Template:Torus Knot Page

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File:Torus Knot Page.jpg	Visit [{{{KnotilusURL}}} Torus Knot Page's page] at Knotilus! Visit Torus Knot Page's page at the original Knot Atlas!
	Torus Knot Page Quick Notes

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. {{{computer_talk}}}

See/edit the Template:Torus Knot Page. See/edit the Torus Knot Splice Template. Back to the top.

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