9 31

9_30

9_32

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 31's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 31 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,1,12,18 X5,13,6,12 X17,7,18,6 X7,14,8,15 X13,16,14,17 X15,8,16,9 X9,2,10,3
Gauss code	-1, 9, -2, 1, -4, 5, -6, 8, -9, 2, -3, 4, -7, 6, -8, 7, -5, 3
Dowker-Thistlethwaite code	4 10 12 14 2 18 16 8 6
Conway Notation	[2111112]

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

[edit Notes on presentations of 9 31]

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 31"];

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 X17,7,18,6 X7,14,8,15 X13,16,14,17 X15,8,16,9 X9,2,10,3

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 12 14 2 18 16 8 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[2111112]

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]=

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

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

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 $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-2] Hyperbolic Volume 11.6863 A-Polynomial See Data:9 31/A-polynomial

[edit Notes for 9 31's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 31's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3}$
Conway polynomial	$z^6+z^4+2 z^2+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 55, -2 }
Jones polynomial	$-q^2+3 q-5+8 q^{-1} -9 q^{-2} +10 q^{-3} -8 q^{-4} +6 q^{-5} -4 q^{-6} + q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^2 a^6-2 z^4 a^4-4 z^2 a^4-2 a^4+z^6 a^2+4 z^4 a^2+7 z^2 a^2+4 a^2-z^4-2 z^2-1$
Kauffman polynomial (db, data sources)	$z^4 a^8+4 z^5 a^7-4 z^3 a^7+6 z^6 a^6-8 z^4 a^6+3 z^2 a^6+4 z^7 a^5+z^5 a^5-8 z^3 a^5+3 z a^5+z^8 a^4+11 z^6 a^4-23 z^4 a^4+13 z^2 a^4-2 a^4+7 z^7 a^3-7 z^5 a^3-5 z^3 a^3+5 z a^3+z^8 a^2+8 z^6 a^2-21 z^4 a^2+15 z^2 a^2-4 a^2+3 z^7 a-3 z^5 a-3 z^3 a+3 z a+3 z^6-7 z^4+5 z^2-1+z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1}$
The A2 invariant	$q^{22}-q^{20}-2 q^{18}+q^{16}-2 q^{14}+q^{12}+q^{10}+3 q^6-q^4+3 q^2- q^{-2} + q^{-4} - q^{-6}$
The G2 invariant	$q^{114}-3 q^{112}+6 q^{110}-10 q^{108}+8 q^{106}-4 q^{104}-5 q^{102}+22 q^{100}-33 q^{98}+45 q^{96}-41 q^{94}+16 q^{92}+17 q^{90}-54 q^{88}+80 q^{86}-86 q^{84}+65 q^{82}-20 q^{80}-33 q^{78}+75 q^{76}-90 q^{74}+70 q^{72}-28 q^{70}-23 q^{68}+48 q^{66}-52 q^{64}+24 q^{62}+26 q^{60}-67 q^{58}+82 q^{56}-60 q^{54}+2 q^{52}+65 q^{50}-121 q^{48}+136 q^{46}-108 q^{44}+48 q^{42}+32 q^{40}-96 q^{38}+129 q^{36}-115 q^{34}+68 q^{32}-4 q^{30}-51 q^{28}+73 q^{26}-55 q^{24}+22 q^{22}+29 q^{20}-57 q^{18}+59 q^{16}-26 q^{14}-24 q^{12}+72 q^{10}-94 q^8+83 q^6-43 q^4-9 q^2+55-80 q^{-2} +81 q^{-4} -55 q^{-6} +18 q^{-8} +13 q^{-10} -35 q^{-12} +38 q^{-14} -31 q^{-16} +19 q^{-18} -5 q^{-20} -5 q^{-22} +7 q^{-24} -8 q^{-26} +5 q^{-28} -2 q^{-30} + q^{-32}$

Further Quantum Invariants

Further quantum knot invariants for 9_31.

A1 Invariants.

