10 4: 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^{33}-q^{32}-q^{31}+3 q^{29}-3 q^{27}-3 q^{26}+5 q^{25}+3 q^{24}-3 q^{23}-7 q^{22}+4 q^{21}+6 q^{20}-q^{19}-8 q^{18}+2 q^{17}+7 q^{16}-7 q^{14}+q^{13}+6 q^{12}-q^{11}-6 q^{10}+q^9+7 q^8-3 q^7-6 q^6+2 q^5+8 q^4-5 q^3-7 q^2+5 q+9-8 q^{-1} -9 q^{-2} +10 q^{-3} +10 q^{-4} -10 q^{-5} -12 q^{-6} +11 q^{-7} +11 q^{-8} -7 q^{-9} -13 q^{-10} +8 q^{-11} +9 q^{-12} -3 q^{-13} -10 q^{-14} +3 q^{-15} +7 q^{-16} - q^{-17} -6 q^{-18} + q^{-19} +4 q^{-20} -4 q^{-22} + q^{-23} + q^{-24} + q^{-25} -2 q^{-26} + q^{-27} </math> |
coloured_jones_3 = <math>q^{33}-q^{32}-q^{31}+3 q^{29}-3 q^{27}-3 q^{26}+5 q^{25}+3 q^{24}-3 q^{23}-7 q^{22}+4 q^{21}+6 q^{20}-q^{19}-8 q^{18}+2 q^{17}+7 q^{16}-7 q^{14}+q^{13}+6 q^{12}-q^{11}-6 q^{10}+q^9+7 q^8-3 q^7-6 q^6+2 q^5+8 q^4-5 q^3-7 q^2+5 q+9-8 q^{-1} -9 q^{-2} +10 q^{-3} +10 q^{-4} -10 q^{-5} -12 q^{-6} +11 q^{-7} +11 q^{-8} -7 q^{-9} -13 q^{-10} +8 q^{-11} +9 q^{-12} -3 q^{-13} -10 q^{-14} +3 q^{-15} +7 q^{-16} - q^{-17} -6 q^{-18} + q^{-19} +4 q^{-20} -4 q^{-22} + q^{-23} + q^{-24} + q^{-25} -2 q^{-26} + q^{-27} </math> |
coloured_jones_4 = <math>q^{56}-q^{55}-q^{54}+4 q^{51}-q^{50}-2 q^{49}-2 q^{48}-4 q^{47}+8 q^{46}+2 q^{45}-3 q^{43}-11 q^{42}+8 q^{41}+3 q^{40}+5 q^{39}+q^{38}-16 q^{37}+6 q^{36}+7 q^{34}+6 q^{33}-17 q^{32}+8 q^{31}-4 q^{30}+5 q^{29}+7 q^{28}-19 q^{27}+12 q^{26}-3 q^{25}+4 q^{24}+7 q^{23}-24 q^{22}+13 q^{21}+6 q^{19}+10 q^{18}-30 q^{17}+9 q^{16}+q^{15}+10 q^{14}+16 q^{13}-32 q^{12}+2 q^{11}-q^{10}+13 q^9+24 q^8-33 q^7-4 q^6-3 q^5+14 q^4+30 q^3-34 q^2-9 q-4+17 q^{-1} +36 q^{-2} -36 q^{-3} -17 q^{-4} -6 q^{-5} +20 q^{-6} +43 q^{-7} -33 q^{-8} -23 q^{-9} -12 q^{-10} +19 q^{-11} +47 q^{-12} -26 q^{-13} -22 q^{-14} -16 q^{-15} +14 q^{-16} +41 q^{-17} -19 q^{-18} -15 q^{-19} -14 q^{-20} +9 q^{-21} +31 q^{-22} -16 q^{-23} -7 q^{-24} -9 q^{-25} +5 q^{-26} +20 q^{-27} -14 q^{-28} -5 q^{-30} +3 q^{-31} +10 q^{-32} -11 q^{-33} +3 q^{-34} -2 q^{-35} +2 q^{-36} +4 q^{-37} -6 q^{-38} +2 q^{-39} - q^{-40} + q^{-41} + q^{-42} -2 q^{-43} + q^{-44} </math> |
coloured_jones_4 = <math>q^{56}-q^{55}-q^{54}+4 q^{51}-q^{50}-2 q^{49}-2 q^{48}-4 q^{47}+8 q^{46}+2 q^{45}-3 q^{43}-11 q^{42}+8 q^{41}+3 q^{40}+5 q^{39}+q^{38}-16 q^{37}+6 q^{36}+7 q^{34}+6 q^{33}-17 q^{32}+8 q^{31}-4 q^{30}+5 q^{29}+7 q^{28}-19 q^{27}+12 q^{26}-3 q^{25}+4 q^{24}+7 q^{23}-24 q^{22}+13 q^{21}+6 q^{19}+10 q^{18}-30 q^{17}+9 q^{16}+q^{15}+10 q^{14}+16 q^{13}-32 q^{12}+2 q^{11}-q^{10}+13 q^9+24 q^8-33 q^7-4 q^6-3 q^5+14 q^4+30 q^3-34 q^2-9 q-4+17 q^{-1} +36 q^{-2} -36 q^{-3} -17 q^{-4} -6 q^{-5} +20 q^{-6} +43 q^{-7} -33 q^{-8} -23 q^{-9} -12 q^{-10} +19 q^{-11} +47 q^{-12} -26 q^{-13} -22 q^{-14} -16 q^{-15} +14 q^{-16} +41 q^{-17} -19 q^{-18} -15 q^{-19} -14 q^{-20} +9 q^{-21} +31 q^{-22} -16 q^{-23} -7 q^{-24} -9 q^{-25} +5 q^{-26} +20 q^{-27} -14 q^{-28} -5 q^{-30} +3 q^{-31} +10 q^{-32} -11 q^{-33} +3 q^{-34} -2 q^{-35} +2 q^{-36} +4 q^{-37} -6 q^{-38} +2 q^{-39} - q^{-40} + q^{-41} + q^{-42} -2 q^{-43} + q^{-44} </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, 4]]</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, 4]]</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[6, 2, 7, 1], X[16, 12, 17, 11], X[12, 3, 13, 4], X[2, 15, 3, 16],
<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[6, 2, 7, 1], X[16, 12, 17, 11], X[12, 3, 13, 4], X[2, 15, 3, 16],
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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, 4]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_4_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, 4]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_4_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, 4]]&) /@ {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, 4]]&) /@ {
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>{Reversible, 2, 2, 2, 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>{Reversible, 2, 2, 2, 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, 4]][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, 4]][t]</nowiki></pre></td></tr>

