T(7,2): Difference between revisions

From Knot Atlas
Jump to navigationJump to search
No edit summary
No edit summary
Line 1: Line 1:
<!-- Script generated - do not edit! -->
<!-- -->


<!-- -->
<!-- -->
Line 10: Line 10:
|- valign=top
|- valign=top
|[[Image:{{PAGENAME}}.jpg]]
|[[Image:{{PAGENAME}}.jpg]]
|Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-4,5,-6,7,-1,2,-3,4,-5,6,-7,1,-2,3/goTop.html {{PAGENAME}}'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!
|{{Torus Knot Site Links|m=7|n=2|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-4,5,-6,7,-1,2,-3,4,-5,6,-7,1,-2,3/goTop.html}}

Visit [http://www.math.toronto.edu/~drorbn/KAtlas/TorusKnots/7.2.html {{PAGENAME}}'s page] at the original [http://www.math.toronto.edu/~drorbn/KAtlas/index.html Knot Atlas]!


{{:{{PAGENAME}} Quick Notes}}
{{:{{PAGENAME}} Quick Notes}}
Line 22: Line 20:


{{Knot Presentations}}
{{Knot Presentations}}

===Knot presentations===

{|
|'''[[Planar Diagrams|Planar diagram presentation]]'''
|style="padding-left: 1em;" | X<sub>5,13,6,12</sub> X<sub>13,7,14,6</sub> X<sub>7,1,8,14</sub> X<sub>1928</sub> X<sub>9,3,10,2</sub> X<sub>3,11,4,10</sub> X<sub>11,5,12,4</sub>
|-
|'''[[Gauss Codes|Gauss code]]'''
|style="padding-left: 1em;" | <math>\{-4,5,-6,7,-1,2,-3,4,-5,6,-7,1,-2,3\}</math>
|-
|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]'''
|style="padding-left: 1em;" | 8 10 12 14 2 4 6
|}

{{Polynomial Invariants}}
{{Polynomial Invariants}}
{{Vassiliev Invariants}}
{{Vassiliev Invariants}}
Line 118: Line 102:
q + q + q t + q t + q t + q t + q t + q t</nowiki></pre></td></tr>
q + q + q t + q t + q t + q t + q t + q t</nowiki></pre></td></tr>
</table>
</table>

{{Category:Knot Page}}

Revision as of 18:42, 28 August 2005


T(5,2).jpg

T(5,2)

T(4,3).jpg

T(4,3)

T(7,2).jpg Visit [[[:Template:KnotilusURL]] T(7,2)'s page] at Knotilus!

Visit T(7,2)'s page at the original Knot Atlas!

See also 7_1.


T(7,2) Further Notes and Views

Knot presentations

Planar diagram presentation X5,13,6,12 X13,7,14,6 X7,1,8,14 X1928 X9,3,10,2 X3,11,4,10 X11,5,12,4
Gauss code -4, 5, -6, 7, -1, 2, -3, 4, -5, 6, -7, 1, -2, 3
Dowker-Thistlethwaite code 8 10 12 14 2 4 6
Conway Notation Data:T(7,2)/Conway Notation

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 7, 6 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
The A2 invariant Data:T(7,2)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(7,2)/QuantumInvariant/G2/1,0

Vassiliev invariants

V2 and V3: (6, 14)
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
Data:T(7,2)/V 2,1 Data:T(7,2)/V 3,1 Data:T(7,2)/V 4,1 Data:T(7,2)/V 4,2 Data:T(7,2)/V 4,3 Data:T(7,2)/V 5,1 Data:T(7,2)/V 5,2 Data:T(7,2)/V 5,3 Data:T(7,2)/V 5,4 Data:T(7,2)/V 6,1 Data:T(7,2)/V 6,2 Data:T(7,2)/V 6,3 Data:T(7,2)/V 6,4 Data:T(7,2)/V 6,5 Data:T(7,2)/V 6,6 Data:T(7,2)/V 6,7 Data:T(7,2)/V 6,8 Data:T(7,2)/V 6,9

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 6 is the signature of T(7,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
01234567χ
21       1-1
19        0
17     11 0
15        0
13   11   0
11        0
9  1     1
71       1
51       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[TorusKnot[7, 2]]
Out[2]=  
7
In[3]:=
PD[TorusKnot[7, 2]]
Out[3]=  
PD[X[5, 13, 6, 12], X[13, 7, 14, 6], X[7, 1, 8, 14], X[1, 9, 2, 8], 
  X[9, 3, 10, 2], X[3, 11, 4, 10], X[11, 5, 12, 4]]
In[4]:=
GaussCode[TorusKnot[7, 2]]
Out[4]=  
GaussCode[-4, 5, -6, 7, -1, 2, -3, 4, -5, 6, -7, 1, -2, 3]
In[5]:=
BR[TorusKnot[7, 2]]
Out[5]=  
BR[2, {1, 1, 1, 1, 1, 1, 1}]
In[6]:=
alex = Alexander[TorusKnot[7, 2]][t]
Out[6]=  
      -3    -2   1        2    3

-1 + t - t + - + t - t + t

t
In[7]:=
Conway[TorusKnot[7, 2]][z]
Out[7]=  
       2      4    6
1 + 6 z  + 5 z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[7, 1]}
In[9]:=
{KnotDet[TorusKnot[7, 2]], KnotSignature[TorusKnot[7, 2]]}
Out[9]=  
{7, 6}
In[10]:=
J=Jones[TorusKnot[7, 2]][q]
Out[10]=  
 3    5    6    7    8    9    10
q  + q  - q  + q  - q  + q  - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[7, 1]}
In[12]:=
A2Invariant[TorusKnot[7, 2]][q]
Out[12]=  
 10    12      14    16    18    26    28    30
q   + q   + 2 q   + q   + q   - q   - q   - q
In[13]:=
Kauffman[TorusKnot[7, 2]][a, z]
Out[13]=  
                                  2       2      2       2    3

-3 4 z z z 3 z z 2 z 7 z 10 z z -- - -- + --- - --- + -- + --- + --- - ---- + ---- + ----- + --- -

8    6    13    11    9    7     12    10      8      6      11

a a a a a a a a a a a

    3      3    4       4      4    5    5    6    6
 3 z    4 z    z     5 z    6 z    z    z    z    z
 ---- - ---- + --- - ---- - ---- + -- + -- + -- + --
   9      7     10     8      6     9    7    8    6
a a a a a a a a a
In[14]:=
{Vassiliev[2][TorusKnot[7, 2]], Vassiliev[3][TorusKnot[7, 2]]}
Out[14]=  
{0, 14}
In[15]:=
Kh[TorusKnot[7, 2]][q, t]
Out[15]=  
 5    7    9  2    13  3    13  4    17  5    17  6    21  7
q  + q  + q  t  + q   t  + q   t  + q   t  + q   t  + q   t
This category should contain all the individual knots pages, like 7_5, K11n67, L8a2 and T(5,3)