T(3,2)

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T(3,2)

T(5,2)

1 Knot presentations
2 Polynomial invariants
3 "Similar" Knots (within the Atlas)
4 Vassiliev invariants
5 Khovanov Homology
6 Computer Talk
7 Modifying This Page

See other torus knots

Visit T(3,2)'s page at Knotilus!

Visit T(3,2)'s page at the original Knot Atlas!

Edit T(3,2) Quick Notes

See 3_1.

Edit T(3,2) Further Notes and Views

[edit] Knot presentations

Planar diagram presentation	X₃₁₄₆ X₁₅₂₄ X₅₃₆₂
Gauss code	-2, 3, -1, 2, -3, 1
Dowker-Thistlethwaite code	4 6 2

Braid presentation

[edit] Polynomial invariants

Alexander polynomial	$t -1 + t -1$
Conway polynomial	$z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 3, 2 }
Jones polynomial	$- q 4 + q 3 + q$
HOMFLY-PT polynomial (db, data sources)	$z 2 a -2 + 2 a -2 - a -4$
Kauffman polynomial (db, data sources)	$z 2 a -2 + z 2 a -4 + z a -3 + z a -5 -2 a -2 - a -4$
The A2 invariant	Data:T(3,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(3,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(3,2) 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.

In[3]:= K = Knot["T(3,2)"];

In[4]:= Alexander[K][t]

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

Out[4]= $t -1 + t -1$

In[5]:= Conway[K][z]

Out[5]= $z 2 + 1$

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]= ${1}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 3, 2 }

In[8]:= Jones[K][q]

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

Out[8]= $- q 4 + q 3 + q$

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= $z 2 a -2 + 2 a -2 - a -4$

In[10]:= Kauffman[K][a, z]

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

Out[10]= $z 2 a -2 + z 2 a -4 + z a -3 + z a -5 -2 a -2 - a -4$

[edit] "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {3_1,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {3_1,}

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.

In[3]:= K = Knot["T(3,2)"];

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]= { $t -1 + t -1$ , $- q 4 + q 3 + q$ }

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]= {3_1,}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

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

Out[6]= {3_1,}

[edit] Vassiliev invariants

V₂ and V₃:

(1, 1)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

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.

[edit] Khovanov Homology

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

2 is the signature of T(3,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$
$r = 1$
$r = 2$	${\mathbb Z}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$