Revision as of 17:47, 31 August 2005

10 3.gif

10_3

10 5.gif

10_5

10 4.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 4 at Knotilus!


Knot presentations

Planar diagram presentation X6271 X16,12,17,11 X12,3,13,4 X2,15,3,16 X14,5,15,6 X20,8,1,7 X18,10,19,9 X4,13,5,14 X10,18,11,17 X8,20,9,19
Gauss code 1, -4, 3, -8, 5, -1, 6, -10, 7, -9, 2, -3, 8, -5, 4, -2, 9, -7, 10, -6
Dowker-Thistlethwaite code 6 12 14 20 18 16 4 2 10 8
Conway Notation [613]


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

Length is 12, width is 5,

Braid index is 5

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

[edit Notes on presentations of 10 4]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-7][-5]
Hyperbolic Volume 5.81713
A-Polynomial See Data:10 4/A-polynomial

[edit Notes for 10 4's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus
Topological 4 genus
Concordance genus
Rasmussen s-Invariant 2

[edit Notes for 10 4's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 27, -2 }
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: (-5, -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 -2 is the signature of 10 4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-10123456χ
11          11
9           0
7        21 1
5       1   -1
3      22   0
1     21    -1
-1    22     0
-3   23      1
-5  11       0
-7 12        1
-9 1         -1
-111          1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials