10 107: Difference between revisions

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{{Rolfsen Knot Page|
{{Rolfsen Knot Page|
n = 10 |
n = 10 |
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coloured_jones_3 = <math>-q^{30}+3 q^{29}-2 q^{28}-3 q^{27}+q^{26}+14 q^{25}-9 q^{24}-32 q^{23}+17 q^{22}+76 q^{21}-24 q^{20}-152 q^{19}+280 q^{17}+60 q^{16}-426 q^{15}-196 q^{14}+573 q^{13}+414 q^{12}-706 q^{11}-665 q^{10}+756 q^9+966 q^8-764 q^7-1225 q^6+682 q^5+1469 q^4-584 q^3-1614 q^2+421 q+1713-263 q^{-1} -1712 q^{-2} +74 q^{-3} +1648 q^{-4} +105 q^{-5} -1499 q^{-6} -272 q^{-7} +1282 q^{-8} +407 q^{-9} -1018 q^{-10} -483 q^{-11} +736 q^{-12} +484 q^{-13} -465 q^{-14} -428 q^{-15} +250 q^{-16} +328 q^{-17} -106 q^{-18} -215 q^{-19} +27 q^{-20} +120 q^{-21} +6 q^{-22} -61 q^{-23} -6 q^{-24} +23 q^{-25} +4 q^{-26} -7 q^{-27} -3 q^{-28} +4 q^{-29} - q^{-30} </math> |
coloured_jones_3 = <math>-q^{30}+3 q^{29}-2 q^{28}-3 q^{27}+q^{26}+14 q^{25}-9 q^{24}-32 q^{23}+17 q^{22}+76 q^{21}-24 q^{20}-152 q^{19}+280 q^{17}+60 q^{16}-426 q^{15}-196 q^{14}+573 q^{13}+414 q^{12}-706 q^{11}-665 q^{10}+756 q^9+966 q^8-764 q^7-1225 q^6+682 q^5+1469 q^4-584 q^3-1614 q^2+421 q+1713-263 q^{-1} -1712 q^{-2} +74 q^{-3} +1648 q^{-4} +105 q^{-5} -1499 q^{-6} -272 q^{-7} +1282 q^{-8} +407 q^{-9} -1018 q^{-10} -483 q^{-11} +736 q^{-12} +484 q^{-13} -465 q^{-14} -428 q^{-15} +250 q^{-16} +328 q^{-17} -106 q^{-18} -215 q^{-19} +27 q^{-20} +120 q^{-21} +6 q^{-22} -61 q^{-23} -6 q^{-24} +23 q^{-25} +4 q^{-26} -7 q^{-27} -3 q^{-28} +4 q^{-29} - q^{-30} </math> |
coloured_jones_4 = <math>q^{50}-3 q^{49}+2 q^{48}+3 q^{47}-6 q^{46}+6 q^{45}-13 q^{44}+15 q^{43}+21 q^{42}-42 q^{41}-q^{40}-45 q^{39}+96 q^{38}+138 q^{37}-149 q^{36}-151 q^{35}-260 q^{34}+330 q^{33}+694 q^{32}-90 q^{31}-609 q^{30}-1281 q^{29}+319 q^{28}+2052 q^{27}+1043 q^{26}-782 q^{25}-3636 q^{24}-1189 q^{23}+3436 q^{22}+3898 q^{21}+884 q^{20}-6369 q^{19}-4885 q^{18}+3065 q^{17}+7369 q^{16}+4989 q^{15}-7525 q^{14}-9427 q^{13}+286 q^{12}+9469 q^{11}+10039 q^{10}-6397 q^9-12745 q^8-3593 q^7+9462 q^6+14012 q^5-3925 q^4-13971 q^3-7011 q^2+7925 q+16069-1101 q^{-1} -13328 q^{-2} -9408 q^{-3} +5386 q^{-4} +16197 q^{-5} +1818 q^{-6} -10982 q^{-7} -10666 q^{-8} +1973 q^{-9} +14263 q^{-10} +4470 q^{-11} -7006 q^{-12} -10208 q^{-13} -1651 q^{-14} +10202 q^{-15} +5740 q^{-16} -2377 q^{-17} -7599 q^{-18} -3896 q^{-19} +5182 q^{-20} +4758 q^{-21} +904 q^{-22} -3850 q^{-23} -3702 q^{-24} +1377 q^{-25} +2416 q^{-26} +1706 q^{-27} -1013 q^{-28} -2035 q^{-29} -111 q^{-30} +597 q^{-31} +996 q^{-32} +39 q^{-33} -661 q^{-34} -181 q^{-35} -17 q^{-36} +305 q^{-37} +101 q^{-38} -133 q^{-39} -34 q^{-40} -45 q^{-41} +54 q^{-42} +26 q^{-43} -22 q^{-44} + q^{-45} -9 q^{-46} +7 q^{-47} +3 q^{-48} -4 q^{-49} + q^{-50} </math> |
coloured_jones_4 = <math>q^{50}-3 q^{49}+2 q^{48}+3 q^{47}-6 q^{46}+6 q^{45}-13 q^{44}+15 q^{43}+21 q^{42}-42 q^{41}-q^{40}-45 q^{39}+96 q^{38}+138 q^{37}-149 q^{36}-151 q^{35}-260 q^{34}+330 q^{33}+694 q^{32}-90 q^{31}-609 q^{30}-1281 q^{29}+319 q^{28}+2052 q^{27}+1043 q^{26}-782 q^{25}-3636 q^{24}-1189 q^{23}+3436 q^{22}+3898 q^{21}+884 q^{20}-6369 q^{19}-4885 q^{18}+3065 q^{17}+7369 q^{16}+4989 q^{15}-7525 q^{14}-9427 q^{13}+286 q^{12}+9469 q^{11}+10039 q^{10}-6397 q^9-12745 q^8-3593 q^7+9462 q^6+14012 q^5-3925 q^4-13971 q^3-7011 q^2+7925 q+16069-1101 q^{-1} -13328 q^{-2} -9408 q^{-3} +5386 q^{-4} +16197 q^{-5} +1818 q^{-6} -10982 q^{-7} -10666 q^{-8} +1973 q^{-9} +14263 q^{-10} +4470 q^{-11} -7006 q^{-12} -10208 q^{-13} -1651 q^{-14} +10202 q^{-15} +5740 q^{-16} -2377 q^{-17} -7599 q^{-18} -3896 q^{-19} +5182 q^{-20} +4758 q^{-21} +904 q^{-22} -3850 q^{-23} -3702 q^{-24} +1377 q^{-25} +2416 q^{-26} +1706 q^{-27} -1013 q^{-28} -2035 q^{-29} -111 q^{-30} +597 q^{-31} +996 q^{-32} +39 q^{-33} -661 q^{-34} -181 q^{-35} -17 q^{-36} +305 q^{-37} +101 q^{-38} -133 q^{-39} -34 q^{-40} -45 q^{-41} +54 q^{-42} +26 q^{-43} -22 q^{-44} + q^{-45} -9 q^{-46} +7 q^{-47} +3 q^{-48} -4 q^{-49} + q^{-50} </math> |
coloured_jones_5 = |
coloured_jones_5 = <math>\textrm{NotAvailable}(q)</math> |
coloured_jones_6 = |
coloured_jones_6 = <math>\textrm{NotAvailable}(q)</math> |
coloured_jones_7 = |
coloured_jones_7 = <math>\textrm{NotAvailable}(q)</math> |
computer_talk =
computer_talk =
<table>
<table>
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<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
</tr>
</tr>
<tr valign=top><td colspan=2>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</td></tr>
<tr valign=top><td colspan=2>Loading KnotTheory` (version of August 29, 2005, 15:27:48)...</td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[10, 107]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[10, 107]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[7, 14, 8, 15], X[9, 19, 10, 18],
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[7, 14, 8, 15], X[9, 19, 10, 18],
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>5</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>5</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Show[DrawMorseLink[Knot[10, 107]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_107_ML.gif]]</td></tr><tr valign=top><td><tt><font color=blue>Out[8]=</font></tt><td><tt><font color=black>-Graphics-</font></tt></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Show[DrawMorseLink[Knot[10, 107]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_107_ML.gif]]</td></tr><tr valign=top><td><tt><font color=blue>Out[8]=</font></tt><td><tt><font color=black>-Graphics-</font></tt></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>(#[Knot[10, 107]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki> (#[Knot[10, 107]]&) /@ {
SymmetryType, UnknottingNumber, ThreeGenus,
BridgeIndex, SuperBridgeIndex, NakanishiIndex
}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Chiral, 1, 3, 3, NotAvailable, 1}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Chiral, 1, 3, 3, NotAvailable, 1}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[10, 107]][t]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[10, 107]][t]</nowiki></pre></td></tr>

