T(5,3)

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T(9,2)

T(11,2)

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(5,3)'s page at Knotilus!

Visit T(5,3)'s page at the original Knot Atlas!

Edit T(5,3) Quick Notes

Edit T(5,3) Further Notes and Views

[edit] Knot presentations

Planar diagram presentation	X_7,1,8,20 X_14,2,15,1 X_15,9,16,8 X_2,10,3,9 X_3,17,4,16 X_10,18,11,17 X_11,5,12,4 X_18,6,19,5 X_19,13,20,12 X_6,14,7,13
Gauss code	2, -4, -5, 7, 8, -10, -1, 3, 4, -6, -7, 9, 10, -2, -3, 5, 6, -8, -9, 1
Dowker-Thistlethwaite code	14 -16 18 -20 2 -4 6 -8 10 -12

Braid presentation

[edit] Polynomial invariants

Alexander polynomial	$t 4 - t 3 + t -1 + t -1 - t -3 + t -4$
Conway polynomial	$z 8 + 7 z 6 + 14 z 4 + 8 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 1, 8 }
Jones polynomial	$- q 10 + q 6 + q 4$
HOMFLY-PT polynomial (db, data sources)	$z 8 a -8 + 8 z 6 a -8 - z 6 a -10 + 21 z 4 a -8 -7 z 4 a -10 + 21 z 2 a -8 -14 z 2 a -10 + z 2 a -12 + 7 a -8 -8 a -10 + 2 a -12$
Kauffman polynomial (db, data sources)	$z 8 a -8 + z 8 a -10 + z 7 a -9 + z 7 a -11 -8 z 6 a -8 -8 z 6 a -10 -7 z 5 a -9 -7 z 5 a -11 + 21 z 4 a -8 + 21 z 4 a -10 + 14 z 3 a -9 + 14 z 3 a -11 -21 z 2 a -8 -22 z 2 a -10 - z 2 a -12 -8 z a -9 -8 z a -11 + 7 a -8 + 8 a -10 + 2 a -12$
The A2 invariant	Data:T(5,3)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(5,3)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(5,3) 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(5,3)"];

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

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

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

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

Out[5]= $z 8 + 7 z 6 + 14 z 4 + 8 z 2 + 1$

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]= { 1, 8 }

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

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

Out[8]= $- q 10 + q 6 + q 4$

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

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

Out[9]= $z 8 a -8 + 8 z 6 a -8 - z 6 a -10 + 21 z 4 a -8 -7 z 4 a -10 + 21 z 2 a -8 -14 z 2 a -10 + z 2 a -12 + 7 a -8 -8 a -10 + 2 a -12$

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

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

Out[10]= $z 8 a -8 + z 8 a -10 + z 7 a -9 + z 7 a -11 -8 z 6 a -8 -8 z 6 a -10 -7 z 5 a -9 -7 z 5 a -11 + 21 z 4 a -8 + 21 z 4 a -10 + 14 z 3 a -9 + 14 z 3 a -11 -21 z 2 a -8 -22 z 2 a -10 - z 2 a -12 -8 z a -9 -8 z a -11 + 7 a -8 + 8 a -10 + 2 a -12$

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

Same Alexander/Conway Polynomial: {10_124,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {10_124,}

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(5,3)"];

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]= { $t 4 - t 3 + t -1 + t -1 - t -3 + t -4$ , $- q 10 + q 6 + q 4$ }

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]= {10_124,}

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

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

Out[6]= {10_124,}

[edit] Vassiliev invariants

V₂ and V₃:

(8, 20)

[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 =

8 is the signature of T(5,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 5$	$i = 7$	$i = 9$
$r = 0$		${\mathbb Z}$	${\mathbb Z}$
$r = 1$
$r = 2$		${\mathbb Z}$
$r = 3$		${\mathbb Z}_2$	${\mathbb Z}$
$r = 4$	${\mathbb Z}$	${\mathbb Z}$
$r = 5$		${\mathbb Z}$	${\mathbb Z}$
$r = 6$	${\mathbb Z}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$