9 24

9_23

9_25

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 24's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 24 at Knotilus!

Knot presentations

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

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

[edit Notes on presentations of 9 24]

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 24"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 14 2 16 18 6 12 10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[3,21,2+]

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]=

[math]\displaystyle{ \textrm{BR}(4,\{-1,-1,2,-1,-3,2,2,2,-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[{12, 4}, {3, 10}, {8, 11}, {10, 12}, {9, 5}, {4, 8}, {6, 9}, {5, 7}, {2, 6}, {1, 3}, {11, 2}, {7, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-5] Hyperbolic Volume 10.8337 A-Polynomial See Data:9 24/A-polynomial

[edit Notes for 9 24's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math] Topological 4 genus [math]\displaystyle{ 1 }[/math] Concordance genus [math]\displaystyle{ 1 }[/math] Rasmussen s-Invariant 0

[edit Notes for 9 24's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+5 t^2-10 t+13-10 t^{-1} +5 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6-z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 45, 0 }
Jones polynomial	[math]\displaystyle{ q^4-3 q^3+5 q^2-7 q+8-7 q^{-1} +7 q^{-2} -4 q^{-3} +2 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+6 a^2 z^2+2 z^2 a^{-2} -6 z^2-2 a^4+5 a^2+ a^{-2} -3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^8+z^8+2 a^3 z^7+5 a z^7+3 z^7 a^{-1} +2 a^4 z^6+3 a^2 z^6+4 z^6 a^{-2} +5 z^6+a^5 z^5-2 a^3 z^5-7 a z^5-z^5 a^{-1} +3 z^5 a^{-3} -5 a^4 z^4-10 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} -11 z^4-3 a^5 z^3-3 a^3 z^3+a z^3-3 z^3 a^{-1} -4 z^3 a^{-3} +4 a^4 z^2+10 a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} +9 z^2+2 a^5 z+3 a^3 z+2 a z+2 z a^{-1} +z a^{-3} -2 a^4-5 a^2- a^{-2} -3 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{16}-q^{14}-q^{10}+3 q^8+2 q^6+q^4+2 q^2-2+ q^{-2} -2 q^{-4} + q^{-8} - q^{-10} + q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-q^{78}+3 q^{76}-4 q^{74}+3 q^{72}-3 q^{70}-3 q^{68}+9 q^{66}-16 q^{64}+19 q^{62}-19 q^{60}+8 q^{58}+7 q^{56}-26 q^{54}+41 q^{52}-47 q^{50}+34 q^{48}-11 q^{46}-23 q^{44}+45 q^{42}-53 q^{40}+46 q^{38}-16 q^{36}-11 q^{34}+34 q^{32}-36 q^{30}+22 q^{28}+10 q^{26}-32 q^{24}+44 q^{22}-27 q^{20}+2 q^{18}+37 q^{16}-60 q^{14}+71 q^{12}-55 q^{10}+21 q^8+20 q^6-60 q^4+77 q^2-69+40 q^{-2} -4 q^{-4} -32 q^{-6} +46 q^{-8} -43 q^{-10} +18 q^{-12} +8 q^{-14} -31 q^{-16} +33 q^{-18} -15 q^{-20} -14 q^{-22} +41 q^{-24} -50 q^{-26} +44 q^{-28} -20 q^{-30} -11 q^{-32} +35 q^{-34} -48 q^{-36} +47 q^{-38} -29 q^{-40} +9 q^{-42} +9 q^{-44} -21 q^{-46} +24 q^{-48} -19 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_24.