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{{Template:Basic Knot Invariants|name=9_48}}

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<span id="top"></span>

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{{Knot Navigation Links|ext=gif}}

{| align=left
|- valign=top
|[[Image:{{PAGENAME}}.gif]]
|{{Rolfsen Knot Site Links|n=9|k=48|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,4,-3,1,5,-6,2,9,-8,3,-4,-2,6,-5,-7,8,-9,7/goTop.html}}
|{{:{{PAGENAME}} Quick Notes}}
|}

<br style="clear:both" />

{{:{{PAGENAME}} Further Notes and Views}}

{{Knot Presentations}}
{{3D Invariants}}
{{4D Invariants}}
{{Polynomial Invariants}}
{{Vassiliev Invariants}}

===[[Khovanov Homology]]===

The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.

<center><table border=1>
<tr align=center>
<td width=16.6667%><table cellpadding=0 cellspacing=0>
<tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
</table></td>
<td width=8.33333%>-2</td ><td width=8.33333%>-1</td ><td width=8.33333%>0</td ><td width=8.33333%>1</td ><td width=8.33333%>2</td ><td width=8.33333%>3</td ><td width=8.33333%>4</td ><td width=8.33333%>5</td ><td width=16.6667%>&chi;</td></tr>
<tr align=center><td>13</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>2</td><td>-2</td></tr>
<tr align=center><td>11</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>&nbsp;</td><td>1</td></tr>
<tr align=center><td>9</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>3</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>-1</td></tr>
<tr align=center><td>7</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>3</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>2</td></tr>
<tr align=center><td>5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>2</td></tr>
<tr align=center><td>3</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>3</td><td bgcolor=yellow>3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>1</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
<tr align=center><td>-1</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-2</td></tr>
<tr align=center><td>-3</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
</table></center>

{{Computer Talk Header}}

<table>
<tr valign=top>
<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
</tr>
<tr valign=top><td colspan=2><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August 17, 2005, 14:44:34)...</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>Crossings[Knot[9, 48]]</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>9</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[3]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[9, 48]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[1, 4, 2, 5], X[12, 8, 13, 7], X[3, 11, 4, 10], X[11, 3, 12, 2],
X[14, 6, 15, 5], X[6, 14, 7, 13], X[15, 18, 16, 1], X[9, 17, 10, 16],
X[17, 9, 18, 8]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[4]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>GaussCode[Knot[9, 48]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-1, 4, -3, 1, 5, -6, 2, 9, -8, 3, -4, -2, 6, -5, -7, 8, -9, 7]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>BR[Knot[9, 48]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[4, {1, 1, 2, -1, 2, 1, -3, 2, -1, 2, -3}]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[9, 48]][t]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -2 7 2
-11 - t + - + 7 t - t
t</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[7]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Conway[Knot[9, 48]][z]</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> 2 4
1 + 3 z - z</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>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[8]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[9, 48], Knot[11, NonAlternating, 1]}</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>{KnotDet[Knot[9, 48]], KnotSignature[Knot[9, 48]]}</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>{27, 2}</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>J=Jones[Knot[9, 48]][q]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 1 2 3 4 5 6
-3 + - + 4 q - 4 q + 6 q - 4 q + 3 q - 2 q
q</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[11]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[9, 48]}</nowiki></pre></td></tr>
<math>\textrm{Include}(\textrm{ColouredJonesM.mhtml})</math>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[12]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[9, 48]][q]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -4 -2 2 4 6 8 10 12 16 18
-1 + q - q + q - q + 2 q + q + 2 q + 2 q + q - 2 q -
20
2 q</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[13]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Knot[9, 48]][a, z]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 2 2 3 3 3
2 3 4 z 5 z z 2 z 2 z 2 z 3 z 5 z 3 z
-- + -- - --- - --- - -- - z - -- + ---- + ---- + ---- + ---- - ---- -
6 4 7 5 3 6 4 2 7 5 3
a a a a a a a a a a a
3 4 4 5 5 5 6 6 6 7
5 z 4 6 z 5 z z 2 z 3 z z 4 z 3 z z
---- + z - ---- - ---- - -- + ---- + ---- + -- + ---- + ---- + -- +
a 4 2 5 3 a 6 4 2 5
a a a a a a a a
7
z
--
3
a</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[14]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{Vassiliev[2][Knot[9, 48]], Vassiliev[3][Knot[9, 48]]}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{0, 5}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[15]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Knot[9, 48]][q, t]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 3 1 2 q 3 5 5 2 7 2
2 q + 3 q + ----- + --- + - + 3 q t + q t + 3 q t + 3 q t +
3 2 q t t
q t
7 3 9 3 9 4 11 4 13 5
q t + 3 q t + 2 q t + q t + 2 q t</nowiki></pre></td></tr>
</table>

Revision as of 21:45, 27 August 2005


9 47.gif

9_47

9 49.gif

9_49

9 48.gif Visit 9 48's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 48's page at Knotilus!