Weight	Invariant
1	$q^{15}-3 q^{13}+2 q^{11}-2 q^9+2 q^7+q^5-q^3+3 q-2 q^{-1} +2 q^{-3} - q^{-5}$
2	$q^{42}-3 q^{40}-q^{38}+10 q^{36}-7 q^{34}-8 q^{32}+18 q^{30}-7 q^{28}-15 q^{26}+18 q^{24}-q^{22}-15 q^{20}+7 q^{18}+6 q^{16}-5 q^{14}-7 q^{12}+11 q^{10}+8 q^8-17 q^6+7 q^4+15 q^2-18+14 q^{-4} -9 q^{-6} -3 q^{-8} +7 q^{-10} -2 q^{-12} -2 q^{-14} + q^{-16}$
3	$q^{81}-3 q^{79}-q^{77}+7 q^{75}+5 q^{73}-11 q^{71}-18 q^{69}+20 q^{67}+29 q^{65}-22 q^{63}-49 q^{61}+21 q^{59}+71 q^{57}-13 q^{55}-89 q^{53}-q^{51}+101 q^{49}+18 q^{47}-96 q^{45}-38 q^{43}+87 q^{41}+48 q^{39}-63 q^{37}-60 q^{35}+35 q^{33}+56 q^{31}-4 q^{29}-56 q^{27}-27 q^{25}+50 q^{23}+54 q^{21}-36 q^{19}-76 q^{17}+23 q^{15}+94 q^{13}-3 q^{11}-99 q^9-15 q^7+96 q^5+36 q^3-84 q-47 q^{-1} +62 q^{-3} +54 q^{-5} -41 q^{-7} -49 q^{-9} +20 q^{-11} +39 q^{-13} -6 q^{-15} -26 q^{-17} -2 q^{-19} +16 q^{-21} +3 q^{-23} -7 q^{-25} -3 q^{-27} +2 q^{-29} +2 q^{-31} - q^{-33}$
4	$q^{132}-3 q^{130}-q^{128}+7 q^{126}+2 q^{124}+q^{122}-21 q^{120}-10 q^{118}+29 q^{116}+23 q^{114}+19 q^{112}-72 q^{110}-63 q^{108}+58 q^{106}+96 q^{104}+89 q^{102}-148 q^{100}-197 q^{98}+31 q^{96}+212 q^{94}+268 q^{92}-164 q^{90}-396 q^{88}-127 q^{86}+273 q^{84}+507 q^{82}-34 q^{80}-504 q^{78}-358 q^{76}+169 q^{74}+631 q^{72}+184 q^{70}-404 q^{68}-484 q^{66}-45 q^{64}+522 q^{62}+330 q^{60}-156 q^{58}-424 q^{56}-226 q^{54}+256 q^{52}+343 q^{50}+102 q^{48}-247 q^{46}-320 q^{44}-43 q^{42}+286 q^{40}+318 q^{38}-53 q^{36}-359 q^{34}-306 q^{32}+198 q^{30}+474 q^{28}+150 q^{26}-327 q^{24}-521 q^{22}+46 q^{20}+524 q^{18}+348 q^{16}-187 q^{14}-617 q^{12}-162 q^{10}+402 q^8+461 q^6+49 q^4-515 q^2-312+152 q^{-2} +395 q^{-4} +229 q^{-6} -269 q^{-8} -292 q^{-10} -57 q^{-12} +201 q^{-14} +235 q^{-16} -58 q^{-18} -149 q^{-20} -106 q^{-22} +39 q^{-24} +128 q^{-26} +19 q^{-28} -34 q^{-30} -59 q^{-32} -13 q^{-34} +41 q^{-36} +14 q^{-38} +2 q^{-40} -16 q^{-42} -10 q^{-44} +7 q^{-46} +3 q^{-48} +3 q^{-50} -2 q^{-52} -2 q^{-54} + q^{-56}$
5	$q^{195}-3 q^{193}-q^{191}+7 q^{189}+2 q^{187}-2 q^{185}-9 q^{183}-13 q^{181}-q^{179}+29 q^{177}+37 q^{175}-3 q^{173}-55 q^{171}-74 q^{169}-16 q^{167}+86 q^{165}+165 q^{163}+74 q^{161}-160 q^{159}-290 q^{157}-171 q^{155}+177 q^{153}+496 q^{151}+398 q^{149}-184 q^{147}-756 q^{145}-715 q^{143}+53 q^{141}+1009 q^{139}+1206 q^{137}+231 q^{135}-1204 q^{133}-1773 q^{131}-710 q^{129}+1230 q^{127}+2331 q^{125}+1378 q^{123}-1030 q^{121}-2774 q^{119}-2105 q^{117}+580 q^{115}+2947 q^{113}+2787 q^{111}+59 q^{109}-2838 q^{107}-3243 q^{105}-771 q^{103}+2405 q^{101}+3437 q^{99}+1405 q^{97}-1779 q^{95}-3268 q^{93}-1888 q^{91}+1030 q^{89}+2882 q^{87}+2136 q^{85}-324 q^{83}-2268 q^{81}-2189 q^{79}-336 q^{77}+1640 q^{75}+2098 q^{73}+849 q^{71}-986 q^{69}-1933 q^{67}-1298 q^{65}+391 q^{63}+1769 q^{61}+1674 q^{59}+133 q^{57}-1609 q^{55}-2038 q^{53}-646 q^{51}+1454 q^{49}+2397 q^{47}+1152 q^{45}-1274 q^{43}-2698 q^{41}-1691 q^{39}+994 q^{37}+2926 q^{35}+2239 q^{33}-586 q^{31}-2993 q^{29}-2738 q^{27}+37 q^{25}+2839 q^{23}+3121 q^{21}+621 q^{19}-2429 q^{17}-3304 q^{15}-1265 q^{13}+1794 q^{11}+3190 q^9+1818 q^7-1007 q^5-2819 q^3-2140 q+240 q^{-1} +2189 q^{-3} +2180 q^{-5} +426 q^{-7} -1472 q^{-9} -1953 q^{-11} -829 q^{-13} +769 q^{-15} +1529 q^{-17} +981 q^{-19} -222 q^{-21} -1034 q^{-23} -908 q^{-25} -123 q^{-27} +590 q^{-29} +702 q^{-31} +260 q^{-33} -254 q^{-35} -458 q^{-37} -277 q^{-39} +66 q^{-41} +261 q^{-43} +205 q^{-45} +20 q^{-47} -117 q^{-49} -132 q^{-51} -46 q^{-53} +49 q^{-55} +71 q^{-57} +33 q^{-59} -13 q^{-61} -29 q^{-63} -23 q^{-65} -2 q^{-67} +16 q^{-69} +10 q^{-71} -3 q^{-75} -3 q^{-77} -3 q^{-79} +2 q^{-81} +2 q^{-83} - q^{-85}$

