q^{-172} -258 q^{-173} -484 q^{-174} -237 q^{-175} -99 q^{-176} +92 q^{-177} +198 q^{-178} +94 q^{-179} +115 q^{-180} +37 q^{-181} -100 q^{-182} -75 q^{-183} -62 q^{-184} -4 q^{-185} +42 q^{-186} -3 q^{-187} +29 q^{-188} +29 q^{-189} -9 q^{-190} -12 q^{-191} -20 q^{-192} -2 q^{-193} +12 q^{-194} -5 q^{-195} +7 q^{-197} -5 q^{-200} +3 q^{-202} - q^{-203}$

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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