K11n139



(Knotscape image) 
See the full HosteThistlethwaite Table of 11 Crossing Knots. 
K11n139 is also known as the pretzel knot P(5,3,3). 
Knot presentations
Planar diagram presentation  X_{4251} X_{12,3,13,4} X_{16,6,17,5} X_{14,8,15,7} X_{9,21,10,20} X_{11,19,12,18} X_{2,13,3,14} X_{6,16,7,15} X_{17,22,18,1} X_{19,11,20,10} X_{21,9,22,8} 
Gauss code  1, 7, 2, 1, 3, 8, 4, 11, 5, 10, 6, 2, 7, 4, 8, 3, 9, 6, 10, 5, 11, 9 
DowkerThistlethwaite code  4 12 16 14 20 18 2 6 22 10 8 
A Braid Representative   
A Morse Link Presentation 
Three dimensional invariants

Four dimensional invariants

Polynomial invariants
KnotTheory`
, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:=

K = Knot["K11n139"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

In[5]:=

Conway[K][z]

Out[5]=

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 9, 0 } 
In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

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In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

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"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {6_1, 9_46, K11n67, K11n97,}
Same Jones Polynomial (up to mirroring, ): {}
KnotTheory`
. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

In[3]:=

K = Knot["K11n139"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ , } 
In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{6_1, 9_46, K11n67, K11n97,} 
In[6]:=

DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q]  (J /. q > 1/q) === Jones[#][q]) &
],
K
]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{} 
Vassiliev invariants
V_{2} and V_{3}:  (2, 5) 
V_{2,1} through V_{6,9}: 

V_{2,1} through V_{6,9} were provided by Petr DuninBarkowski <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 V_{2} and V_{3}.
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 K11n139. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. 

Integral Khovanov Homology
(db, data source) 

Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`
. See A Sample KnotTheory` Session.
Modifying This Page
Read me first: Modifying Knot Pages.
See/edit the HosteThistlethwaite Knot Page master template (intermediate). See/edit the HosteThistlethwaite_Splice_Base (expert). Back to the top. 