A2 Invariants.

Weight	Invariant
1,0	$q^{22}-q^{20}-2 q^{18}+q^{16}-2 q^{14}+q^{12}+q^{10}+3 q^6-q^4+3 q^2- q^{-2} + q^{-4} - q^{-6}$
1,1	$q^{60}-6 q^{58}+18 q^{56}-38 q^{54}+69 q^{52}-120 q^{50}+186 q^{48}-254 q^{46}+324 q^{44}-382 q^{42}+414 q^{40}-396 q^{38}+326 q^{36}-208 q^{34}+40 q^{32}+156 q^{30}-368 q^{28}+554 q^{26}-708 q^{24}+792 q^{22}-811 q^{20}+756 q^{18}-632 q^{16}+466 q^{14}-253 q^{12}+60 q^{10}+126 q^8-270 q^6+367 q^4-406 q^2+396-354 q^{-2} +292 q^{-4} -218 q^{-6} +150 q^{-8} -96 q^{-10} +54 q^{-12} -28 q^{-14} +12 q^{-16} -4 q^{-18} + q^{-20}$
2,0	$q^{56}-q^{54}-3 q^{52}+5 q^{48}+3 q^{46}-6 q^{44}+8 q^{40}-9 q^{36}+7 q^{32}-5 q^{30}-10 q^{28}+2 q^{26}+2 q^{24}-7 q^{22}+3 q^{20}+7 q^{18}-q^{16}+q^{14}+10 q^{12}+2 q^{10}-8 q^8+2 q^6+10 q^4-4 q^2-8+5 q^{-2} +5 q^{-4} -4 q^{-6} -3 q^{-8} +2 q^{-10} +2 q^{-12} -2 q^{-14} - q^{-16} + q^{-18}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{48}-3 q^{46}+7 q^{42}-8 q^{40}+q^{38}+13 q^{36}-13 q^{34}-3 q^{32}+14 q^{30}-12 q^{28}-6 q^{26}+10 q^{24}-5 q^{22}-6 q^{20}+6 q^{16}-6 q^{12}+15 q^{10}+7 q^8-13 q^6+12 q^4+5 q^2-14+6 q^{-2} +3 q^{-4} -8 q^{-6} +3 q^{-8} + q^{-10} -2 q^{-12} + q^{-14}$
1,0,0	$q^{29}-q^{27}-2 q^{23}+q^{21}-3 q^{19}+q^{17}-q^{15}+q^{13}+q^{11}+2 q^9+3 q^7+3 q^3-q+ q^{-1} -2 q^{-3} + q^{-5} - q^{-7}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{62}-q^{60}-3 q^{58}+q^{56}+4 q^{54}-q^{52}-4 q^{50}+6 q^{48}+8 q^{46}-6 q^{44}-8 q^{42}+9 q^{40}+2 q^{38}-13 q^{36}-2 q^{34}+7 q^{32}-8 q^{30}-12 q^{28}+4 q^{26}+q^{24}-10 q^{22}+4 q^{20}+17 q^{18}-2 q^{16}+19 q^{12}+9 q^{10}-9 q^8+2 q^6+9 q^4-4 q^2-10+ q^{-2} +3 q^{-4} -5 q^{-6} -2 q^{-8} +3 q^{-10} - q^{-14} + q^{-16}$
1,0,0,0	$q^{36}-q^{34}-2 q^{28}+q^{26}-3 q^{24}-q^{20}-q^{18}+q^{16}+q^{14}+3 q^{12}+2 q^{10}+4 q^8+3 q^4-q^2-2 q^{-4} + q^{-6} - q^{-8}$

B2 Invariants.

Weight	Invariant
0,1	$q^{48}-3 q^{46}+6 q^{44}-9 q^{42}+12 q^{40}-15 q^{38}+15 q^{36}-15 q^{34}+11 q^{32}-8 q^{30}+8 q^{26}-16 q^{24}+23 q^{22}-26 q^{20}+30 q^{18}-28 q^{16}+26 q^{14}-18 q^{12}+11 q^{10}-3 q^8-3 q^6+10 q^4-13 q^2+16-14 q^{-2} +13 q^{-4} -10 q^{-6} +7 q^{-8} -5 q^{-10} +2 q^{-12} - q^{-14}$
1,0	$q^{78}-3 q^{74}-3 q^{72}+3 q^{70}+8 q^{68}+q^{66}-10 q^{64}-7 q^{62}+10 q^{60}+14 q^{58}-2 q^{56}-17 q^{54}-7 q^{52}+12 q^{50}+12 q^{48}-8 q^{46}-15 q^{44}+q^{42}+12 q^{40}+2 q^{38}-12 q^{36}-5 q^{34}+9 q^{32}+7 q^{30}-8 q^{28}-8 q^{26}+6 q^{24}+11 q^{22}-3 q^{20}-10 q^{18}+3 q^{16}+16 q^{14}+5 q^{12}-13 q^{10}-10 q^8+11 q^6+16 q^4-2 q^2-16-6 q^{-2} +10 q^{-4} +10 q^{-6} -4 q^{-8} -9 q^{-10} -2 q^{-12} +5 q^{-14} +3 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-24}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{66}-3 q^{64}+3 q^{62}-4 q^{60}+8 q^{58}-10 q^{56}+10 q^{54}-10 q^{52}+14 q^{50}-12 q^{48}+8 q^{46}-9 q^{44}+8 q^{42}-2 q^{40}-5 q^{38}+4 q^{36}-11 q^{34}+15 q^{32}-21 q^{30}+16 q^{28}-25 q^{26}+23 q^{24}-20 q^{22}+20 q^{20}-15 q^{18}+18 q^{16}-q^{14}+9 q^{12}+2 q^{10}-2 q^8+10 q^6-9 q^4+9 q^2-14+11 q^{-2} -11 q^{-4} +8 q^{-6} -9 q^{-8} +6 q^{-10} -4 q^{-12} +3 q^{-14} -2 q^{-16} + q^{-18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{114}-3 q^{112}+6 q^{110}-10 q^{108}+8 q^{106}-4 q^{104}-5 q^{102}+22 q^{100}-33 q^{98}+45 q^{96}-41 q^{94}+16 q^{92}+17 q^{90}-54 q^{88}+80 q^{86}-86 q^{84}+65 q^{82}-20 q^{80}-33 q^{78}+75 q^{76}-90 q^{74}+70 q^{72}-28 q^{70}-23 q^{68}+48 q^{66}-52 q^{64}+24 q^{62}+26 q^{60}-67 q^{58}+82 q^{56}-60 q^{54}+2 q^{52}+65 q^{50}-121 q^{48}+136 q^{46}-108 q^{44}+48 q^{42}+32 q^{40}-96 q^{38}+129 q^{36}-115 q^{34}+68 q^{32}-4 q^{30}-51 q^{28}+73 q^{26}-55 q^{24}+22 q^{22}+29 q^{20}-57 q^{18}+59 q^{16}-26 q^{14}-24 q^{12}+72 q^{10}-94 q^8+83 q^6-43 q^4-9 q^2+55-80 q^{-2} +81 q^{-4} -55 q^{-6} +18 q^{-8} +13 q^{-10} -35 q^{-12} +38 q^{-14} -31 q^{-16} +19 q^{-18} -5 q^{-20} -5 q^{-22} +7 q^{-24} -8 q^{-26} +5 q^{-28} -2 q^{-30} + q^{-32}$

. </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 31"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^6+z^4+2 z^2+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 55, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^2+3 q-5+8 q^{-1} -9 q^{-2} +10 q^{-3} -8 q^{-4} +6 q^{-5} -4 q^{-6} + q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^2 a^6-2 z^4 a^4-4 z^2 a^4-2 a^4+z^6 a^2+4 z^4 a^2+7 z^2 a^2+4 a^2-z^4-2 z^2-1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^4 a^8+4 z^5 a^7-4 z^3 a^7+6 z^6 a^6-8 z^4 a^6+3 z^2 a^6+4 z^7 a^5+z^5 a^5-8 z^3 a^5+3 z a^5+z^8 a^4+11 z^6 a^4-23 z^4 a^4+13 z^2 a^4-2 a^4+7 z^7 a^3-7 z^5 a^3-5 z^3 a^3+5 z a^3+z^8 a^2+8 z^6 a^2-21 z^4 a^2+15 z^2 a^2-4 a^2+3 z^7 a-3 z^5 a-3 z^3 a+3 z a+3 z^6-7 z^4+5 z^2-1+z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n11, K11n22, K11n112, K11n127,}

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 31"];

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]=

{ $t^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3}$, $-q^2+3 q-5+8 q^{-1} -9 q^{-2} +10 q^{-3} -8 q^{-4} +6 q^{-5} -4 q^{-6} + q^{-7}$ }

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]=

{K11n11, K11n22, K11n112, K11n127,}

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:

(2, -2)

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 $8$ $-16$ $32$ $\frac{172}{3}$ $\frac{20}{3}$ $-128$ $-\frac{640}{3}$ $-\frac{160}{3}$ $-16$ $\frac{256}{3}$ $128$ $\frac{1376}{3}$ $\frac{160}{3}$ $\frac{11911}{15}$ $\frac{916}{15}$ $\frac{12604}{45}$ $-\frac{55}{9}$ $\frac{631}{15}$

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 31. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

<p>

$n$	$J_n$
2	$q^7-3 q^6+10 q^4-13 q^3-6 q^2+33 q-27-24 q^{-1} +66 q^{-2} -35 q^{-3} -48 q^{-4} +91 q^{-5} -32 q^{-6} -66 q^{-7} +93 q^{-8} -21 q^{-9} -65 q^{-10} +71 q^{-11} -7 q^{-12} -46 q^{-13} +38 q^{-14} + q^{-15} -21 q^{-16} +12 q^{-17} +2 q^{-18} -4 q^{-19} + q^{-20}$
3	$-q^{15}+3 q^{14}-5 q^{12}-5 q^{11}+13 q^{10}+13 q^9-23 q^8-29 q^7+33 q^6+58 q^5-42 q^4-98 q^3+41 q^2+153 q-34-207 q^{-1} +4 q^{-2} +273 q^{-3} +26 q^{-4} -318 q^{-5} -80 q^{-6} +369 q^{-7} +123 q^{-8} -389 q^{-9} -179 q^{-10} +409 q^{-11} +213 q^{-12} -393 q^{-13} -256 q^{-14} +380 q^{-15} +265 q^{-16} -333 q^{-17} -277 q^{-18} +285 q^{-19} +262 q^{-20} -222 q^{-21} -238 q^{-22} +160 q^{-23} +204 q^{-24} -108 q^{-25} -155 q^{-26} +58 q^{-27} +116 q^{-28} -32 q^{-29} -71 q^{-30} +8 q^{-31} +46 q^{-32} -5 q^{-33} -20 q^{-34} - q^{-35} +8 q^{-36} +2 q^{-37} -4 q^{-38} + q^{-39}$
4	$q^{26}-3 q^{25}+5 q^{23}+5 q^{21}-20 q^{20}-6 q^{19}+23 q^{18}+12 q^{17}+32 q^{16}-74 q^{15}-52 q^{14}+48 q^{13}+65 q^{12}+141 q^{11}-163 q^{10}-197 q^9+5 q^8+156 q^7+434 q^6-197 q^5-455 q^4-230 q^3+179 q^2+932 q-31-698 q^{-1} -694 q^{-2} -24 q^{-3} +1496 q^{-4} +381 q^{-5} -757 q^{-6} -1258 q^{-7} -479 q^{-8} +1926 q^{-9} +916 q^{-10} -581 q^{-11} -1736 q^{-12} -1046 q^{-13} +2120 q^{-14} +1393 q^{-15} -257 q^{-16} -2012 q^{-17} -1550 q^{-18} +2067 q^{-19} +1699 q^{-20} +114 q^{-21} -2044 q^{-22} -1879 q^{-23} +1790 q^{-24} +1772 q^{-25} +463 q^{-26} -1803 q^{-27} -1966 q^{-28} +1308 q^{-29} +1574 q^{-30} +731 q^{-31} -1317 q^{-32} -1774 q^{-33} +741 q^{-34} +1135 q^{-35} +811 q^{-36} -729 q^{-37} -1327 q^{-38} +279 q^{-39} +608 q^{-40} +665 q^{-41} -259 q^{-42} -786 q^{-43} +45 q^{-44} +208 q^{-45} +396 q^{-46} -27 q^{-47} -354 q^{-48} -11 q^{-49} +27 q^{-50} +168 q^{-51} +22 q^{-52} -117 q^{-53} -4 q^{-54} -11 q^{-55} +47 q^{-56} +13 q^{-57} -26 q^{-58} -5 q^{-60} +8 q^{-61} +2 q^{-62} -4 q^{-63} + q^{-64}$
5	$-q^{40}+3 q^{39}-5 q^{37}+2 q^{34}+13 q^{33}+6 q^{32}-23 q^{31}-21 q^{30}-6 q^{29}+18 q^{28}+59 q^{27}+44 q^{26}-45 q^{25}-116 q^{24}-92 q^{23}+33 q^{22}+196 q^{21}+229 q^{20}+11 q^{19}-311 q^{18}-435 q^{17}-148 q^{16}+400 q^{15}+743 q^{14}+453 q^{13}-423 q^{12}-1148 q^{11}-933 q^{10}+274 q^9+1555 q^8+1656 q^7+125 q^6-1908 q^5-2531 q^4-850 q^3+2036 q^2+3554 q+1879-1899 q^{-1} -4480 q^{-2} -3230 q^{-3} +1357 q^{-4} +5366 q^{-5} +4704 q^{-6} -527 q^{-7} -5876 q^{-8} -6289 q^{-9} -682 q^{-10} +6241 q^{-11} +7754 q^{-12} +1973 q^{-13} -6158 q^{-14} -9091 q^{-15} -3457 q^{-16} +5986 q^{-17} +10161 q^{-18} +4798 q^{-19} -5471 q^{-20} -11023 q^{-21} -6142 q^{-22} +4979 q^{-23} +11585 q^{-24} +7224 q^{-25} -4226 q^{-26} -11966 q^{-27} -8242 q^{-28} +3587 q^{-29} +12014 q^{-30} +8966 q^{-31} -2685 q^{-32} -11871 q^{-33} -9620 q^{-34} +1898 q^{-35} +11379 q^{-36} +9913 q^{-37} -850 q^{-38} -10622 q^{-39} -10078 q^{-40} -78 q^{-41} +9526 q^{-42} +9834 q^{-43} +1094 q^{-44} -8162 q^{-45} -9332 q^{-46} -1930 q^{-47} +6608 q^{-48} +8454 q^{-49} +2583 q^{-50} -4978 q^{-51} -7300 q^{-52} -2962 q^{-53} +3432 q^{-54} +5982 q^{-55} +2988 q^{-56} -2081 q^{-57} -4572 q^{-58} -2802 q^{-59} +1065 q^{-60} +3297 q^{-61} +2319 q^{-62} -337 q^{-63} -2164 q^{-64} -1849 q^{-65} -36 q^{-66} +1357 q^{-67} +1256 q^{-68} +232 q^{-69} -729 q^{-70} -874 q^{-71} -233 q^{-72} +401 q^{-73} +488 q^{-74} +191 q^{-75} -157 q^{-76} -292 q^{-77} -135 q^{-78} +82 q^{-79} +140 q^{-80} +72 q^{-81} -27 q^{-82} -58 q^{-83} -44 q^{-84} +3 q^{-85} +38 q^{-86} +14 q^{-87} -8 q^{-88} -6 q^{-89} -4 q^{-90} -5 q^{-91} +8 q^{-92} +2 q^{-93} -4 q^{-94} + q^{-95}$
