Template:Torus Knot Page

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File:Torus Knot Page.jpg	See other torus knots Visit Torus Knot Page at Knotilus!
	Edit Torus Knot Page Quick Notes

Edit Torus Knot Page Further Notes and Views

Knot presentations

Planar diagram presentation	Data:Torus Knot Page/PD Presentation
Gauss code	Data:Torus Knot Page/Gauss Code
Dowker-Thistlethwaite code	Data:Torus Knot Page/DT Code

Braid presentation

{{{braid_table}}}

Polynomial invariants

Alexander polynomial	Data:Torus Knot Page/Alexander Polynomial
Conway polynomial	Data:Torus Knot Page/Conway Polynomial
2nd Alexander ideal (db, data sources)	Data:Torus Knot Page/2nd AlexanderIdeal
Determinant and Signature	{ Data:Torus Knot Page/Determinant, Data:Torus Knot Page/Signature }
Jones polynomial	Data:Torus Knot Page/Jones Polynomial
HOMFLY-PT polynomial (db, data sources)	Data:Torus Knot Page/HOMFLYPT Polynomial
Kauffman polynomial (db, data sources)	Data:Torus Knot Page/Kauffman Polynomial
The A2 invariant	Data:Torus Knot Page/QuantumInvariant/A2/1,0
The G2 invariant	Data:Torus Knot Page/QuantumInvariant/G2/1,0

Further Quantum Invariants

Torus 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["Torus Knot Page"];

In[4]:=

Alexander[K][t]

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

Out[4]=

Data:Torus Knot Page/Alexander Polynomial

In[5]:=

Conway[K][z]

Out[5]=

Data:Torus 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:Torus Knot Page/2nd AlexanderIdeal

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ Data:Torus Knot Page/Determinant, Data:Torus Knot Page/Signature }

In[8]:=

Jones[K][q]

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

Out[8]=

Data:Torus Knot Page/Jones Polynomial

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

Data:Torus Knot Page/HOMFLYPT Polynomial

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

Data:Torus Knot Page/Kauffman Polynomial

"Similar" Knots (within the Atlas)

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

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {{{{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["Torus 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:Torus Knot Page/Alexander Polynomial, Data:Torus 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:Torus Knot Page/V 2, Data:Torus 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:Torus Knot Page/V 2,1 Data:Torus Knot Page/V 3,1 Data:Torus Knot Page/V 4,1 Data:Torus Knot Page/V 4,2 Data:Torus Knot Page/V 4,3 Data:Torus Knot Page/V 5,1 Data:Torus Knot Page/V 5,2 Data:Torus Knot Page/V 5,3 Data:Torus Knot Page/V 5,4 Data:Torus Knot Page/V 6,1 Data:Torus Knot Page/V 6,2 Data:Torus Knot Page/V 6,3 Data:Torus Knot Page/V 6,4 Data:Torus Knot Page/V 6,5 Data:Torus Knot Page/V 6,6 Data:Torus Knot Page/V 6,7 Data:Torus Knot Page/V 6,8 Data:Torus Knot Page/V 6,9

V_2,1 through V_6,9 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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$Data:Torus Knot Page/Signature is the signature of Torus Knot Page. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:Torus Knot Page/KhovanovTable

Integral Khovanov Homology (db, data source)

Data:Torus 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.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Torus Knot Page master template (intermediate). See/edit the Torus Knot_Splice_Base (expert). Back to the top.

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