10 54

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10 53.gif

10_53

10 55.gif

10_55

10 54.gif
(KnotPlot image)

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[edit 10 54 Quick Notes]
[edit 10 54 Further Notes and Views]


Knot presentations

Planar diagram presentation X1425 X3,10,4,11 X7,12,8,13 X11,8,12,9 X13,19,14,18 X5,17,6,16 X17,7,18,6 X15,1,16,20 X19,15,20,14 X9,2,10,3
Gauss code -1, 10, -2, 1, -6, 7, -3, 4, -10, 2, -4, 3, -5, 9, -8, 6, -7, 5, -9, 8
Dowker-Thistlethwaite code 4 10 16 12 2 8 18 20 6 14
Conway Notation [23,3,2]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif

Length is 11, width is 4,

Braid index is 4

10 54 ML.gif 10 54 AP.gif
[{12, 4}, {3, 10}, {9, 11}, {10, 12}, {11, 8}, {6, 9}, {5, 7}, {4, 6}, {2, 5}, {1, 3}, {8, 2}, {7, 1}]

[edit Notes on presentations of 10 54]


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 54"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X3,10,4,11 X7,12,8,13 X11,8,12,9 X13,19,14,18 X5,17,6,16 X17,7,18,6 X15,1,16,20 X19,15,20,14 X9,2,10,3
In[5]:= GaussCode[K]
Out[5]= -1, 10, -2, 1, -6, 7, -3, 4, -10, 2, -4, 3, -5, 9, -8, 6, -7, 5, -9, 8
In[6]:= DTCode[K]
Out[6]= 4 10 16 12 2 8 18 20 6 14

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [23,3,2]
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,\{1,1,1,-2,1,1,-2,-3,2,-3,-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, 11, 4 }
In[11]:= Show[BraidPlot[br]]
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif
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."
10 54 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{12, 4}, {3, 10}, {9, 11}, {10, 12}, {11, 8}, {6, 9}, {5, 7}, {4, 6}, {2, 5}, {1, 3}, {8, 2}, {7, 1}]
In[14]:= Draw[ap]
10 54 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number [math]\displaystyle{ \{2,3\} }[/math]
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-4][-8]
Hyperbolic Volume 10.5913
A-Polynomial See Data:10 54/A-polynomial

[edit Notes for 10 54'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 2

[edit Notes for 10 54's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^3-6 t^2+10 t-11+10 t^{-1} -6 t^{-2} +2 t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ 2 z^6+6 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 47, 2 }
Jones polynomial [math]\displaystyle{ -q^6+2 q^5-4 q^4+6 q^3-7 q^2+8 q-6+6 q^{-1} -4 q^{-2} +2 q^{-3} - q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-2} +z^6-a^2 z^4+4 z^4 a^{-2} -z^4 a^{-4} +4 z^4-3 a^2 z^2+5 z^2 a^{-2} -3 z^2 a^{-4} +5 z^2-2 a^2+2 a^{-2} -2 a^{-4} +3 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a z^9+z^9 a^{-1} +2 a^2 z^8+3 z^8 a^{-2} +5 z^8+a^3 z^7+3 z^7 a^{-1} +4 z^7 a^{-3} -9 a^2 z^6-5 z^6 a^{-2} +4 z^6 a^{-4} -18 z^6-5 a^3 z^5-13 a z^5-18 z^5 a^{-1} -7 z^5 a^{-3} +3 z^5 a^{-5} +12 a^2 z^4-3 z^4 a^{-2} -6 z^4 a^{-4} +2 z^4 a^{-6} +17 z^4+8 a^3 z^3+20 a z^3+17 z^3 a^{-1} +2 z^3 a^{-3} -2 z^3 a^{-5} +z^3 a^{-7} -6 a^2 z^2+5 z^2 a^{-2} +5 z^2 a^{-4} -z^2 a^{-6} -7 z^2-4 a^3 z-8 a z-5 z a^{-1} +z a^{-3} +z a^{-5} -z a^{-7} +2 a^2-2 a^{-2} -2 a^{-4} +3 }[/math]
The A2 invariant [math]\displaystyle{ -q^{12}-q^8-q^6+q^4+3+2 q^{-2} + q^{-4} +2 q^{-6} - q^{-8} + q^{-10} - q^{-12} - q^{-14} - q^{-18} }[/math]
The G2 invariant [math]\displaystyle{ q^{60}-q^{58}+4 q^{56}-6 q^{54}+6 q^{52}-5 q^{50}-3 q^{48}+13 q^{46}-22 q^{44}+24 q^{42}-19 q^{40}+2 q^{38}+16 q^{36}-34 q^{34}+38 q^{32}-29 q^{30}+7 q^{28}+12 q^{26}-30 q^{24}+29 q^{22}-18 q^{20}+q^{18}+15 q^{16}-22 q^{14}+17 q^{12}-q^{10}-15 q^8+28 q^6-28 q^4+24 q^2-2-14 q^{-2} +35 q^{-4} -40 q^{-6} +40 q^{-8} -18 q^{-10} -5 q^{-12} +27 q^{-14} -36 q^{-16} +34 q^{-18} -16 q^{-20} - q^{-22} +18 q^{-24} -22 q^{-26} +14 q^{-28} -15 q^{-32} +21 q^{-34} -16 q^{-36} +3 q^{-38} +11 q^{-40} -19 q^{-42} +21 q^{-44} -15 q^{-46} +7 q^{-48} + q^{-50} -11 q^{-52} +13 q^{-54} -14 q^{-56} +12 q^{-58} -8 q^{-60} +3 q^{-62} + q^{-64} -8 q^{-66} +9 q^{-68} -12 q^{-70} +9 q^{-72} -5 q^{-74} + q^{-76} +2 q^{-78} -6 q^{-80} +6 q^{-82} -5 q^{-84} +4 q^{-86} -2 q^{-88} + q^{-92} -2 q^{-94} +2 q^{-96} - q^{-98} + q^{-100} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_54.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^9+q^7-2 q^5+2 q^3+2 q^{-1} + q^{-3} - q^{-5} +2 q^{-7} -2 q^{-9} + q^{-11} - q^{-13} }[/math]
2 [math]\displaystyle{ q^{28}-q^{26}-2 q^{24}+4 q^{22}+q^{20}-7 q^{18}+3 q^{16}+5 q^{14}-8 q^{12}-q^{10}+7 q^8-4 q^6-4 q^4+7 q^2+2-4 q^{-2} +3 q^{-4} +6 q^{-6} -3 q^{-8} -3 q^{-10} +6 q^{-12} + q^{-14} -8 q^{-16} +3 q^{-18} +3 q^{-20} -6 q^{-22} +2 q^{-24} + q^{-26} -3 q^{-28} +2 q^{-30} - q^{-34} + q^{-36} }[/math]
3 [math]\displaystyle{ -q^{57}+q^{55}+2 q^{53}-5 q^{49}-3 q^{47}+7 q^{45}+9 q^{43}-5 q^{41}-15 q^{39}-2 q^{37}+19 q^{35}+11 q^{33}-15 q^{31}-20 q^{29}+6 q^{27}+25 q^{25}+3 q^{23}-22 q^{21}-16 q^{19}+16 q^{17}+20 q^{15}-8 q^{13}-26 q^{11}+q^9+23 q^7+9 q^5-23 q^3-9 q+24 q^{-1} +17 q^{-3} -20 q^{-5} -19 q^{-7} +18 q^{-9} +23 q^{-11} -8 q^{-13} -24 q^{-15} +20 q^{-19} +15 q^{-21} -14 q^{-23} -24 q^{-25} + q^{-27} +27 q^{-29} +6 q^{-31} -23 q^{-33} -16 q^{-35} +17 q^{-37} +13 q^{-39} -9 q^{-41} -10 q^{-43} +3 q^{-45} +6 q^{-47} -2 q^{-51} + q^{-57} + q^{-59} - q^{-61} + q^{-67} - q^{-69} }[/math]
4 [math]\displaystyle{ q^{96}-q^{94}-2 q^{92}+q^{88}+7 q^{86}+q^{84}-7 q^{82}-9 q^{80}-9 q^{78}+16 q^{76}+20 q^{74}+8 q^{72}-14 q^{70}-42 q^{68}-10 q^{66}+22 q^{64}+48 q^{62}+33 q^{60}-40 q^{58}-56 q^{56}-41 q^{54}+31 q^{52}+88 q^{50}+38 q^{48}-25 q^{46}-97 q^{44}-64 q^{42}+47 q^{40}+93 q^{38}+80 q^{36}-44 q^{34}-123 q^{32}-66 q^{30}+40 q^{28}+135 q^{26}+66 q^{24}-73 q^{22}-128 q^{20}-60 q^{18}+98 q^{16}+130 q^{14}+13 q^{12}-111 q^{10}-109 q^8+37 q^6+125 q^4+55 q^2-76-107 q^{-2} +9 q^{-4} +112 q^{-6} +60 q^{-8} -69 q^{-10} -104 q^{-12} -8 q^{-14} +111 q^{-16} +85 q^{-18} -41 q^{-20} -110 q^{-22} -70 q^{-24} +62 q^{-26} +120 q^{-28} +53 q^{-30} -55 q^{-32} -141 q^{-34} -67 q^{-36} +81 q^{-38} +149 q^{-40} +75 q^{-42} -118 q^{-44} -169 q^{-46} -34 q^{-48} +132 q^{-50} +164 q^{-52} -19 q^{-54} -149 q^{-56} -98 q^{-58} +44 q^{-60} +132 q^{-62} +35 q^{-64} -64 q^{-66} -70 q^{-68} -8 q^{-70} +62 q^{-72} +26 q^{-74} -15 q^{-76} -25 q^{-78} -12 q^{-80} +21 q^{-82} +6 q^{-84} - q^{-86} -4 q^{-88} -7 q^{-90} +7 q^{-92} - q^{-94} -3 q^{-100} +3 q^{-102} - q^{-104} - q^{-110} + q^{-112} }[/math]
5 [math]\displaystyle{ -q^{145}+q^{143}+2 q^{141}-q^{137}-3 q^{135}-5 q^{133}-q^{131}+9 q^{129}+11 q^{127}+6 q^{125}-4 q^{123}-20 q^{121}-26 q^{119}-9 q^{117}+23 q^{115}+43 q^{113}+40 q^{111}+4 q^{109}-50 q^{107}-80 q^{105}-55 q^{103}+19 q^{101}+93 q^{99}+119 q^{97}+62 q^{95}-54 q^{93}-157 q^{91}-161 q^{89}-48 q^{87}+114 q^{85}+226 q^{83}+195 q^{81}+14 q^{79}-206 q^{77}-304 q^{75}-196 q^{73}+60 q^{71}+314 q^{69}+373 q^{67}+163 q^{65}-192 q^{63}-437 q^{61}-394 q^{59}-59 q^{57}+365 q^{55}+553 q^{53}+338 q^{51}-145 q^{49}-556 q^{47}-585 q^{45}-162 q^{43}+421 q^{41}+704 q^{39}+458 q^{37}-162 q^{35}-692 q^{33}-684 q^{31}-128 q^{29}+544 q^{27}+789 q^{25}+395 q^{23}-337 q^{21}-778 q^{19}-572 q^{17}+105 q^{15}+681 q^{13}+665 q^{11}+74 q^9-545 q^7-650 q^5-188 q^3+407 q+600 q^{-1} +224 q^{-3} -317 q^{-5} -511 q^{-7} -200 q^{-9} +280 q^{-11} +452 q^{-13} +162 q^{-15} -292 q^{-17} -443 q^{-19} -135 q^{-21} +321 q^{-23} +475 q^{-25} +181 q^{-27} -305 q^{-29} -541 q^{-31} -302 q^{-33} +212 q^{-35} +561 q^{-37} +477 q^{-39} +6 q^{-41} -497 q^{-43} -646 q^{-45} -308 q^{-47} +297 q^{-49} +732 q^{-51} +625 q^{-53} +16 q^{-55} -667 q^{-57} -877 q^{-59} -385 q^{-61} +473 q^{-63} +968 q^{-65} +687 q^{-67} -166 q^{-69} -903 q^{-71} -878 q^{-73} -117 q^{-75} +704 q^{-77} +887 q^{-79} +329 q^{-81} -449 q^{-83} -778 q^{-85} -416 q^{-87} +235 q^{-89} +588 q^{-91} +398 q^{-93} -84 q^{-95} -393 q^{-97} -316 q^{-99} +4 q^{-101} +241 q^{-103} +218 q^{-105} +20 q^{-107} -134 q^{-109} -128 q^{-111} -23 q^{-113} +67 q^{-115} +74 q^{-117} +16 q^{-119} -34 q^{-121} -38 q^{-123} -7 q^{-125} +13 q^{-127} +15 q^{-129} +6 q^{-131} -5 q^{-133} -9 q^{-135} -2 q^{-137} +4 q^{-139} +2 q^{-145} -2 q^{-147} +2 q^{-151} - q^{-153} - q^{-155} + q^{-157} + q^{-163} - q^{-165} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{12}-q^8-q^6+q^4+3+2 q^{-2} + q^{-4} +2 q^{-6} - q^{-8} + q^{-10} - q^{-12} - q^{-14} - q^{-18} }[/math]
1,1 [math]\displaystyle{ q^{36}-2 q^{34}+8 q^{32}-18 q^{30}+33 q^{28}-54 q^{26}+76 q^{24}-96 q^{22}+108 q^{20}-108 q^{18}+90 q^{16}-62 q^{14}+18 q^{12}+24 q^{10}-78 q^8+116 q^6-150 q^4+170 q^2-172+172 q^{-2} -134 q^{-4} +112 q^{-6} -56 q^{-8} +28 q^{-10} +15 q^{-12} -40 q^{-14} +50 q^{-16} -62 q^{-18} +49 q^{-20} -48 q^{-22} +38 q^{-24} -36 q^{-26} +30 q^{-28} -28 q^{-30} +26 q^{-32} -26 q^{-34} +22 q^{-36} -18 q^{-38} +16 q^{-40} -12 q^{-42} +9 q^{-44} -6 q^{-46} +4 q^{-48} -2 q^{-50} + q^{-52} }[/math]
2,0 [math]\displaystyle{ q^{34}-q^{30}+2 q^{26}+q^{24}-2 q^{22}-2 q^{20}+2 q^{18}-4 q^{14}-3 q^{12}-q^{10}-q^8-5 q^6-q^4+4 q^2+3+4 q^{-2} +7 q^{-4} +4 q^{-6} +3 q^{-8} +5 q^{-10} +5 q^{-12} - q^{-14} - q^{-16} +2 q^{-18} - q^{-20} -9 q^{-22} -4 q^{-24} -3 q^{-28} -2 q^{-30} +2 q^{-34} + q^{-40} + q^{-46} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{26}-q^{24}+2 q^{22}+q^{20}-3 q^{18}+2 q^{16}-3 q^{14}-7 q^{12}-4 q^8-4 q^6+7 q^4+3 q^2+2+9 q^{-2} +5 q^{-4} +3 q^{-8} +3 q^{-10} +2 q^{-12} -4 q^{-14} + q^{-16} + q^{-18} -8 q^{-20} - q^{-22} +3 q^{-24} -6 q^{-26} - q^{-28} +5 q^{-30} -3 q^{-32} -2 q^{-34} +3 q^{-36} - q^{-40} + q^{-42} }[/math]
1,0,0 [math]\displaystyle{ -q^{15}-2 q^{11}-2 q^7+q^5+3 q+2 q^{-1} +3 q^{-3} +2 q^{-5} + q^{-7} +2 q^{-9} - q^{-11} + q^{-13} -2 q^{-15} -2 q^{-19} - q^{-23} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{32}+q^{28}+3 q^{26}+q^{24}+2 q^{20}-3 q^{18}-8 q^{16}-6 q^{14}-8 q^{12}-12 q^{10}-8 q^8+3 q^6+4 q^4+2 q^2+13+19 q^{-2} +7 q^{-4} +6 q^{-6} +12 q^{-8} +4 q^{-10} -2 q^{-12} +3 q^{-14} +2 q^{-16} -3 q^{-18} -4 q^{-20} + q^{-22} -4 q^{-24} -9 q^{-26} -3 q^{-28} -6 q^{-32} -4 q^{-34} +2 q^{-36} + q^{-38} - q^{-40} - q^{-42} +2 q^{-44} +2 q^{-46} + q^{-52} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{18}-2 q^{14}-q^{12}-q^{10}-2 q^8+q^6+3 q^2+2+3 q^{-2} +3 q^{-4} +2 q^{-6} +2 q^{-8} + q^{-10} +2 q^{-12} - q^{-14} + q^{-16} -2 q^{-18} - q^{-20} - q^{-22} -2 q^{-24} - q^{-28} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{26}+q^{24}-4 q^{22}+5 q^{20}-7 q^{18}+8 q^{16}-9 q^{14}+9 q^{12}-8 q^{10}+6 q^8-2 q^6-q^4+7 q^2-10+15 q^{-2} -15 q^{-4} +18 q^{-6} -15 q^{-8} +15 q^{-10} -10 q^{-12} +8 q^{-14} -3 q^{-16} - q^{-18} +4 q^{-20} -7 q^{-22} +7 q^{-24} -8 q^{-26} +7 q^{-28} -7 q^{-30} +5 q^{-32} -4 q^{-34} +3 q^{-36} -2 q^{-38} + q^{-40} - q^{-42} }[/math]
1,0 [math]\displaystyle{ q^{44}-q^{40}-q^{38}+3 q^{36}+3 q^{34}-3 q^{32}-5 q^{30}+q^{28}+6 q^{26}+q^{24}-9 q^{22}-7 q^{20}+4 q^{18}+7 q^{16}-3 q^{14}-11 q^{12}-3 q^{10}+8 q^8+8 q^6-3 q^4-5 q^2+3+9 q^{-2} +3 q^{-4} -2 q^{-6} +7 q^{-10} +3 q^{-12} -4 q^{-14} -3 q^{-16} +6 q^{-18} +6 q^{-20} -3 q^{-22} -8 q^{-24} +7 q^{-28} + q^{-30} -8 q^{-32} -6 q^{-34} +3 q^{-36} +6 q^{-38} - q^{-40} -7 q^{-42} -4 q^{-44} +3 q^{-46} +6 q^{-48} -4 q^{-52} -3 q^{-54} + q^{-56} +3 q^{-58} + q^{-60} - q^{-62} - q^{-64} + q^{-68} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{34}-q^{32}+3 q^{30}-3 q^{28}+6 q^{26}-6 q^{24}+5 q^{22}-9 q^{20}+5 q^{18}-12 q^{16}+q^{14}-9 q^{12}+2 q^{10}-2 q^8+7 q^4+15-5 q^{-2} +16 q^{-4} -9 q^{-6} +16 q^{-8} -11 q^{-10} +14 q^{-12} -9 q^{-14} +10 q^{-16} -6 q^{-18} +5 q^{-20} -3 q^{-22} -2 q^{-24} -7 q^{-28} +2 q^{-30} -7 q^{-32} +5 q^{-34} -7 q^{-36} +4 q^{-38} -5 q^{-40} +6 q^{-42} -4 q^{-44} +2 q^{-46} -3 q^{-48} +3 q^{-50} - q^{-52} + q^{-54} - q^{-56} + q^{-58} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{60}-q^{58}+4 q^{56}-6 q^{54}+6 q^{52}-5 q^{50}-3 q^{48}+13 q^{46}-22 q^{44}+24 q^{42}-19 q^{40}+2 q^{38}+16 q^{36}-34 q^{34}+38 q^{32}-29 q^{30}+7 q^{28}+12 q^{26}-30 q^{24}+29 q^{22}-18 q^{20}+q^{18}+15 q^{16}-22 q^{14}+17 q^{12}-q^{10}-15 q^8+28 q^6-28 q^4+24 q^2-2-14 q^{-2} +35 q^{-4} -40 q^{-6} +40 q^{-8} -18 q^{-10} -5 q^{-12} +27 q^{-14} -36 q^{-16} +34 q^{-18} -16 q^{-20} - q^{-22} +18 q^{-24} -22 q^{-26} +14 q^{-28} -15 q^{-32} +21 q^{-34} -16 q^{-36} +3 q^{-38} +11 q^{-40} -19 q^{-42} +21 q^{-44} -15 q^{-46} +7 q^{-48} + q^{-50} -11 q^{-52} +13 q^{-54} -14 q^{-56} +12 q^{-58} -8 q^{-60} +3 q^{-62} + q^{-64} -8 q^{-66} +9 q^{-68} -12 q^{-70} +9 q^{-72} -5 q^{-74} + q^{-76} +2 q^{-78} -6 q^{-80} +6 q^{-82} -5 q^{-84} +4 q^{-86} -2 q^{-88} + q^{-92} -2 q^{-94} +2 q^{-96} - q^{-98} + q^{-100} }[/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 54"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^3-6 t^2+10 t-11+10 t^{-1} -6 t^{-2} +2 t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 2 z^6+6 z^4+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]= { 47, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^6+2 q^5-4 q^4+6 q^3-7 q^2+8 q-6+6 q^{-1} -4 q^{-2} +2 q^{-3} - q^{-4} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^6 a^{-2} +z^6-a^2 z^4+4 z^4 a^{-2} -z^4 a^{-4} +4 z^4-3 a^2 z^2+5 z^2 a^{-2} -3 z^2 a^{-4} +5 z^2-2 a^2+2 a^{-2} -2 a^{-4} +3 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a z^9+z^9 a^{-1} +2 a^2 z^8+3 z^8 a^{-2} +5 z^8+a^3 z^7+3 z^7 a^{-1} +4 z^7 a^{-3} -9 a^2 z^6-5 z^6 a^{-2} +4 z^6 a^{-4} -18 z^6-5 a^3 z^5-13 a z^5-18 z^5 a^{-1} -7 z^5 a^{-3} +3 z^5 a^{-5} +12 a^2 z^4-3 z^4 a^{-2} -6 z^4 a^{-4} +2 z^4 a^{-6} +17 z^4+8 a^3 z^3+20 a z^3+17 z^3 a^{-1} +2 z^3 a^{-3} -2 z^3 a^{-5} +z^3 a^{-7} -6 a^2 z^2+5 z^2 a^{-2} +5 z^2 a^{-4} -z^2 a^{-6} -7 z^2-4 a^3 z-8 a z-5 z a^{-1} +z a^{-3} +z a^{-5} -z a^{-7} +2 a^2-2 a^{-2} -2 a^{-4} +3 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_12,}