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{11}+q^9-2 q^7+3 q^5+q+ q^{-1} -2 q^{-3} +2 q^{-5} -2 q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{32}-q^{30}-q^{28}+4 q^{26}-3 q^{24}-6 q^{22}+9 q^{20}-q^{18}-12 q^{16}+11 q^{14}+5 q^{12}-12 q^{10}+5 q^8+7 q^6-5 q^4-2 q^2+5+5 q^{-2} -10 q^{-4} +12 q^{-8} -11 q^{-10} -4 q^{-12} +13 q^{-14} -5 q^{-16} -5 q^{-18} +6 q^{-20} - q^{-22} -2 q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ -q^{63}+q^{61}+q^{59}-q^{57}-3 q^{55}+3 q^{53}+7 q^{51}-3 q^{49}-13 q^{47}+q^{45}+21 q^{43}+5 q^{41}-30 q^{39}-17 q^{37}+34 q^{35}+30 q^{33}-30 q^{31}-48 q^{29}+26 q^{27}+54 q^{25}-11 q^{23}-59 q^{21}-q^{19}+55 q^{17}+12 q^{15}-46 q^{13}-18 q^{11}+34 q^9+25 q^7-16 q^5-30 q^3+3 q+32 q^{-1} +17 q^{-3} -36 q^{-5} -32 q^{-7} +32 q^{-9} +49 q^{-11} -27 q^{-13} -56 q^{-15} +16 q^{-17} +57 q^{-19} -3 q^{-21} -53 q^{-23} -7 q^{-25} +41 q^{-27} +13 q^{-29} -26 q^{-31} -14 q^{-33} +14 q^{-35} +11 q^{-37} -8 q^{-39} -5 q^{-41} +3 q^{-43} +3 q^{-45} - q^{-47} -2 q^{-49} + q^{-51} }[/math]
4	[math]\displaystyle{ q^{104}-q^{102}-q^{100}+q^{98}+3 q^{94}-5 q^{92}-5 q^{90}+5 q^{88}+5 q^{86}+13 q^{84}-13 q^{82}-24 q^{80}+18 q^{76}+51 q^{74}-7 q^{72}-62 q^{70}-48 q^{68}+6 q^{66}+121 q^{64}+61 q^{62}-67 q^{60}-138 q^{58}-92 q^{56}+148 q^{54}+182 q^{52}+34 q^{50}-177 q^{48}-244 q^{46}+62 q^{44}+244 q^{42}+183 q^{40}-104 q^{38}-320 q^{36}-78 q^{34}+190 q^{32}+264 q^{30}+10 q^{28}-275 q^{26}-155 q^{24}+89 q^{22}+236 q^{20}+81 q^{18}-164 q^{16}-163 q^{14}-5 q^{12}+160 q^{10}+121 q^8-41 q^6-149 q^4-95 q^2+70+162 q^{-2} +100 q^{-4} -130 q^{-6} -197 q^{-8} -46 q^{-10} +180 q^{-12} +245 q^{-14} -61 q^{-16} -257 q^{-18} -180 q^{-20} +124 q^{-22} +327 q^{-24} +56 q^{-26} -203 q^{-28} -251 q^{-30} -3 q^{-32} +269 q^{-34} +135 q^{-36} -65 q^{-38} -202 q^{-40} -92 q^{-42} +129 q^{-44} +108 q^{-46} +32 q^{-48} -86 q^{-50} -80 q^{-52} +28 q^{-54} +36 q^{-56} +36 q^{-58} -18 q^{-60} -31 q^{-62} +6 q^{-64} +3 q^{-66} +12 q^{-68} -3 q^{-70} -8 q^{-72} +3 q^{-74} +3 q^{-78} - q^{-80} -2 q^{-82} + q^{-84} }[/math]
5	[math]\displaystyle{ -q^{155}+q^{153}+q^{151}-q^{149}-q^{143}+3 q^{141}+4 q^{139}-5 q^{137}-8 q^{135}-3 q^{133}+3 q^{131}+15 q^{129}+18 q^{127}-4 q^{125}-33 q^{123}-37 q^{121}-7 q^{119}+45 q^{117}+80 q^{115}+44 q^{113}-51 q^{111}-137 q^{109}-117 q^{107}+22 q^{105}+191 q^{103}+236 q^{101}+75 q^{99}-208 q^{97}-385 q^{95}-252 q^{93}+141 q^{91}+507 q^{89}+504 q^{87}+49 q^{85}-542 q^{83}-782 q^{81}-365 q^{79}+452 q^{77}+991 q^{75}+743 q^{73}-182 q^{71}-1079 q^{69}-1137 q^{67}-171 q^{65}+1004 q^{63}+1392 q^{61}+604 q^{59}-770 q^{57}-1536 q^{55}-963 q^{53}+466 q^{51}+1481 q^{49}+1212 q^{47}-122 q^{45}-1326 q^{43}-1310 q^{41}-155 q^{39}+1078 q^{37}+1280 q^{35}+345 q^{33}-813 q^{31}-1161 q^{29}-460 q^{27}+570 q^{25}+999 q^{23}+517 q^{21}-356 q^{19}-836 q^{17}-557 q^{15}+171 q^{13}+691 q^{11}+615 q^9+34 q^7-577 q^5-700 q^3-242 q+443 q^{-1} +820 q^{-3} +521 q^{-5} -300 q^{-7} -942 q^{-9} -808 q^{-11} +70 q^{-13} +1011 q^{-15} +1140 q^{-17} +203 q^{-19} -1002 q^{-21} -1395 q^{-23} -550 q^{-25} +857 q^{-27} +1563 q^{-29} +895 q^{-31} -596 q^{-33} -1567 q^{-35} -1165 q^{-37} +236 q^{-39} +1391 q^{-41} +1310 q^{-43} +137 q^{-45} -1074 q^{-47} -1286 q^{-49} -423 q^{-51} +677 q^{-53} +1100 q^{-55} +588 q^{-57} -303 q^{-59} -818 q^{-61} -603 q^{-63} +32 q^{-65} +508 q^{-67} +498 q^{-69} +118 q^{-71} -248 q^{-73} -349 q^{-75} -160 q^{-77} +94 q^{-79} +196 q^{-81} +126 q^{-83} -7 q^{-85} -90 q^{-87} -85 q^{-89} -12 q^{-91} +40 q^{-93} +36 q^{-95} +11 q^{-97} -11 q^{-99} -15 q^{-101} -8 q^{-103} +6 q^{-105} +8 q^{-107} -2 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^{14}-q^{10}+3 q^8+2 q^6+q^4+2 q^2-2+ q^{-2} -2 q^{-4} + q^{-8} - q^{-10} + q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{44}-2 q^{42}+6 q^{40}-12 q^{38}+25 q^{36}-42 q^{34}+66 q^{32}-100 q^{30}+130 q^{28}-168 q^{26}+186 q^{24}-196 q^{22}+177 q^{20}-132 q^{18}+72 q^{16}+26 q^{14}-114 q^{12}+216 q^{10}-294 q^8+352 q^6-385 q^4+376 q^2-338+268 q^{-2} -177 q^{-4} +80 q^{-6} +16 q^{-8} -96 q^{-10} +156 q^{-12} -186 q^{-14} +192 q^{-16} -178 q^{-18} +152 q^{-20} -120 q^{-22} +86 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}+q^{40}-q^{36}+q^{34}+q^{32}-4 q^{30}-6 q^{28}+q^{24}-4 q^{22}+8 q^{18}+7 q^{16}-3 q^{14}+2 q^{12}+5 q^{10}-3 q^8-3 q^6+3 q^4-4+2 q^{-2} +4 q^{-4} -4 q^{-6} -3 q^{-8} +6 q^{-10} +2 q^{-12} -6 q^{-14} + q^{-16} +5 q^{-18} - q^{-20} -3 q^{-22} +2 q^{-26} - q^{-28} - q^{-30} + q^{-32} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{34}-q^{32}+q^{30}+2 q^{28}-5 q^{26}+2 q^{22}-12 q^{20}+q^{18}+8 q^{16}-8 q^{14}+8 q^{12}+13 q^{10}-q^8+q^6+4 q^4-2 q^2-6-5 q^{-2} +5 q^{-4} -3 q^{-6} -7 q^{-8} +12 q^{-10} -8 q^{-14} +9 q^{-16} -6 q^{-20} +4 q^{-22} -2 q^{-26} + q^{-28} }[/math]
1,0,0	[math]\displaystyle{ -q^{21}-q^{19}-2 q^{17}-q^{13}+4 q^{11}+2 q^9+4 q^7+q^5+q^3-q-2 q^{-1} -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^{42}+q^{40}+q^{38}+q^{36}-2 q^{34}-5 q^{32}-4 q^{30}-4 q^{28}-11 q^{26}-7 q^{24}+6 q^{22}+5 q^{20}+2 q^{18}+14 q^{16}+20 q^{14}+6 q^{12}+5 q^8-q^6-16 q^4-6 q^2+1-9 q^{-2} -5 q^{-4} +10 q^{-6} +2 q^{-8} -5 q^{-10} +6 q^{-12} +8 q^{-14} -3 q^{-16} -5 q^{-18} +5 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^{24}-2 q^{22}-2 q^{20}-q^{16}+4 q^{14}+3 q^{12}+4 q^{10}+4 q^8+q^6+q^4-2 q^2-1-3 q^{-2} -2 q^{-6} + q^{-8} +2 q^{-14} - q^{-16} + q^{-18} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{34}+q^{32}-3 q^{30}+4 q^{28}-7 q^{26}+8 q^{24}-10 q^{22}+10 q^{20}-9 q^{18}+8 q^{16}-2 q^{14}+9 q^{10}-11 q^8+17 q^6-18 q^4+20 q^2-20+15 q^{-2} -11 q^{-4} +5 q^{-6} - q^{-8} -4 q^{-10} +8 q^{-12} -10 q^{-14} +11 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}-q^{52}-q^{50}+2 q^{48}+3 q^{46}-q^{44}-6 q^{42}-3 q^{40}+5 q^{38}+6 q^{36}-5 q^{34}-12 q^{32}-3 q^{30}+10 q^{28}+8 q^{26}-7 q^{24}-8 q^{22}+4 q^{20}+13 q^{18}+3 q^{16}-5 q^{14}-q^{12}+8 q^{10}+4 q^8-5 q^6-6 q^4+3 q^2+6-4 q^{-2} -9 q^{-4} +8 q^{-8} + q^{-10} -9 q^{-12} -5 q^{-14} +9 q^{-16} +10 q^{-18} -4 q^{-20} -11 q^{-22} - q^{-24} +10 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}-q^{44}+2 q^{42}-2 q^{40}+4 q^{38}-6 q^{36}+4 q^{34}-9 q^{32}+5 q^{30}-12 q^{28}+4 q^{26}-8 q^{24}+7 q^{22}-q^{20}+4 q^{18}+9 q^{16}+3 q^{14}+15 q^{12}-8 q^{10}+14 q^8-13 q^6+14 q^4-19 q^2+9-17 q^{-2} +9 q^{-4} -8 q^{-6} +3 q^{-8} -3 q^{-10} + q^{-12} +7 q^{-14} -4 q^{-16} +7 q^{-18} -8 q^{-20} +10 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}-q^{78}+3 q^{76}-4 q^{74}+3 q^{72}-3 q^{70}-3 q^{68}+9 q^{66}-16 q^{64}+19 q^{62}-19 q^{60}+8 q^{58}+7 q^{56}-26 q^{54}+41 q^{52}-47 q^{50}+34 q^{48}-11 q^{46}-23 q^{44}+45 q^{42}-53 q^{40}+46 q^{38}-16 q^{36}-11 q^{34}+34 q^{32}-36 q^{30}+22 q^{28}+10 q^{26}-32 q^{24}+44 q^{22}-27 q^{20}+2 q^{18}+37 q^{16}-60 q^{14}+71 q^{12}-55 q^{10}+21 q^8+20 q^6-60 q^4+77 q^2-69+40 q^{-2} -4 q^{-4} -32 q^{-6} +46 q^{-8} -43 q^{-10} +18 q^{-12} +8 q^{-14} -31 q^{-16} +33 q^{-18} -15 q^{-20} -14 q^{-22} +41 q^{-24} -50 q^{-26} +44 q^{-28} -20 q^{-30} -11 q^{-32} +35 q^{-34} -48 q^{-36} +47 q^{-38} -29 q^{-40} +9 q^{-42} +9 q^{-44} -21 q^{-46} +24 q^{-48} -19 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 24"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+5 t^2-10 t+13-10 t^{-1} +5 t^{-2} - t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

In[6]:=

Alexander[K, 2][t]

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

Out[6]=

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

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 45, 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+8-7 q^{-1} +7 q^{-2} -4 q^{-3} +2 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+6 a^2 z^2+2 z^2 a^{-2} -6 z^2-2 a^4+5 a^2+ a^{-2} -3 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^2 z^8+z^8+2 a^3 z^7+5 a z^7+3 z^7 a^{-1} +2 a^4 z^6+3 a^2 z^6+4 z^6 a^{-2} +5 z^6+a^5 z^5-2 a^3 z^5-7 a z^5-z^5 a^{-1} +3 z^5 a^{-3} -5 a^4 z^4-10 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} -11 z^4-3 a^5 z^3-3 a^3 z^3+a z^3-3 z^3 a^{-1} -4 z^3 a^{-3} +4 a^4 z^2+10 a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} +9 z^2+2 a^5 z+3 a^3 z+2 a z+2 z a^{-1} +z a^{-3} -2 a^4-5 a^2- a^{-2} -3 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_18, K11n85, K11n164,}

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

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 24"];

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^2-10 t+13-10 t^{-1} +5 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ q^4-3 q^3+5 q^2-7 q+8-7 q^{-1} +7 q^{-2} -4 q^{-3} +2 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]=

{8_18, K11n85, K11n164,}

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:

