9 43: Difference between revisions

From Knot Atlas
Jump to navigationJump to search
No edit summary
No edit summary
Line 1: Line 1:
<!-- WARNING! WARNING! WARNING!
<!-- This page was generated from the splice template "Rolfsen_Splice_Template". Please do not edit! -->
<!-- This page was generated from the splice template [[Rolfsen_Splice_Base]]. Please do not edit!
<!-- --> <!--
<!-- You probably want to edit the template referred to immediately below. (See [[Category:Knot Page Template]].)
-->
<!-- This page itself was created by running [[Media:KnotPageSpliceRobot.nb]] on [[Rolfsen_Splice_Base]]. -->
<!-- <math>\text{Null}</math> -->
<!-- <math>\text{Null}</math> -->
{{Rolfsen Knot Page|
{{Rolfsen Knot Page|
n = 9 |
n = 9 |
Line 47: Line 50:
<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[9, 43]]</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[9, 43]]</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[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10],
<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[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10],
Line 65: Line 68:
<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>4</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>4</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[9, 43]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:9_43_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[9, 43]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:9_43_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[9, 43]]&) /@ {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[9, 43]]&) /@ {
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, 3, 3, {4, 5}, 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, 3, 3, {4, 5}, 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[9, 43]][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[9, 43]][t]</nowiki></pre></td></tr>

Revision as of 17:53, 31 August 2005

9 42.gif

9_42

9 44.gif

9_44

9 43.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 9 43's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 43 at Knotilus!


Knot presentations

Planar diagram presentation X4251 X10,6,11,5 X8394 X2,9,3,10 X14,8,15,7 X15,1,16,18 X11,17,12,16 X17,13,18,12 X6,14,7,13
Gauss code 1, -4, 3, -1, 2, -9, 5, -3, 4, -2, -7, 8, 9, -5, -6, 7, -8, 6
Dowker-Thistlethwaite code 4 8 10 14 2 -16 6 -18 -12
Conway Notation [211,3,2-]


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

Length is 9, width is 4,

Braid index is 4

9 43 ML.gif 9 43 AP.gif
[{5, 10}, {9, 1}, {10, 8}, {6, 9}, {4, 7}, {3, 6}, {2, 5}, {1, 4}, {7, 2}, {8, 3}]

[edit Notes on presentations of 9 43]


Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index
Nakanishi index 1
Maximal Thurston-Bennequin number [1][-10]
Hyperbolic Volume 5.90409
A-Polynomial See Data:9 43/A-polynomial

[edit Notes for 9 43's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 43's four dimensional invariants]

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 13, 4 }
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, ): {K11n12,}

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

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 4 is the signature of 9 43. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-1012345χ
15       1-1
13      1 1
11     11 0
9    11  0
7   11   0
5  11    0
3 12     1
1        0
-11       1
Integral Khovanov Homology

(db, data source)

  

The Coloured Jones Polynomials