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["10 54"];
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{ 2 t^3-6 t^2+10 t-11+10 t^{-1} -6 t^{-2} +2 t^{-3} }[/math], [math]\displaystyle{ -q^6+2 q^5-4 q^4+6 q^3-7 q^2+8 q-6+6 q^{-1} -4 q^{-2} +2 q^{-3} - q^{-4} }[/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]= {10_12,}
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: (4, 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{ 16 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{488}{3} }[/math] [math]\displaystyle{ \frac{40}{3} }[/math] [math]\displaystyle{ 256 }[/math] [math]\displaystyle{ \frac{1120}{3} }[/math] [math]\displaystyle{ \frac{160}{3} }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{7808}{3} }[/math] [math]\displaystyle{ \frac{640}{3} }[/math] [math]\displaystyle{ \frac{43862}{15} }[/math] [math]\displaystyle{ \frac{4952}{15} }[/math] [math]\displaystyle{ \frac{31688}{45} }[/math] [math]\displaystyle{ \frac{394}{9} }[/math] [math]\displaystyle{ \frac{182}{15} }[/math]

V2,1 through V6,9 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]2 is the signature of 10 54. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-1012345χ
13          1-1
11         1 1
9        31 -2
7       31  2
5      43   -1
3     43    1
1    35     2
-1   33      0
-3  13       2
-5 13        -2
-7 1         1
-91          -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=3 }[/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}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=5 }[/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^{17}-2 q^{16}+q^{15}+3 q^{14}-7 q^{13}+5 q^{12}+4 q^{11}-15 q^{10}+14 q^9+4 q^8-26 q^7+23 q^6+9 q^5-35 q^4+23 q^3+18 q^2-38 q+16+24 q^{-1} -33 q^{-2} +5 q^{-3} +24 q^{-4} -22 q^{-5} -3 q^{-6} +17 q^{-7} -9 q^{-8} -5 q^{-9} +7 q^{-10} - q^{-11} -2 q^{-12} + q^{-13} }[/math]
3 [math]\displaystyle{ -q^{33}+2 q^{32}-q^{31}-2 q^{29}+4 q^{28}-q^{27}-q^{26}-2 q^{25}+2 q^{24}+q^{23}+5 q^{22}-5 q^{21}-11 q^{20}+2 q^{19}+27 q^{18}-q^{17}-44 q^{16}-5 q^{15}+56 q^{14}+20 q^{13}-70 q^{12}-30 q^{11}+66 q^{10}+49 q^9-65 q^8-50 q^7+42 q^6+65 q^5-34 q^4-55 q^3+5 q^2+64 q+3-48 q^{-1} -28 q^{-2} +50 q^{-3} +35 q^{-4} -34 q^{-5} -50 q^{-6} +23 q^{-7} +53 q^{-8} -6 q^{-9} -54 q^{-10} -9 q^{-11} +47 q^{-12} +19 q^{-13} -32 q^{-14} -28 q^{-15} +21 q^{-16} +24 q^{-17} -6 q^{-18} -20 q^{-19} +11 q^{-21} +4 q^{-22} -6 q^{-23} -2 q^{-24} + q^{-25} +2 q^{-26} - q^{-27} }[/math]
4 [math]\displaystyle{ q^{54}-2 q^{53}+q^{52}-q^{50}+5 q^{49}-8 q^{48}+4 q^{47}-2 q^{45}+13 q^{44}-22 q^{43}+7 q^{42}+3 q^{41}+5 q^{40}+28 q^{39}-55 q^{38}-6 q^{37}+13 q^{36}+46 q^{35}+64 q^{34}-125 q^{33}-68 q^{32}+19 q^{31}+145 q^{30}+161 q^{29}-213 q^{28}-210 q^{27}-32 q^{26}+275 q^{25}+344 q^{24}-245 q^{23}-376 q^{22}-167 q^{21}+326 q^{20}+537 q^{19}-171 q^{18}-444 q^{17}-315 q^{16}+252 q^{15}+623 q^{14}-63 q^{13}-377 q^{12}-373 q^{11}+120 q^{10}+583 q^9+6 q^8-251 q^7-347 q^6+q^5+487 q^4+41 q^3-122 q^2-295 q-102+371 q^{-1} +72 q^{-2} +9 q^{-3} -225 q^{-4} -190 q^{-5} +225 q^{-6} +70 q^{-7} +133 q^{-8} -108 q^{-9} -222 q^{-10} +67 q^{-11} +2 q^{-12} +188 q^{-13} +31 q^{-14} -153 q^{-15} -28 q^{-16} -104 q^{-17} +131 q^{-18} +110 q^{-19} -29 q^{-20} -15 q^{-21} -150 q^{-22} +20 q^{-23} +77 q^{-24} +43 q^{-25} +48 q^{-26} -100 q^{-27} -37 q^{-28} +5 q^{-29} +28 q^{-30} +64 q^{-31} -27 q^{-32} -22 q^{-33} -21 q^{-34} -4 q^{-35} +32 q^{-36} + q^{-37} -9 q^{-39} -8 q^{-40} +7 q^{-41} + q^{-42} +2 q^{-43} - q^{-44} -2 q^{-45} + q^{-46} }[/math]
5 [math]\displaystyle{ -q^{80}+2 q^{79}-q^{78}+q^{76}-2 q^{75}-q^{74}+5 q^{73}-3 q^{72}-2 q^{71}+5 q^{70}-4 q^{69}-q^{68}+9 q^{67}-9 q^{66}-9 q^{65}+9 q^{64}+7 q^{63}+8 q^{62}+7 q^{61}-29 q^{60}-40 q^{59}+13 q^{58}+57 q^{57}+66 q^{56}-119 q^{54}-145 q^{53}+7 q^{52}+211 q^{51}+264 q^{50}+23 q^{49}-356 q^{48}-465 q^{47}-70 q^{46}+520 q^{45}+746 q^{44}+213 q^{43}-709 q^{42}-1116 q^{41}-432 q^{40}+849 q^{39}+1524 q^{38}+771 q^{37}-892 q^{36}-1937 q^{35}-1193 q^{34}+824 q^{33}+2261 q^{32}+1624 q^{31}-611 q^{30}-2432 q^{29}-2051 q^{28}+332 q^{27}+2471 q^{26}+2307 q^{25}-2 q^{24}-2325 q^{23}-2486 q^{22}-273 q^{21}+2133 q^{20}+2456 q^{19}+501 q^{18}-1854 q^{17}-2402 q^{16}-622 q^{15}+1619 q^{14}+2217 q^{13}+737 q^{12}-1368 q^{11}-2108 q^{10}-776 q^9+1163 q^8+1909 q^7+888 q^6-914 q^5-1818 q^4-948 q^3+683 q^2+1598 q+1082-373 q^{-1} -1442 q^{-2} -1141 q^{-3} +88 q^{-4} +1136 q^{-5} +1187 q^{-6} +246 q^{-7} -851 q^{-8} -1125 q^{-9} -488 q^{-10} +459 q^{-11} +981 q^{-12} +687 q^{-13} -119 q^{-14} -731 q^{-15} -733 q^{-16} -213 q^{-17} +425 q^{-18} +679 q^{-19} +411 q^{-20} -111 q^{-21} -487 q^{-22} -496 q^{-23} -158 q^{-24} +256 q^{-25} +440 q^{-26} +300 q^{-27} -4 q^{-28} -281 q^{-29} -346 q^{-30} -168 q^{-31} +105 q^{-32} +257 q^{-33} +241 q^{-34} +74 q^{-35} -136 q^{-36} -227 q^{-37} -149 q^{-38} + q^{-39} +133 q^{-40} +172 q^{-41} +84 q^{-42} -46 q^{-43} -118 q^{-44} -111 q^{-45} -29 q^{-46} +59 q^{-47} +88 q^{-48} +57 q^{-49} -2 q^{-50} -54 q^{-51} -55 q^{-52} -15 q^{-53} +14 q^{-54} +32 q^{-55} +28 q^{-56} -19 q^{-58} -12 q^{-59} -6 q^{-60} +11 q^{-62} +6 q^{-63} -3 q^{-64} -2 q^{-65} - q^{-66} -2 q^{-67} + q^{-68} +2 q^{-69} - q^{-70} }[/math]