6	$q^{57}-3 q^{56}+5 q^{54}-7 q^{51}+5 q^{50}-13 q^{49}-6 q^{48}+32 q^{47}+12 q^{46}+6 q^{45}-35 q^{44}-3 q^{43}-62 q^{42}-36 q^{41}+109 q^{40}+97 q^{39}+80 q^{38}-84 q^{37}-53 q^{36}-283 q^{35}-223 q^{34}+216 q^{33}+380 q^{32}+471 q^{31}+50 q^{30}-75 q^{29}-914 q^{28}-1017 q^{27}-59 q^{26}+798 q^{25}+1581 q^{24}+1094 q^{23}+642 q^{22}-1834 q^{21}-3035 q^{20}-1903 q^{19}+288 q^{18}+3153 q^{17}+3922 q^{16}+3835 q^{15}-1464 q^{14}-5813 q^{13}-6496 q^{12}-3466 q^{11}+2952 q^{10}+7804 q^9+10912 q^8+3157 q^7-6415 q^6-12771 q^5-12013 q^4-2596 q^3+9173 q^2+20363 q+13631-834 q^{-1} -16582 q^{-2} -23412 q^{-3} -14998 q^{-4} +3854 q^{-5} +27561 q^{-6} +27611 q^{-7} +12152 q^{-8} -13740 q^{-9} -32809 q^{-10} -31464 q^{-11} -8919 q^{-12} +28404 q^{-13} +40118 q^{-14} +29361 q^{-15} -3836 q^{-16} -36402 q^{-17} -46928 q^{-18} -25659 q^{-19} +22695 q^{-20} +47556 q^{-21} +45798 q^{-22} +9710 q^{-23} -34155 q^{-24} -57914 q^{-25} -41627 q^{-26} +13471 q^{-27} +49748 q^{-28} +58226 q^{-29} +22824 q^{-30} -28580 q^{-31} -63964 q^{-32} -54014 q^{-33} +3859 q^{-34} +48411 q^{-35} +66103 q^{-36} +33477 q^{-37} -21818 q^{-38} -66031 q^{-39} -62441 q^{-40} -5048 q^{-41} +44639 q^{-42} +69948 q^{-43} +41739 q^{-44} -14105 q^{-45} -64351 q^{-46} -67262 q^{-47} -13791 q^{-48} +37884 q^{-49} +69463 q^{-50} +47919 q^{-51} -4580 q^{-52} -57815 q^{-53} -67781 q^{-54} -22494 q^{-55} +27105 q^{-56} +63153 q^{-57} +50837 q^{-58} +6528 q^{-59} -45404 q^{-60} -62142 q^{-61} -29244 q^{-62} +13089 q^{-63} +50088 q^{-64} +48063 q^{-65} +16453 q^{-66} -28624 q^{-67} -49484 q^{-68} -30788 q^{-69} -383 q^{-70} +32408 q^{-71} +38503 q^{-72} +21191 q^{-73} -12206 q^{-74} -32402 q^{-75} -25661 q^{-76} -8609 q^{-77} +15450 q^{-78} +24803 q^{-79} +19092 q^{-80} -1246 q^{-81} -16397 q^{-82} -16395 q^{-83} -9916 q^{-84} +4182 q^{-85} +12156 q^{-86} +12711 q^{-87} +2804 q^{-88} -5902 q^{-89} -7710 q^{-90} -6870 q^{-91} -463 q^{-92} +4208 q^{-93} +6368 q^{-94} +2500 q^{-95} -1271 q^{-96} -2500 q^{-97} -3313 q^{-98} -1094 q^{-99} +860 q^{-100} +2462 q^{-101} +1162 q^{-102} -53 q^{-103} -461 q^{-104} -1170 q^{-105} -578 q^{-106} +4 q^{-107} +770 q^{-108} +344 q^{-109} +46 q^{-110} +7 q^{-111} -307 q^{-112} -188 q^{-113} -65 q^{-114} +209 q^{-115} +62 q^{-116} +9 q^{-117} +30 q^{-118} -62 q^{-119} -38 q^{-120} -27 q^{-121} +52 q^{-122} +5 q^{-123} -7 q^{-124} +12 q^{-125} -10 q^{-126} -4 q^{-127} -5 q^{-128} +8 q^{-129} +2 q^{-130} -4 q^{-131} + q^{-132}$
7	$-q^{77}+3 q^{76}-5 q^{74}+7 q^{71}-5 q^{69}+13 q^{68}-3 q^{67}-23 q^{66}-12 q^{65}-6 q^{64}+35 q^{63}+31 q^{62}-5 q^{61}+43 q^{60}-17 q^{59}-90 q^{58}-87 q^{57}-83 q^{56}+96 q^{55}+172 q^{54}+120 q^{53}+200 q^{52}-6 q^{51}-287 q^{50}-409 q^{49}-517 q^{48}-29 q^{47}+477 q^{46}+689 q^{45}+1028 q^{44}+499 q^{43}-427 q^{42}-1271 q^{41}-2126 q^{40}-1402 q^{39}+161 q^{38}+1744 q^{37}+3579 q^{36}+3273 q^{35}+1186 q^{34}-1803 q^{33}-5643 q^{32}-6380 q^{31}-3948 q^{30}+722 q^{29}+7458 q^{28}+10657 q^{27}+9072 q^{26}+2824 q^{25}-8147 q^{24}-15843 q^{23}-16819 q^{22}-9805 q^{21}+6061 q^{20}+20328 q^{19}+26846 q^{18}+21458 q^{17}+863 q^{16}-22298 q^{15}-38050 q^{14}-37695 q^{13}-14077 q^{12}+18959 q^{11}+47751 q^{10}+57676 q^9+34751 q^8-7965 q^7-53130 q^6-78875 q^5-62159 q^4-12618 q^3+50763 q^2+98221 q+94484+42739 q^{-1} -38228 q^{-2} -111581 q^{-3} -128455 q^{-4} -81621 q^{-5} +14371 q^{-6} +116494 q^{-7} +160268 q^{-8} +125608 q^{-9} +20176 q^{-10} -110076 q^{-11} -186238 q^{-12} -172048 q^{-13} -63419 q^{-14} +93241 q^{-15} +204020 q^{-16} +215933 q^{-17} +111601 q^{-18} -66028 q^{-19} -211910 q^{-20} -255401 q^{-21} -161687 q^{-22} +32323 q^{-23} +211005 q^{-24} +287116 q^{-25} +209431 q^{-26} +5971 q^{-27} -201908 q^{-28} -311536 q^{-29} -253245 q^{-30} -44536 q^{-31} +187731 q^{-32} +327952 q^{-33} +290651 q^{-34} +82021 q^{-35} -169976 q^{-36} -338536 q^{-37} -322041 q^{-38} -115679 q^{-39} +151346 q^{-40} +343582 q^{-41} +347125 q^{-42} +145965 q^{-43} -132536 q^{-44} -345612 q^{-45} -367246 q^{-46} -171729 q^{-47} +114478 q^{-48} +344198 q^{-49} +382831 q^{-50} +195129 q^{-51} -96510 q^{-52} -341020 q^{-53} -395109 q^{-54} -215553 q^{-55} +78338 q^{-56} +334089 q^{-57} +403707 q^{-58} +235404 q^{-59} -58083 q^{-60} -324132 q^{-61} -409006 q^{-62} -253431 q^{-63} +35629 q^{-64} +308177 q^{-65} +408982 q^{-66} +270954 q^{-67} -9332 q^{-68} -286451 q^{-69} -403208 q^{-70} -285424 q^{-71} -19511 q^{-72} +256729 q^{-73} +388976 q^{-74} +296150 q^{-75} +50789 q^{-76} -220045 q^{-77} -365865 q^{-78} -300125 q^{-79} -81393 q^{-80} +176913 q^{-81} +332631 q^{-82} +295683 q^{-83} +109056 q^{-84} -130107 q^{-85} -290581 q^{-86} -281264 q^{-87} -130280 q^{-88} +83046 q^{-89} +241727 q^{-90} +256697 q^{-91} +142628 q^{-92} -39569 q^{-93} -189620 q^{-94} -223562 q^{-95} -144791 q^{-96} +3495 q^{-97} +138565 q^{-98} +184498 q^{-99} +136842 q^{-100} +22975 q^{-101} -92456 q^{-102} -143625 q^{-103} -120792 q^{-104} -38596 q^{-105} +54577 q^{-106} +104380 q^{-107} +99484 q^{-108} +44780 q^{-109} -26370 q^{-110} -70626 q^{-111} -76524 q^{-112} -42871 q^{-113} +7899 q^{-114} +43408 q^{-115} +54726 q^{-116} +36778 q^{-117} +2513 q^{-118} -24388 q^{-119} -36511 q^{-120} -28010 q^{-121} -6640 q^{-122} +11484 q^{-123} +22388 q^{-124} +20015 q^{-125} +7342 q^{-126} -4537 q^{-127} -12926 q^{-128} -12795 q^{-129} -5882 q^{-130} +769 q^{-131} +6644 q^{-132} +7817 q^{-133} +4294 q^{-134} +506 q^{-135} -3319 q^{-136} -4364 q^{-137} -2533 q^{-138} -821 q^{-139} +1377 q^{-140} +2273 q^{-141} +1479 q^{-142} +738 q^{-143} -598 q^{-144} -1183 q^{-145} -724 q^{-146} -417 q^{-147} +219 q^{-148} +498 q^{-149} +312 q^{-150} +306 q^{-151} -45 q^{-152} -291 q^{-153} -153 q^{-154} -111 q^{-155} +54 q^{-156} +91 q^{-157} +13 q^{-158} +84 q^{-159} +12 q^{-160} -58 q^{-161} -32 q^{-162} -21 q^{-163} +22 q^{-164} +19 q^{-165} -16 q^{-166} +13 q^{-167} +8 q^{-168} -10 q^{-169} -4 q^{-170} -5 q^{-171} +8 q^{-172} +2 q^{-173} -4 q^{-174} + q^{-175}$

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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