(1, -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 [math]\displaystyle{ 4 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{110}{3} }[/math] [math]\displaystyle{ \frac{34}{3} }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ -\frac{352}{3} }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ -48 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{440}{3} }[/math] [math]\displaystyle{ \frac{136}{3} }[/math] [math]\displaystyle{ \frac{13471}{30} }[/math] [math]\displaystyle{ -\frac{782}{15} }[/math] [math]\displaystyle{ \frac{10862}{45} }[/math] [math]\displaystyle{ \frac{545}{18} }[/math] [math]\displaystyle{ \frac{991}{30} }[/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 24. 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           4 3       1 -1         4 5         1 -3       3 3           0 -5     1 4             3 -7   1 3               -2 -9   1                 1 -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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+q^7+26 q^6-31 q^5-6 q^4+49 q^3-43 q^2-16 q+64-43 q^{-1} -23 q^{-2} +61 q^{-3} -31 q^{-4} -25 q^{-5} +44 q^{-6} -14 q^{-7} -19 q^{-8} +21 q^{-9} -3 q^{-10} -9 q^{-11} +6 q^{-12} -2 q^{-14} + q^{-15} }[/math]
3	[math]\displaystyle{ q^{24}-3 q^{23}+q^{22}+4 q^{21}+q^{20}-11 q^{19}-2 q^{18}+23 q^{17}+4 q^{16}-39 q^{15}-14 q^{14}+62 q^{13}+32 q^{12}-87 q^{11}-60 q^{10}+112 q^9+92 q^8-128 q^7-132 q^6+141 q^5+168 q^4-145 q^3-196 q^2+137 q+221-130 q^{-1} -225 q^{-2} +104 q^{-3} +235 q^{-4} -89 q^{-5} -216 q^{-6} +52 q^{-7} +207 q^{-8} -31 q^{-9} -173 q^{-10} -4 q^{-11} +149 q^{-12} +17 q^{-13} -108 q^{-14} -32 q^{-15} +75 q^{-16} +35 q^{-17} -48 q^{-18} -28 q^{-19} +24 q^{-20} +22 q^{-21} -13 q^{-22} -12 q^{-23} +4 q^{-24} +8 q^{-25} -3 q^{-26} -2 q^{-27} +2 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}+6 q^{33}+19 q^{32}-12 q^{31}+7 q^{30}-51 q^{29}+19 q^{28}+73 q^{27}-12 q^{26}-q^{25}-159 q^{24}+13 q^{23}+191 q^{22}+64 q^{21}+20 q^{20}-380 q^{19}-97 q^{18}+328 q^{17}+264 q^{16}+154 q^{15}-652 q^{14}-345 q^{13}+376 q^{12}+523 q^{11}+425 q^{10}-855 q^9-649 q^8+299 q^7+719 q^6+731 q^5-920 q^4-875 q^3+148 q^2+786 q+961-858 q^{-1} -967 q^{-2} -17 q^{-3} +732 q^{-4} +1069 q^{-5} -696 q^{-6} -928 q^{-7} -182 q^{-8} +574 q^{-9} +1068 q^{-10} -451 q^{-11} -773 q^{-12} -329 q^{-13} +330 q^{-14} +948 q^{-15} -166 q^{-16} -519 q^{-17} -403 q^{-18} +62 q^{-19} +706 q^{-20} +50 q^{-21} -232 q^{-22} -342 q^{-23} -120 q^{-24} +400 q^{-25} +117 q^{-26} -21 q^{-27} -194 q^{-28} -154 q^{-29} +160 q^{-30} +71 q^{-31} +50 q^{-32} -66 q^{-33} -94 q^{-34} +45 q^{-35} +17 q^{-36} +36 q^{-37} -11 q^{-38} -36 q^{-39} +12 q^{-40} - q^{-41} +12 q^{-42} -10 q^{-44} +4 q^{-45} - q^{-46} +2 q^{-47} -2 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}+2 q^{52}+14 q^{51}-5 q^{50}-14 q^{49}-6 q^{48}-2 q^{47}+24 q^{46}+39 q^{45}-q^{44}-66 q^{43}-79 q^{42}-7 q^{41}+107 q^{40}+172 q^{39}+69 q^{38}-168 q^{37}-333 q^{36}-196 q^{35}+208 q^{34}+538 q^{33}+449 q^{32}-158 q^{31}-809 q^{30}-831 q^{29}-7 q^{28}+1053 q^{27}+1340 q^{26}+354 q^{25}-1232 q^{24}-1931 q^{23}-870 q^{22}+1265 q^{21}+2551 q^{20}+1527 q^{19}-1151 q^{18}-3086 q^{17}-2271 q^{16}+863 q^{15}+3522 q^{14}+3018 q^{13}-483 q^{12}-3792 q^{11}-3678 q^{10}+18 q^9+3915 q^8+4223 q^7+454 q^6-3921 q^5-4619 q^4-860 q^3+3781 q^2+4865 q+1275-3622 q^{-1} -4996 q^{-2} -1545 q^{-3} +3323 q^{-4} +4988 q^{-5} +1886 q^{-6} -3041 q^{-7} -4920 q^{-8} -2065 q^{-9} +2595 q^{-10} +4709 q^{-11} +2366 q^{-12} -2168 q^{-13} -4438 q^{-14} -2494 q^{-15} +1565 q^{-16} +4008 q^{-17} +2714 q^{-18} -1010 q^{-19} -3503 q^{-20} -2696 q^{-21} +332 q^{-22} +2853 q^{-23} +2698 q^{-24} +194 q^{-25} -2169 q^{-26} -2427 q^{-27} -683 q^{-28} +1424 q^{-29} +2125 q^{-30} +960 q^{-31} -795 q^{-32} -1639 q^{-33} -1071 q^{-34} +249 q^{-35} +1159 q^{-36} +1018 q^{-37} +102 q^{-38} -714 q^{-39} -823 q^{-40} -290 q^{-41} +342 q^{-42} +601 q^{-43} +342 q^{-44} -123 q^{-45} -368 q^{-46} -287 q^{-47} -24 q^{-48} +208 q^{-49} +209 q^{-50} +54 q^{-51} -85 q^{-52} -126 q^{-53} -69 q^{-54} +39 q^{-55} +70 q^{-56} +34 q^{-57} + q^{-58} -31 q^{-59} -33 q^{-60} +4 q^{-61} +18 q^{-62} +4 q^{-63} +5 q^{-64} -2 q^{-65} -11 q^{-66} + q^{-67} +6 q^{-68} -2 q^{-69} + q^{-71} -2 q^{-72} +2 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}-3 q^{75}+21 q^{74}-15 q^{73}-8 q^{72}-11 q^{71}+36 q^{70}-9 q^{69}-2 q^{68}+54 q^{67}-59 q^{66}-65 q^{65}-60 q^{64}+110 q^{63}+45 q^{62}+83 q^{61}+181 q^{60}-163 q^{59}-293 q^{58}-343 q^{57}+118 q^{56}+196 q^{55}+480 q^{54}+747 q^{53}-88 q^{52}-761 q^{51}-1276 q^{50}-510 q^{49}+26 q^{48}+1292 q^{47}+2386 q^{46}+1063 q^{45}-834 q^{44}-2971 q^{43}-2645 q^{42}-1728 q^{41}+1621 q^{40}+5148 q^{39}+4384 q^{38}+1092 q^{37}-4217 q^{36}-6238 q^{35}-6260 q^{34}-381 q^{33}+7486 q^{32}+9593 q^{31}+6158 q^{30}-2964 q^{29}-9446 q^{28}-12930 q^{27}-5661 q^{26}+7256 q^{25}+14508 q^{24}+13314 q^{23}+1504 q^{22}-10137 q^{21}-19221 q^{20}-12791 q^{19}+3969 q^{18}+16960 q^{17}+19867 q^{16}+7571 q^{15}-8007 q^{14}-22977 q^{13}-19097 q^{12}-756 q^{11}+16666 q^{10}+23858 q^9+12843 q^8-4544 q^7-23940 q^6-22954 q^5-4949 q^4+14828 q^3+25188 q^2+16161 q-1272-23090 q^{-1} -24468 q^{-2} -7896 q^{-3} +12518 q^{-4} +24741 q^{-5} +17851 q^{-6} +1461 q^{-7} -21188 q^{-8} -24457 q^{-9} -10067 q^{-10} +9789 q^{-11} +23057 q^{-12} +18658 q^{-13} +4229 q^{-14} -18113 q^{-15} -23290 q^{-16} -12085 q^{-17} +6069 q^{-18} +19884 q^{-19} +18710 q^{-20} +7474 q^{-21} -13340 q^{-22} -20537 q^{-23} -13752 q^{-24} +1184 q^{-25} +14726 q^{-26} +17254 q^{-27} +10578 q^{-28} -6988 q^{-29} -15602 q^{-30} -13882 q^{-31} -3759 q^{-32} +7912 q^{-33} +13444 q^{-34} +11899 q^{-35} -611 q^{-36} -8900 q^{-37} -11356 q^{-38} -6693 q^{-39} +1289 q^{-40} +7705 q^{-41} +10190 q^{-42} +3436 q^{-43} -2472 q^{-44} -6675 q^{-45} -6341 q^{-46} -2739 q^{-47} +2197 q^{-48} +6189 q^{-49} +4001 q^{-50} +1342 q^{-51} -2113 q^{-52} -3676 q^{-53} -3349 q^{-54} -896 q^{-55} +2304 q^{-56} +2309 q^{-57} +2036 q^{-58} +348 q^{-59} -1054 q^{-60} -1998 q^{-61} -1390 q^{-62} +255 q^{-63} +595 q^{-64} +1151 q^{-65} +750 q^{-66} +157 q^{-67} -683 q^{-68} -757 q^{-69} -188 q^{-70} -111 q^{-71} +337 q^{-72} +378 q^{-73} +295 q^{-74} -124 q^{-75} -238 q^{-76} -78 q^{-77} -154 q^{-78} +34 q^{-79} +98 q^{-80} +145 q^{-81} -10 q^{-82} -54 q^{-83} +5 q^{-84} -62 q^{-85} -11 q^{-86} +11 q^{-87} +48 q^{-88} -4 q^{-89} -15 q^{-90} +14 q^{-91} -15 q^{-92} -4 q^{-93} -2 q^{-94} +13 q^{-95} -2 q^{-96} -7 q^{-97} +6 q^{-98} -2 q^{-99} - q^{-101} +2 q^{-102} -2 