Visit 9 48's page at the original Knot Atlas!

9 48 Quick Notes


9 48 Further Notes and Views

Knot presentations

Planar diagram presentation X1425 X12,8,13,7 X3,11,4,10 X11,3,12,2 X14,6,15,5 X6,14,7,13 X15,18,16,1 X9,17,10,16 X17,9,18,8
Gauss code -1, 4, -3, 1, 5, -6, 2, 9, -8, 3, -4, -2, 6, -5, -7, 8, -9, 7
Dowker-Thistlethwaite code 4 10 -14 -12 16 2 -6 18 8
Conway Notation [21,21,21-]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 3
Super bridge index
Nakanishi index 2
Maximal Thurston-Bennequin number [-1][-8]
Hyperbolic Volume 9.53188
A-Polynomial See Data:9 48/A-polynomial

[edit Notes for 9 48's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 9 48'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

Vassiliev invariants

V2 and V3: (3, 5)
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 9 48. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
-2-1012345χ
13       2-2
11      1 1
9     32 -1
7    31  2
5   13   2
3  33    0
1 12     1
-1 2      -2
-31       1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=    
<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=
Crossings[Knot[9, 48]]
Out[2]=  
9
In[3]:=
PD[Knot[9, 48]]
Out[3]=  
PD[X[1, 4, 2, 5], X[12, 8, 13, 7], X[3, 11, 4, 10], X[11, 3, 12, 2], 
 X[14, 6, 15, 5], X[6, 14, 7, 13], X[15, 18, 16, 1], X[9, 17, 10, 16], 

X[17, 9, 18, 8]]
In[4]:=
GaussCode[Knot[9, 48]]
Out[4]=  
GaussCode[-1, 4, -3, 1, 5, -6, 2, 9, -8, 3, -4, -2, 6, -5, -7, 8, -9, 7]
In[5]:=
BR[Knot[9, 48]]
Out[5]=  
BR[4, {1, 1, 2, -1, 2, 1, -3, 2, -1, 2, -3}]
In[6]:=
alex = Alexander[Knot[9, 48]][t]
Out[6]=  
       -2   7          2

-11 - t + - + 7 t - t

t
In[7]:=
Conway[Knot[9, 48]][z]
Out[7]=  
       2    4
1 + 3 z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[9, 48], Knot[11, NonAlternating, 1]}
In[9]:=
{KnotDet[Knot[9, 48]], KnotSignature[Knot[9, 48]]}
Out[9]=  
{27, 2}
In[10]:=
J=Jones[Knot[9, 48]][q]
Out[10]=  
     1            2      3      4      5      6

-3 + - + 4 q - 4 q + 6 q - 4 q + 3 q - 2 q

q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[9, 48]}
In[12]:=
A2Invariant[Knot[9, 48]][q]
Out[12]=  
      -4    -2    2    4      6    8      10      12    16      18

-1 + q - q + q - q + 2 q + q + 2 q + 2 q + q - 2 q -

    20
2 q
In[13]:=
Kauffman[Knot[9, 48]][a, z]
Out[13]=  
                                 2      2      2      3      3      3

2 3 4 z 5 z z 2 z 2 z 2 z 3 z 5 z 3 z -- + -- - --- - --- - -- - z - -- + ---- + ---- + ---- + ---- - ---- -

6    4    7     5     3         6     4      2      7      5      3

a a a a a a a a a a a

    3           4      4    5      5      5    6      6      6    7
 5 z     4   6 z    5 z    z    2 z    3 z    z    4 z    3 z    z
 ---- + z  - ---- - ---- - -- + ---- + ---- + -- + ---- + ---- + -- + 
  a            4      2     5     3     a      6     4      2     5
              a      a     a     a            a     a      a     a

  7
 z
 --
  3
a
In[14]:=
{Vassiliev[2][Knot[9, 48]], Vassiliev[3][Knot[9, 48]]}
Out[14]=  
{0, 5}
In[15]:=
Kh[Knot[9, 48]][q, t]
Out[15]=  
         3     1      2    q      3      5        5  2      7  2

2 q + 3 q + ----- + --- + - + 3 q t + q t + 3 q t + 3 q t +

             3  2   q t   t
            q  t

  7  3      9  3      9  4    11  4      13  5
q t + 3 q t + 2 q t + q t + 2 q t