Revision as of 17:49, 31 August 2005

10 106.gif

10_106

10 108.gif

10_108

10 107.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 107's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 107 at Knotilus!


Knot presentations

Planar diagram presentation X1425 X3,12,4,13 X7,14,8,15 X9,19,10,18 X19,7,20,6 X5,17,6,16 X17,11,18,10 X13,8,14,9 X15,1,16,20 X11,2,12,3
Gauss code -1, 10, -2, 1, -6, 5, -3, 8, -4, 7, -10, 2, -8, 3, -9, 6, -7, 4, -5, 9
Dowker-Thistlethwaite code 4 12 16 14 18 2 8 20 10 6
Conway Notation [210:2:20]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif

Length is 12, width is 5,

Braid index is 5

10 107 ML.gif 10 107 AP.gif
[{12, 2}, {1, 10}, {6, 11}, {10, 12}, {3, 7}, {2, 5}, {9, 6}, {7, 4}, {11, 8}, {5, 9}, {8, 3}, {4, 1}]

[edit Notes on presentations of 10 107]


Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-6][-6]
Hyperbolic Volume 15.3529
A-Polynomial See Data:10 107/A-polynomial

[edit Notes for 10 107's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus
Topological 4 genus
Concordance genus
Rasmussen s-Invariant 0

[edit Notes for 10 107's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 93, 0 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
The A2 invariant
The G2 invariant

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, ): {}

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

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 are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 0 is the signature of 10 107. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-1012345χ
11          1-1
9         2 2
7        51 -4
5       72  5
3      75   -2
1     97    2
-1    78     1
-3   58      -3
-5  37       4
-7 15        -4
-9 3         3
-111          -1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials