T(4,3)

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T(7,2)

T(9,2)

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(4,3)'s page at Knotilus!

Visit T(4,3)'s page at the original Knot Atlas!

Edit T(4,3) Quick Notes

[edit] Knot presentations

Planar diagram presentation	X_5,11,6,10 X_16,12,1,11 X₁₇₂₆ X_12,8,13,7 X_13,3,14,2 X₈₄₉₃ X_9,15,10,14 X_4,16,5,15
Gauss code	-3, 5, 6, -8, -1, 3, 4, -6, -7, 1, 2, -4, -5, 7, 8, -2
Dowker-Thistlethwaite code	6 -8 10 -12 14 -16 2 -4

Braid presentation

[edit] Polynomial invariants

Alexander polynomial	$t 3 - t 2 + 1- t -2 + t -3$
Conway polynomial	$z 6 + 5 z 4 + 5 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 3, 6 }
Jones polynomial	$- q 8 + q 5 + q 3$
HOMFLY-PT polynomial (db, data sources)	$z 6 a -6 + 6 z 4 a -6 - z 4 a -8 + 10 z 2 a -6 -5 z 2 a -8 + 5 a -6 -5 a -8 + a -10$
Kauffman polynomial (db, data sources)	$z 6 a -6 + z 6 a -8 + z 5 a -7 + z 5 a -9 -6 z 4 a -6 -6 z 4 a -8 -5 z 3 a -7 -5 z 3 a -9 + 10 z 2 a -6 + 10 z 2 a -8 + 5 z a -7 + 5 z a -9 -5 a -6 -5 a -8 - a -10$
The A2 invariant	Data:T(4,3)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(4,3)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(4,3) 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(4,3)"];

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

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

Out[4]= $t 3 - t 2 + 1- t -2 + t -3$

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

Out[5]= $z 6 + 5 z 4 + 5 z 2 + 1$

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, 6 }

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

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

Out[8]= $- q 8 + q 5 + q 3$

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

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

Out[9]= $z 6 a -6 + 6 z 4 a -6 - z 4 a -8 + 10 z 2 a -6 -5 z 2 a -8 + 5 a -6 -5 a -8 + a -10$

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

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

Out[10]= $z 6 a -6 + z 6 a -8 + z 5 a -7 + z 5 a -9 -6 z 4 a -6 -6 z 4 a -8 -5 z 3 a -7 -5 z 3 a -9 + 10 z 2 a -6 + 10 z 2 a -8 + 5 z a -7 + 5 z a -9 -5 a -6 -5 a -8 - a -10$

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

Same Alexander/Conway Polynomial: {8_19,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {8_19,}

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(4,3)"];

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]= { $t 3 - t 2 + 1- t -2 + t -3$ , $- q 8 + q 5 + q 3$ }

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]= {8_19,}

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

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

Out[6]= {8_19,}

[edit] Vassiliev invariants

V₂ and V₃:

(5, 10)

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 =

6 is the signature of T(4,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 3$	$i = 5$	$i = 7$
$r = 0$		${\mathbb Z}$	${\mathbb Z}$
$r = 1$
$r = 2$		${\mathbb Z}$
$r = 3$		${\mathbb Z}_2$	${\mathbb Z}$
$r = 4$	${\mathbb Z}$	${\mathbb Z}$
$r = 5$		${\mathbb Z}$	${\mathbb Z}$