10 60

10_59

10_61

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 10 60's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 60 at Knotilus!

Two figure 8 knots on a loop, interlinked.

Knotscape

Knot presentations

Planar diagram presentation	X4251 X10,6,11,5 X8394 X2,9,3,10 X16,12,17,11 X14,7,15,8 X6,15,7,16 X20,18,1,17 X18,13,19,14 X12,19,13,20
Gauss code	1, -4, 3, -1, 2, -7, 6, -3, 4, -2, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8
Dowker-Thistlethwaite code	4 8 10 14 2 16 18 6 20 12
Conway Notation	[211,211,2]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 12, width is 5, Braid index is 5		[{2, 13}, {1, 10}, {12, 6}, {13, 11}, {9, 3}, {10, 8}, {7, 9}, {8, 12}, {5, 2}, {6, 4}, {3, 5}, {4, 7}, {11, 1}]

[edit Notes on presentations of 10 60]

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["10 60"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X10,6,11,5 X8394 X2,9,3,10 X16,12,17,11 X14,7,15,8 X6,15,7,16 X20,18,1,17 X18,13,19,14 X12,19,13,20

In[5]:=

GaussCode[K]

Out[5]=

1, -4, 3, -1, 2, -7, 6, -3, 4, -2, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8

In[6]:=

DTCode[K]

Out[6]=

4 8 10 14 2 16 18 6 20 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[211,211,2]

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}(5,\{-1,2,-1,2,2,-3,2,-3,-2,-4,3,-4\}) }[/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]=

{ 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[{2, 13}, {1, 10}, {12, 6}, {13, 11}, {9, 3}, {10, 8}, {7, 9}, {8, 12}, {5, 2}, {6, 4}, {3, 5}, {4, 7}, {11, 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 Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-5] Hyperbolic Volume 13.98 A-Polynomial See Data:10 60/A-polynomial

[edit Notes for 10 60'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{ 3 }[/math] Rasmussen s-Invariant 0

[edit Notes for 10 60's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+7 t^2-20 t+29-20 t^{-1} +7 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	{ 85, 0 }
Jones polynomial	[math]\displaystyle{ q^4-4 q^3+8 q^2-11 q+14-14 q^{-1} +13 q^{-2} -10 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^6-3 z^2 a^4-3 a^4+3 z^4 a^2+6 z^2 a^2+4 a^2-z^6-3 z^4-5 z^2-2+z^4 a^{-2} +z^2 a^{-2} + a^{-2} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^3 z^9+a z^9+3 a^4 z^8+8 a^2 z^8+5 z^8+3 a^5 z^7+10 a^3 z^7+16 a z^7+9 z^7 a^{-1} +a^6 z^6-3 a^4 z^6-7 a^2 z^6+8 z^6 a^{-2} +5 z^6-9 a^5 z^5-32 a^3 z^5-38 a z^5-11 z^5 a^{-1} +4 z^5 a^{-3} -3 a^6 z^4-8 a^4 z^4-17 a^2 z^4-9 z^4 a^{-2} +z^4 a^{-4} -22 z^4+9 a^5 z^3+27 a^3 z^3+25 a z^3+5 z^3 a^{-1} -2 z^3 a^{-3} +3 a^6 z^2+11 a^4 z^2+18 a^2 z^2+4 z^2 a^{-2} +14 z^2-3 a^5 z-7 a^3 z-6 a z-2 z a^{-1} -a^6-3 a^4-4 a^2- a^{-2} -2 }[/math]
The A2 invariant	[math]\displaystyle{ q^{20}+q^{18}-2 q^{16}-3 q^{10}+3 q^8+q^4+2 q^2-2+3 q^{-2} -3 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{94}-2 q^{92}+6 q^{90}-10 q^{88}+12 q^{86}-11 q^{84}+22 q^{80}-45 q^{78}+69 q^{76}-72 q^{74}+47 q^{72}+7 q^{70}-83 q^{68}+155 q^{66}-189 q^{64}+162 q^{62}-75 q^{60}-59 q^{58}+184 q^{56}-259 q^{54}+248 q^{52}-149 q^{50}-2 q^{48}+141 q^{46}-220 q^{44}+196 q^{42}-90 q^{40}-48 q^{38}+160 q^{36}-186 q^{34}+112 q^{32}+35 q^{30}-189 q^{28}+289 q^{26}-278 q^{24}+162 q^{22}+33 q^{20}-231 q^{18}+361 q^{16}-373 q^{14}+267 q^{12}-75 q^{10}-130 q^8+270 q^6-303 q^4+226 q^2-75-81 q^{-2} +172 q^{-4} -168 q^{-6} +73 q^{-8} +61 q^{-10} -170 q^{-12} +207 q^{-14} -146 q^{-16} +18 q^{-18} +122 q^{-20} -225 q^{-22} +252 q^{-24} -194 q^{-26} +83 q^{-28} +41 q^{-30} -136 q^{-32} +177 q^{-34} -161 q^{-36} +108 q^{-38} -38 q^{-40} -20 q^{-42} +53 q^{-44} -68 q^{-46} +57 q^{-48} -35 q^{-50} +17 q^{-52} + q^{-54} -8 q^{-56} +10 q^{-58} -10 q^{-60} +6 q^{-62} -3 q^{-64} + q^{-66} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_60.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{13}-2 q^{11}+3 q^9-4 q^7+3 q^5-q^3+3 q^{-1} -3 q^{-3} +4 q^{-5} -3 q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{38}-2 q^{36}-2 q^{34}+8 q^{32}-4 q^{30}-12 q^{28}+20 q^{26}+q^{24}-30 q^{22}+24 q^{20}+16 q^{18}-38 q^{16}+12 q^{14}+26 q^{12}-25 q^{10}-7 q^8+20 q^6+2 q^4-20 q^2+3+29 q^{-2} -23 q^{-4} -17 q^{-6} +39 q^{-8} -13 q^{-10} -24 q^{-12} +28 q^{-14} -2 q^{-16} -15 q^{-18} +9 q^{-20} + q^{-22} -3 q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ q^{75}-2 q^{73}-2 q^{71}+3 q^{69}+8 q^{67}-4 q^{65}-19 q^{63}+2 q^{61}+35 q^{59}+10 q^{57}-56 q^{55}-36 q^{53}+74 q^{51}+77 q^{49}-79 q^{47}-130 q^{45}+61 q^{43}+187 q^{41}-22 q^{39}-224 q^{37}-39 q^{35}+239 q^{33}+105 q^{31}-228 q^{29}-159 q^{27}+186 q^{25}+200 q^{23}-132 q^{21}-216 q^{19}+68 q^{17}+212 q^{15}-3 q^{13}-185 q^{11}-66 q^9+151 q^7+127 q^5-96 q^3-183 q+32 q^{-1} +223 q^{-3} +37 q^{-5} -239 q^{-7} -105 q^{-9} +230 q^{-11} +154 q^{-13} -187 q^{-15} -181 q^{-17} +136 q^{-19} +177 q^{-21} -80 q^{-23} -150 q^{-25} +38 q^{-27} +105 q^{-29} -8 q^{-31} -68 q^{-33} -3 q^{-35} +38 q^{-37} +2 q^{-39} -14 q^{-41} -3 q^{-43} +6 q^{-45} + q^{-47} -3 q^{-49} + q^{-51} }[/math]
4	[math]\displaystyle{ q^{124}-2 q^{122}-2 q^{120}+3 q^{118}+3 q^{116}+8 q^{114}-11 q^{112}-19 q^{110}+2 q^{108}+17 q^{106}+53 q^{104}-13 q^{102}-80 q^{100}-55 q^{98}+18 q^{96}+191 q^{94}+87 q^{92}-146 q^{90}-262 q^{88}-158 q^{86}+363 q^{84}+436 q^{82}+32 q^{80}-526 q^{78}-711 q^{76}+212 q^{74}+881 q^{72}+712 q^{70}-382 q^{68}-1405 q^{66}-552 q^{64}+848 q^{62}+1571 q^{60}+454 q^{58}-1559 q^{56}-1531 q^{54}+65 q^{52}+1887 q^{50}+1500 q^{48}-920 q^{46}-1987 q^{44}-950 q^{42}+1434 q^{40}+2035 q^{38}+23 q^{36}-1731 q^{34}-1557 q^{32}+643 q^{30}+1904 q^{28}+747 q^{26}-1098 q^{24}-1650 q^{22}-125 q^{20}+1387 q^{18}+1223 q^{16}-330 q^{14}-1451 q^{12}-878 q^{10}+624 q^8+1548 q^6+608 q^4-963 q^2-1604-430 q^{-2} +1540 q^{-4} +1568 q^{-6} -71 q^{-8} -1901 q^{-10} -1520 q^{-12} +929 q^{-14} +2007 q^{-16} +961 q^{-18} -1423 q^{-20} -2014 q^{-22} +1563 q^{-26} +1455 q^{-28} -508 q^{-30} -1607 q^{-32} -557 q^{-34} +679 q^{-36} +1159 q^{-38} +104 q^{-40} -801 q^{-42} -489 q^{-44} +82 q^{-46} +562 q^{-48} +199 q^{-50} -247 q^{-52} -206 q^{-54} -64 q^{-56} +174 q^{-58} +86 q^{-60} -51 q^{-62} -41 q^{-64} -35 q^{-66} +35 q^{-68} +18 q^{-70} -10 q^{-72} -2 q^{-74} -6 q^{-76} +6 q^{-78} + q^{-80} -3 q^{-82} + q^{-84} }[/math]
5	[math]\displaystyle{ q^{185}-2 q^{183}-2 q^{181}+3 q^{179}+3 q^{177}+3 q^{175}+q^{173}-11 q^{171}-19 q^{169}+2 q^{167}+26 q^{165}+35 q^{163}+21 q^{161}-37 q^{159}-96 q^{157}-74 q^{155}+49 q^{153}+178 q^{151}+192 q^{149}+14 q^{147}-285 q^{145}-436 q^{143}-199 q^{141}+354 q^{139}+784 q^{137}+623 q^{135}-217 q^{133}-1192 q^{131}-1379 q^{129}-298 q^{127}+1450 q^{125}+2397 q^{123}+1412 q^{121}-1197 q^{119}-3475 q^{117}-3188 q^{115}+99 q^{113}+4143 q^{111}+5399 q^{109}+2083 q^{107}-3824 q^{105}-7540 q^{103}-5277 q^{101}+2111 q^{99}+8914 q^{97}+8920 q^{95}+1106 q^{93}-8786 q^{91}-12301 q^{89}-5470 q^{87}+6927 q^{85}+14531 q^{83}+10158 q^{81}-3404 q^{79}-15031 q^{77}-14349 q^{75}-1132 q^{73}+13740 q^{71}+17219 q^{69}+5794 q^{67}-10917 q^{65}-18399 q^{63}-9867 q^{61}+7313 q^{59}+17977 q^{57}+12704 q^{55}-3602 q^{53}-16274 q^{51}-14229 q^{49}+318 q^{47}+13932 q^{45}+14556 q^{43}+2264 q^{41}-11369 q^{39}-14124 q^{37}-4186 q^{35}+8885 q^{33}+13315 q^{31}+5752 q^{29}-6516 q^{27}-12480 q^{25}-7229 q^{23}+4096 q^{21}+11575 q^{19}+8978 q^{17}-1323 q^{15}-10563 q^{13}-10941 q^{11}-1993 q^9+9010 q^7+12920 q^5+5974 q^3-6650 q-14461 q^{-1} -10268 q^{-3} +3254 q^{-5} +14958 q^{-7} +14366 q^{-9} +1074 q^{-11} -14016 q^{-13} -17544 q^{-15} -5758 q^{-17} +11455 q^{-19} +19058 q^{-21} +10134 q^{-23} -7620 q^{-25} -18599 q^{-27} -13299 q^{-29} +3218 q^{-31} +16217 q^{-33} +14720 q^{-35} +880 q^{-37} -12534 q^{-39} -14220 q^{-41} -3941 q^{-43} +8410 q^{-45} +12217 q^{-47} +5483 q^{-49} -4658 q^{-51} -9318 q^{-53} -5678 q^{-55} +1833 q^{-57} +6393 q^{-59} +4845 q^{-61} -169 q^{-63} -3844 q^{-65} -3594 q^{-67} -599 q^{-69} +2057 q^{-71} +2369 q^{-73} +716 q^{-75} -970 q^{-77} -1356 q^{-79} -574 q^{-81} +372 q^{-83} +713 q^{-85} +381 q^{-87} -149 q^{-89} -329 q^{-91} -184 q^{-93} +31 q^{-95} +137 q^{-97} +92 q^{-99} -6 q^{-101} -61 q^{-103} -32 q^{-105} +9 q^{-107} +15 q^{-109} +6 q^{-111} +2 q^{-113} -5 q^{-115} -6 q^{-117} +6 q^{-119} + q^{-121} -3 q^{-123} + q^{-125} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{20}+q^{18}-2 q^{16}-3 q^{10}+3 q^8+q^4+2 q^2-2+3 q^{-2} -3 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} + q^{-12} }[/math]
2,0	[math]\displaystyle{ q^{52}+q^{50}-q^{48}-5 q^{46}-2 q^{44}+6 q^{42}+5 q^{40}-6 q^{38}-6 q^{36}+11 q^{34}+12 q^{32}-12 q^{30}-15 q^{28}+10 q^{26}+11 q^{24}-13 q^{22}-12 q^{20}+15 q^{18}+10 q^{16}-11 q^{14}-q^{12}+9 q^{10}-6 q^8-2 q^6+9 q^4-6 q^2-10+11 q^{-2} +15 q^{-4} -17 q^{-6} -12 q^{-8} +20 q^{-10} +10 q^{-12} -18 q^{-14} -5 q^{-16} +14 q^{-18} +3 q^{-20} -8 q^{-22} -3 q^{-24} +5 q^{-26} -2 q^{-30} + q^{-32} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{40}-2 q^{38}+2 q^{36}+3 q^{34}-9 q^{32}+7 q^{30}+6 q^{28}-20 q^{26}+14 q^{24}+11 q^{22}-28 q^{20}+17 q^{18}+14 q^{16}-27 q^{14}+8 q^{12}+13 q^{10}-12 q^8-5 q^6+7 q^4+11 q^2-11-6 q^{-2} +27 q^{-4} -15 q^{-6} -17 q^{-8} +30 q^{-10} -11 q^{-12} -17 q^{-14} +22 q^{-16} -4 q^{-18} -10 q^{-20} +9 q^{-22} -3 q^{-26} + q^{-28} }[/math]
1,0,0	[math]\displaystyle{ q^{27}+q^{25}+q^{23}-2 q^{21}-3 q^{17}-3 q^{13}+4 q^{11}+3 q^7+q^5+q^3-2 q^{-1} +2 q^{-3} -3 q^{-5} +2 q^{-7} - q^{-9} +3 q^{-11} -2 q^{-13} + q^{-15} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{40}-2 q^{38}+6 q^{36}-9 q^{34}+15 q^{32}-21 q^{30}+26 q^{28}-32 q^{26}+32 q^{24}-31 q^{22}+22 q^{20}-13 q^{18}-2 q^{16}+19 q^{14}-34 q^{12}+51 q^{10}-58 q^8+65 q^6-61 q^4+55 q^2-43+26 q^{-2} -9 q^{-4} -7 q^{-6} +19 q^{-8} -28 q^{-10} +33 q^{-12} -33 q^{-14} +30 q^{-16} -24 q^{-18} +18 q^{-20} -11 q^{-22} +6 q^{-24} -3 q^{-26} + q^{-28} }[/math]
1,0	[math]\displaystyle{ q^{66}-2 q^{62}-2 q^{60}+4 q^{58}+6 q^{56}-3 q^{54}-12 q^{52}-4 q^{50}+16 q^{48}+15 q^{46}-12 q^{44}-27 q^{42}-q^{40}+32 q^{38}+19 q^{36}-25 q^{34}-31 q^{32}+9 q^{30}+36 q^{28}+7 q^{26}-30 q^{24}-17 q^{22}+19 q^{20}+20 q^{18}-13 q^{16}-20 q^{14}+7 q^{12}+21 q^{10}-4 q^8-22 q^6-q^4+25 q^2+10-23 q^{-2} -17 q^{-4} +22 q^{-6} +27 q^{-8} -12 q^{-10} -34 q^{-12} -3 q^{-14} +33 q^{-16} +19 q^{-18} -22 q^{-20} -29 q^{-22} +5 q^{-24} +26 q^{-26} +9 q^{-28} -14 q^{-30} -14 q^{-32} +3 q^{-34} +10 q^{-36} +3 q^{-38} -3 q^{-40} -3 q^{-42} + q^{-46} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{94}-2 q^{92}+6 q^{90}-10 q^{88}+12 q^{86}-11 q^{84}+22 q^{80}-45 q^{78}+69 q^{76}-72 q^{74}+47 q^{72}+7 q^{70}-83 q^{68}+155 q^{66}-189 q^{64}+162 q^{62}-75 q^{60}-59 q^{58}+184 q^{56}-259 q^{54}+248 q^{52}-149 q^{50}-2 q^{48}+141 q^{46}-220 q^{44}+196 q^{42}-90 q^{40}-48 q^{38}+160 q^{36}-186 q^{34}+112 q^{32}+35 q^{30}-189 q^{28}+289 q^{26}-278 q^{24}+162 q^{22}+33 q^{20}-231 q^{18}+361 q^{16}-373 q^{14}+267 q^{12}-75 q^{10}-130 q^8+270 q^6-303 q^4+226 q^2-75-81 q^{-2} +172 q^{-4} -168 q^{-6} +73 q^{-8} +61 q^{-10} -170 q^{-12} +207 q^{-14} -146 q^{-16} +18 q^{-18} +122 q^{-20} -225 q^{-22} +252 q^{-24} -194 q^{-26} +83 q^{-28} +41 q^{-30} -136 q^{-32} +177 q^{-34} -161 q^{-36} +108 q^{-38} -38 q^{-40} -20 q^{-42} +53 q^{-44} -68 q^{-46} +57 q^{-48} -35 q^{-50} +17 q^{-52} + q^{-54} -8 q^{-56} +10 q^{-58} -10 q^{-60} +6 q^{-62} -3 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["10 60"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+7 t^2-20 t+29-20 t^{-1} +7 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]=

{ 85, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^4-4 q^3+8 q^2-11 q+14-14 q^{-1} +13 q^{-2} -10 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^3 z^9+a z^9+3 a^4 z^8+8 a^2 z^8+5 z^8+3 a^5 z^7+10 a^3 z^7+16 a z^7+9 z^7 a^{-1} +a^6 z^6-3 a^4 z^6-7 a^2 z^6+8 z^6 a^{-2} +5 z^6-9 a^5 z^5-32 a^3 z^5-38 a z^5-11 z^5 a^{-1} +4 z^5 a^{-3} -3 a^6 z^4-8 a^4 z^4-17 a^2 z^4-9 z^4 a^{-2} +z^4 a^{-4} -22 z^4+9 a^5 z^3+27 a^3 z^3+25 a z^3+5 z^3 a^{-1} -2 z^3 a^{-3} +3 a^6 z^2+11 a^4 z^2+18 a^2 z^2+4 z^2 a^{-2} +14 z^2-3 a^5 z-7 a^3 z-6 a z-2 z a^{-1} -a^6-3 a^4-4 a^2- a^{-2} -2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n165,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {10_86,}

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["10 60"];

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+7 t^2-20 t+29-20 t^{-1} +7 t^{-2} - t^{-3} }[/math], [math]\displaystyle{ q^4-4 q^3+8 q^2-11 q+14-14 q^{-1} +13 q^{-2} -10 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6} }[/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]=

{K11n165,}

In[6]:=

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

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

Out[6]=

{10_86,}

Vassiliev invariants

V2 and V3:

(-1, 1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ -4 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{62}{3} }[/math] [math]\displaystyle{ -\frac{10}{3} }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ \frac{80}{3} }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{248}{3} }[/math] [math]\displaystyle{ \frac{40}{3} }[/math] [math]\displaystyle{ \frac{2609}{30} }[/math] [math]\displaystyle{ \frac{1382}{15} }[/math] [math]\displaystyle{ -\frac{3062}{45} }[/math] [math]\displaystyle{ -\frac{17}{18} }[/math] [math]\displaystyle{ -\frac{271}{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 10 60. 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 4 χ 9                     1 1 7                   3   -3 5                 5 1   4 3               6 3     -3 1             8 5       3 -1           7 7         0 -3         6 7           -1 -5       4 7             3 -7     2 6               -4 -9   1 4                 3 -11   2                   -2 -13 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}-4 q^{11}+4 q^{10}+9 q^9-28 q^8+17 q^7+39 q^6-80 q^5+28 q^4+91 q^3-136 q^2+22 q+143-162 q^{-1} - q^{-2} +165 q^{-3} -144 q^{-4} -28 q^{-5} +147 q^{-6} -93 q^{-7} -42 q^{-8} +97 q^{-9} -39 q^{-10} -34 q^{-11} +43 q^{-12} -8 q^{-13} -15 q^{-14} +11 q^{-15} -3 q^{-17} + q^{-18} }[/math]
3	[math]\displaystyle{ q^{24}-4 q^{23}+4 q^{22}+5 q^{21}-8 q^{20}-15 q^{19}+20 q^{18}+41 q^{17}-49 q^{16}-80 q^{15}+80 q^{14}+154 q^{13}-116 q^{12}-268 q^{11}+150 q^{10}+411 q^9-157 q^8-585 q^7+144 q^6+752 q^5-81 q^4-920 q^3+10 q^2+1028 q+105-1111 q^{-1} -205 q^{-2} +1115 q^{-3} +328 q^{-4} -1087 q^{-5} -422 q^{-6} +996 q^{-7} +510 q^{-8} -872 q^{-9} -566 q^{-10} +712 q^{-11} +594 q^{-12} -540 q^{-13} -580 q^{-14} +367 q^{-15} +525 q^{-16} -207 q^{-17} -446 q^{-18} +89 q^{-19} +340 q^{-20} -5 q^{-21} -237 q^{-22} -37 q^{-23} +149 q^{-24} +46 q^{-25} -81 q^{-26} -40 q^{-27} +39 q^{-28} +26 q^{-29} -15 q^{-30} -15 q^{-31} +6 q^{-32} +5 q^{-33} -3 q^{-35} + q^{-36} }[/math]
4	[math]\displaystyle{ q^{40}-4 q^{39}+4 q^{38}+5 q^{37}-12 q^{36}+5 q^{35}-12 q^{34}+32 q^{33}+22 q^{32}-82 q^{31}-q^{30}-22 q^{29}+169 q^{28}+110 q^{27}-320 q^{26}-143 q^{25}-63 q^{24}+615 q^{23}+473 q^{22}-800 q^{21}-714 q^{20}-375 q^{19}+1520 q^{18}+1528 q^{17}-1280 q^{16}-1950 q^{15}-1425 q^{14}+2619 q^{13}+3491 q^{12}-1172 q^{11}-3513 q^{10}-3439 q^9+3210 q^8+5875 q^7-126 q^6-4591 q^5-5888 q^4+2829 q^3+7705 q^2+1513 q-4619-7858 q^{-1} +1655 q^{-2} +8346 q^{-3} +3084 q^{-4} -3679 q^{-5} -8782 q^{-6} +153 q^{-7} +7773 q^{-8} +4205 q^{-9} -2126 q^{-10} -8618 q^{-11} -1359 q^{-12} +6248 q^{-13} +4757 q^{-14} -281 q^{-15} -7461 q^{-16} -2620 q^{-17} +4048 q^{-18} +4583 q^{-19} +1473 q^{-20} -5449 q^{-21} -3221 q^{-22} +1664 q^{-23} +3546 q^{-24} +2540 q^{-25} -3029 q^{-26} -2834 q^{-27} -158 q^{-28} +1950 q^{-29} +2512 q^{-30} -1016 q^{-31} -1717 q^{-32} -881 q^{-33} +550 q^{-34} +1659 q^{-35} +7 q^{-36} -623 q^{-37} -712 q^{-38} -119 q^{-39} +736 q^{-40} +192 q^{-41} -65 q^{-42} -308 q^{-43} -192 q^{-44} +215 q^{-45} +88 q^{-46} +51 q^{-47} -75 q^{-48} -88 q^{-49} +42 q^{-50} +15 q^{-51} +26 q^{-52} -8 q^{-53} -22 q^{-54} +6 q^{-55} +5 q^{-57} -3 q^{-59} + q^{-60} }[/math]
5	[math]\displaystyle{ q^{60}-4 q^{59}+4 q^{58}+5 q^{57}-12 q^{56}+q^{55}+8 q^{54}+13 q^{52}-q^{51}-53 q^{50}-28 q^{49}+63 q^{48}+98 q^{47}+58 q^{46}-107 q^{45}-268 q^{44}-173 q^{43}+243 q^{42}+628 q^{41}+390 q^{40}-448 q^{39}-1214 q^{38}-955 q^{37}+629 q^{36}+2314 q^{35}+2043 q^{34}-760 q^{33}-3870 q^{32}-3950 q^{31}+379 q^{30}+5989 q^{29}+7057 q^{28}+788 q^{27}-8430 q^{26}-11461 q^{25}-3261 q^{24}+10649 q^{23}+17198 q^{22}+7522 q^{21}-12237 q^{20}-23812 q^{19}-13540 q^{18}+12335 q^{17}+30612 q^{16}+21362 q^{15}-10740 q^{14}-36811 q^{13}-30057 q^{12}+7035 q^{11}+41591 q^{10}+39116 q^9-1816 q^8-44414 q^7-47270 q^6-4751 q^5+45119 q^4+54206 q^3+11476 q^2-43822 q-58974-18273 q^{-1} +40926 q^{-2} +62017 q^{-3} +24100 q^{-4} -36876 q^{-5} -62884 q^{-6} -29276 q^{-7} +31978 q^{-8} +62395 q^{-9} +33340 q^{-10} -26575 q^{-11} -60287 q^{-12} -36755 q^{-13} +20653 q^{-14} +57144 q^{-15} +39304 q^{-16} -14307 q^{-17} -52724 q^{-18} -41185 q^{-19} +7582 q^{-20} +47206 q^{-21} +42059 q^{-22} -674 q^{-23} -40432 q^{-24} -41809 q^{-25} -6032 q^{-26} +32659 q^{-27} +40014 q^{-28} +11998 q^{-29} -24126 q^{-30} -36536 q^{-31} -16696 q^{-32} +15479 q^{-33} +31482 q^{-34} +19480 q^{-35} -7415 q^{-36} -25111 q^{-37} -20175 q^{-38} +607 q^{-39} +18265 q^{-40} +18798 q^{-41} +4212 q^{-42} -11549 q^{-43} -15802 q^{-44} -6997 q^{-45} +5868 q^{-46} +11967 q^{-47} +7727 q^{-48} -1657 q^{-49} -7988 q^{-50} -7003 q^{-51} -935 q^{-52} +4579 q^{-53} +5464 q^{-54} +2059 q^{-55} -2081 q^{-56} -3687 q^{-57} -2191 q^{-58} +535 q^{-59} +2177 q^{-60} +1772 q^{-61} +197 q^{-62} -1078 q^{-63} -1206 q^{-64} -412 q^{-65} +429 q^{-66} +691 q^{-67} +384 q^{-68} -103 q^{-69} -366 q^{-70} -251 q^{-71} - q^{-72} +138 q^{-73} +147 q^{-74} +48 q^{-75} -67 q^{-76} -73 q^{-77} -15 q^{-78} +9 q^{-79} +24 q^{-80} +26 q^{-81} -8 q^{-82} -15 q^{-83} - q^{-84} +5 q^{-87} -3 q^{-89} + q^{-90} }[/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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