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}+4 q^{102}+11 q^{101}-9 q^{100}-2 q^{99}-12 q^{98}+19 q^{97}+37 q^{96}-32 q^{95}-2 q^{94}-48 q^{92}-14 q^{91}-40 q^{90}+72 q^{89}+168 q^{88}+24 q^{87}+16 q^{86}-91 q^{85}-259 q^{84}-181 q^{83}-188 q^{82}+155 q^{81}+590 q^{80}+477 q^{79}+407 q^{78}-137 q^{77}-875 q^{76}-1031 q^{75}-1120 q^{74}-203 q^{73}+1281 q^{72}+1989 q^{71}+2345 q^{70}+1079 q^{69}-1362 q^{68}-3110 q^{67}-4428 q^{66}-3118 q^{65}+618 q^{64}+4273 q^{63}+7457 q^{62}+6604 q^{61}+1553 q^{60}-4636 q^{59}-10927 q^{58}-11983 q^{57}-6169 q^{56}+3303 q^{55}+14408 q^{54}+19031 q^{53}+13506 q^{52}+831 q^{51}-16442 q^{50}-26995 q^{49}-23947 q^{48}-8676 q^{47}+15902 q^{46}+34749 q^{45}+36681 q^{44}+20480 q^{43}-11524 q^{42}-40702 q^{41}-50508 q^{40}-35873 q^{39}+2765 q^{38}+43450 q^{37}+63855 q^{36}+53648 q^{35}+10102 q^{34}-42189 q^{33}-75024 q^{32}-71983 q^{31}-26135 q^{30}+36687 q^{29}+82781 q^{28}+89294 q^{27}+43774 q^{26}-27768 q^{25}-86691 q^{24}-103909 q^{23}-61115 q^{22}+16431 q^{21}+86756 q^{20}+115077 q^{19}+76927 q^{18}-4269 q^{17}-83961 q^{16}-122499 q^{15}-89954 q^{14}-7438 q^{13}+79102 q^{12}+126507 q^{11}+99926 q^{10}+17883 q^9-73296 q^8-127958 q^7-106970 q^6-26389 q^5+67443 q^4+127324 q^3+111375 q^2+33314 q-61653-125691 q^{-1} -114184 q^{-2} -38579 q^{-3} +56492 q^{-4} +123025 q^{-5} +115435 q^{-6} +43195 q^{-7} -51048 q^{-8} -119959 q^{-9} -116286 q^{-10} -47264 q^{-11} +45652 q^{-12} +115863 q^{-13} +116152 q^{-14} +51725 q^{-15} -38925 q^{-16} -110840 q^{-17} -115841 q^{-18} -56394 q^{-19} +31289 q^{-20} +103934 q^{-21} +114155 q^{-22} +61680 q^{-23} -21528 q^{-24} -95151 q^{-25} -111441 q^{-26} -66761 q^{-27} +10615 q^{-28} +83603 q^{-29} +106145 q^{-30} +71494 q^{-31} +1936 q^{-32} -69811 q^{-33} -98629 q^{-34} -74307 q^{-35} -14326 q^{-36} +53509 q^{-37} +87585 q^{-38} +74888 q^{-39} +26325 q^{-40} -36303 q^{-41} -74054 q^{-42} -71793 q^{-43} -35554 q^{-44} +18850 q^{-45} +57750 q^{-46} +65265 q^{-47} +41743 q^{-48} -3220 q^{-49} -40863 q^{-50} -55196 q^{-51} -43277 q^{-52} -9314 q^{-53} +24128 q^{-54} +42753 q^{-55} +40895 q^{-56} +17656 q^{-57} -9884 q^{-58} -29576 q^{-59} -34702 q^{-60} -21292 q^{-61} -1049 q^{-62} +17056 q^{-63} +26466 q^{-64} +21009 q^{-65} +7799 q^{-66} -6956 q^{-67} -17714 q^{-68} -17580 q^{-69} -10597 q^{-70} -194 q^{-71} +9806 q^{-72} +12797 q^{-73} +10460 q^{-74} +4065 q^{-75} -3942 q^{-76} -7862 q^{-77} -8324 q^{-78} -5364 q^{-79} +153 q^{-80} +3845 q^{-81} +5689 q^{-82} +4915 q^{-83} +1542 q^{-84} -1117 q^{-85} -3181 q^{-86} -3682 q^{-87} -1983 q^{-88} -293 q^{-89} +1431 q^{-90} +2297 q^{-91} +1615 q^{-92} +851 q^{-93} -342 q^{-94} -1288 q^{-95} -1099 q^{-96} -782 q^{-97} -83 q^{-98} +543 q^{-99} +559 q^{-100} +625 q^{-101} +267 q^{-102} -237 q^{-103} -296 q^{-104} -346 q^{-105} -172 q^{-106} +44 q^{-107} +50 q^{-108} +208 q^{-109} +172 q^{-110} -16 q^{-111} -40 q^{-112} -97 q^{-113} -48 q^{-114} +3 q^{-115} -38 q^{-116} +38 q^{-117} +60 q^{-118} +4 q^{-119} - q^{-120} -27 q^{-121} -5 q^{-122} +13 q^{-123} -21 q^{-124} + q^{-125} +14 q^{-126} +2 q^{-127} +2 q^{-128} -9 q^{-129} +8 q^{-131} -5 q^{-132} -2 q^{-133} +2 q^{-134} + q^{-136} -2 q^{-137} +2 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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