Template:Hoste-Thistlethwaite Knot Page: Difference between revisions

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Revision as of 10:22, 26 May 2007

[[Image:Data:Hoste-Thistlethwaite Knot Page/Previous Knot.gif|80px|link=Data:Hoste-Thistlethwaite Knot Page/Previous Knot]]

[[Data:Hoste-Thistlethwaite Knot Page/Previous Knot]]

[[Image:Data:Hoste-Thistlethwaite Knot Page/Next Knot.gif|80px|link=Data:Hoste-Thistlethwaite Knot Page/Next Knot]]

[[Data:Hoste-Thistlethwaite Knot Page/Next Knot]]

File:Hoste-Thistlethwaite Knot Page.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit [{{{KnotilusURL}}} Hoste-Thistlethwaite Knot Page's page] at Knotilus!

Visit Hoste-Thistlethwaite Knot Page's page at the original Knot Atlas!

[edit Hoste-Thistlethwaite Knot Page Quick Notes]

<ifpageexists>Hoste-Thistlethwaite Knot Page Quick Notes</ifpageexists>


[edit Hoste-Thistlethwaite Knot Page Further Notes and Views]

<ifpageexists>Hoste-Thistlethwaite Knot Page Further Notes and Views</ifpageexists>

Knot presentations

Planar diagram presentation Data:Hoste-Thistlethwaite Knot Page/PD Presentation
Gauss code Data:Hoste-Thistlethwaite Knot Page/Gauss Code
Dowker-Thistlethwaite code Data:Hoste-Thistlethwaite Knot Page/DT Code
A Braid Representative {{{braid_table}}}
A Morse Link Presentation File:Hoste-Thistlethwaite Knot Page ML.gif

Four dimensional invariants

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

[edit Notes for Hoste-Thistlethwaite Knot Page's four dimensional invariants]

Polynomial invariants

Alexander polynomial Data:Hoste-Thistlethwaite Knot Page/Alexander Polynomial
Conway polynomial Data:Hoste-Thistlethwaite Knot Page/Conway Polynomial
2nd Alexander ideal (db, data sources) Data:Hoste-Thistlethwaite Knot Page/2nd AlexanderIdeal
Determinant and Signature { Data:Hoste-Thistlethwaite Knot Page/Determinant, Data:Hoste-Thistlethwaite Knot Page/Signature }
Jones polynomial Data:Hoste-Thistlethwaite Knot Page/Jones Polynomial
HOMFLY-PT polynomial (db, data sources) Data:Hoste-Thistlethwaite Knot Page/HOMFLYPT Polynomial
Kauffman polynomial (db, data sources) Data:Hoste-Thistlethwaite Knot Page/Kauffman Polynomial
The A2 invariant Data:Hoste-Thistlethwaite Knot Page/QuantumInvariant/A2/1,0
The G2 invariant Data:Hoste-Thistlethwaite Knot Page/QuantumInvariant/G2/1,0
Further Quantum Invariants
Hoste-Thistlethwaite Knot Page Quantum Invariants.
Computer Talk
The above data is available with the Mathematica package 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.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["Hoste-Thistlethwaite Knot Page"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= Data:Hoste-Thistlethwaite Knot Page/Alexander Polynomial
In[5]:= Conway[K][z]
Out[5]= Data:Hoste-Thistlethwaite Knot Page/Conway Polynomial
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]= Data:Hoste-Thistlethwaite Knot Page/2nd AlexanderIdeal
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { Data:Hoste-Thistlethwaite Knot Page/Determinant, Data:Hoste-Thistlethwaite Knot Page/Signature }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= Data:Hoste-Thistlethwaite Knot Page/Jones Polynomial
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= Data:Hoste-Thistlethwaite Knot Page/HOMFLYPT Polynomial
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= Data:Hoste-Thistlethwaite Knot Page/Kauffman Polynomial

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {{{{same_alexander}}}}

Same Jones Polynomial (up to mirroring, ): {{{{same_jones}}}}

Computer Talk
The above data is available with the Mathematica package 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.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["Hoste-Thistlethwaite Knot Page"];
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]= { Data:Hoste-Thistlethwaite Knot Page/Alexander Polynomial, Data:Hoste-Thistlethwaite Knot Page/Jones Polynomial }
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]= {{{{same_alexander}}}}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {{{{same_jones}}}}

Vassiliev invariants

V2 and V3: (Data:Hoste-Thistlethwaite Knot Page/V 2, Data:Hoste-Thistlethwaite Knot Page/V 3)
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:Hoste-Thistlethwaite Knot Page/V 2,1 Data:Hoste-Thistlethwaite Knot Page/V 3,1 Data:Hoste-Thistlethwaite Knot Page/V 4,1 Data:Hoste-Thistlethwaite Knot Page/V 4,2 Data:Hoste-Thistlethwaite Knot Page/V 4,3 Data:Hoste-Thistlethwaite Knot Page/V 5,1 Data:Hoste-Thistlethwaite Knot Page/V 5,2 Data:Hoste-Thistlethwaite Knot Page/V 5,3 Data:Hoste-Thistlethwaite Knot Page/V 5,4 Data:Hoste-Thistlethwaite Knot Page/V 6,1 Data:Hoste-Thistlethwaite Knot Page/V 6,2 Data:Hoste-Thistlethwaite Knot Page/V 6,3 Data:Hoste-Thistlethwaite Knot Page/V 6,4 Data:Hoste-Thistlethwaite Knot Page/V 6,5 Data:Hoste-Thistlethwaite Knot Page/V 6,6 Data:Hoste-Thistlethwaite Knot Page/V 6,7 Data:Hoste-Thistlethwaite Knot Page/V 6,8 Data:Hoste-Thistlethwaite Knot Page/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 Data:Hoste-Thistlethwaite Knot Page/Signature is the signature of Hoste-Thistlethwaite Knot Page. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:Hoste-Thistlethwaite Knot Page/KhovanovTable
Integral Khovanov Homology

(db, data source)

   Data:Hoste-Thistlethwaite Knot Page/Integral Khovanov Homology

Computer Talk

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

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