6 [math]\displaystyle{ q^{111}-2 q^{110}+q^{109}-q^{107}+2 q^{106}-2 q^{105}+4 q^{104}-6 q^{103}+5 q^{102}-q^{101}-8 q^{100}+9 q^{99}-3 q^{98}+8 q^{97}-11 q^{96}+14 q^{95}-6 q^{94}-28 q^{93}+20 q^{92}+3 q^{91}+18 q^{90}-11 q^{89}+30 q^{88}-30 q^{87}-78 q^{86}+31 q^{85}+30 q^{84}+66 q^{83}+19 q^{82}+45 q^{81}-125 q^{80}-226 q^{79}+23 q^{78}+144 q^{77}+278 q^{76}+181 q^{75}+28 q^{74}-473 q^{73}-699 q^{72}-111 q^{71}+491 q^{70}+996 q^{69}+803 q^{68}+48 q^{67}-1376 q^{66}-2011 q^{65}-785 q^{64}+1052 q^{63}+2660 q^{62}+2554 q^{61}+617 q^{60}-2788 q^{59}-4658 q^{58}-2817 q^{57}+1109 q^{56}+5048 q^{55}+5911 q^{54}+2738 q^{53}-3642 q^{52}-8100 q^{51}-6606 q^{50}-584 q^{49}+6652 q^{48}+9938 q^{47}+6682 q^{46}-2449 q^{45}-10386 q^{44}-10831 q^{43}-4152 q^{42}+5889 q^{41}+12395 q^{40}+10773 q^{39}+661 q^{38}-10008 q^{37}-13141 q^{36}-7669 q^{35}+3223 q^{34}+12044 q^{33}+12757 q^{32}+3635 q^{31}-7772 q^{30}-12726 q^{29}-9196 q^{28}+719 q^{27}+10012 q^{26}+12321 q^{25}+4956 q^{24}-5607 q^{23}-10963 q^{22}-8916 q^{21}-566 q^{20}+8026 q^{19}+10991 q^{18}+5165 q^{17}-4175 q^{16}-9334 q^{15}-8315 q^{14}-1347 q^{13}+6443 q^{12}+9918 q^{11}+5532 q^{10}-2727 q^9-7915 q^8-8143 q^7-2609 q^6+4544 q^5+8900 q^4+6380 q^3-563 q^2-5955 q-7898-4355 q^{-1} +1812 q^{-2} +7113 q^{-3} +6961 q^{-4} +2077 q^{-5} -3006 q^{-6} -6632 q^{-7} -5650 q^{-8} -1381 q^{-9} +4114 q^{-10} +6224 q^{-11} +4103 q^{-12} +454 q^{-13} -3892 q^{-14} -5340 q^{-15} -3804 q^{-16} +495 q^{-17} +3739 q^{-18} +4257 q^{-19} +3031 q^{-20} -422 q^{-21} -3054 q^{-22} -4123 q^{-23} -2151 q^{-24} +464 q^{-25} +2278 q^{-26} +3337 q^{-27} +1982 q^{-28} - q^{-29} -2245 q^{-30} -2439 q^{-31} -1640 q^{-32} -350 q^{-33} +1529 q^{-34} +2018 q^{-35} +1730 q^{-36} +91 q^{-37} -807 q^{-38} -1461 q^{-39} -1546 q^{-40} -442 q^{-41} +453 q^{-42} +1317 q^{-43} +933 q^{-44} +690 q^{-45} -80 q^{-46} -886 q^{-47} -890 q^{-48} -667 q^{-49} +99 q^{-50} +269 q^{-51} +759 q^{-52} +633 q^{-53} +131 q^{-54} -209 q^{-55} -512 q^{-56} -340 q^{-57} -403 q^{-58} +102 q^{-59} +328 q^{-60} +330 q^{-61} +242 q^{-62} +18 q^{-63} -52 q^{-64} -350 q^{-65} -178 q^{-66} -65 q^{-67} +57 q^{-68} +133 q^{-69} +142 q^{-70} +152 q^{-71} -76 q^{-72} -65 q^{-73} -95 q^{-74} -61 q^{-75} -27 q^{-76} +32 q^{-77} +98 q^{-78} +13 q^{-79} +20 q^{-80} -15 q^{-81} -23 q^{-82} -37 q^{-83} -14 q^{-84} +24 q^{-85} +2 q^{-86} +14 q^{-87} +5 q^{-88} +3 q^{-89} -11 q^{-90} -8 q^{-91} +5 q^{-92} -2 q^{-93} +2 q^{-94} + q^{-95} +2 q^{-96} - q^{-97} -2 q^{-98} + q^{-99} }[/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.

10 53.gif

10_53

10 55.gif

10_55

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