T(11,4)

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T(16,3)

T(33,2)

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(11,4)'s page at Knotilus!

Visit T(11,4)'s page at the original Knot Atlas!

Edit T(11,4) Quick Notes

Edit T(11,4) Further Notes and Views

[edit] Knot presentations

Planar diagram presentation	X_3,53,4,52 X_20,54,21,53 X_37,55,38,54 X_21,5,22,4 X_38,6,39,5 X_55,7,56,6 X_39,23,40,22 X_56,24,57,23 X_7,25,8,24 X_57,41,58,40 X_8,42,9,41 X_25,43,26,42 X_9,59,10,58 X_26,60,27,59 X_43,61,44,60 X_27,11,28,10 X_44,12,45,11 X_61,13,62,12 X_45,29,46,28 X_62,30,63,29 X_13,31,14,30 X_63,47,64,46 X_14,48,15,47 X_31,49,32,48 X_15,65,16,64 X_32,66,33,65 X_49,1,50,66 X_33,17,34,16 X_50,18,51,17 X_1,19,2,18 X_51,35,52,34 X_2,36,3,35 X_19,37,20,36
Gauss code	-30, -32, -1, 4, 5, 6, -9, -11, -13, 16, 17, 18, -21, -23, -25, 28, 29, 30, -33, -2, -4, 7, 8, 9, -12, -14, -16, 19, 20, 21, -24, -26, -28, 31, 32, 33, -3, -5, -7, 10, 11, 12, -15, -17, -19, 22, 23, 24, -27, -29, -31, 1, 2, 3, -6, -8, -10, 13, 14, 15, -18, -20, -22, 25, 26, 27
Dowker-Thistlethwaite code	18 52 -38 24 58 -44 30 64 -50 36 4 -56 42 10 -62 48 16 -2 54 22 -8 60 28 -14 66 34 -20 6 40 -26 12 46 -32

Braid presentation

[edit] Polynomial invariants

Alexander polynomial	$t 15 - t 14 + t 11 - t 10 + t 7 - t 6 + t 4 - t 2 + 1- t -2 + t -4 - t -6 + t -7 - t -10 + t -11 - t -14 + t -15$
Conway polynomial	$z 30 + 29 z 28 + 377 z 26 + 2900 z 24 + 14675 z 22 + 51380 z 20 + 127470 z 18 + 225760 z 16 + 283951 z 14 + 249288 z 12 + 148070 z 10 + 56577 z 8 + 12825 z 6 + 1510 z 4 + 75 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 11, 22 }
Jones polynomial	$- q 28 - q 26 + q 25 - q 24 + q 23 - q 22 + q 21 - q 20 + q 19 + q 17 + q 15$
HOMFLY-PT polynomial (db, data sources)	Data:T(11,4)/HOMFLYPT Polynomial
Kauffman polynomial (db, data sources)	Data:T(11,4)/Kauffman Polynomial
The A2 invariant	Data:T(11,4)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(11,4)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(11,4) 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(11,4)"];

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

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

Out[4]= $t 15 - t 14 + t 11 - t 10 + t 7 - t 6 + t 4 - t 2 + 1- t -2 + t -4 - t -6 + t -7 - t -10 + t -11 - t -14 + t -15$

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

Out[5]= $z 30 + 29 z 28 + 377 z 26 + 2900 z 24 + 14675 z 22 + 51380 z 20 + 127470 z 18 + 225760 z 16 + 283951 z 14 + 249288 z 12 + 148070 z 10 + 56577 z 8 + 12825 z 6 + 1510 z 4 + 75 z 2 + 1$

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]= { 11, 22 }

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

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

Out[8]= $- q 28 - q 26 + q 25 - q 24 + q 23 - q 22 + q 21 - q 20 + q 19 + q 17 + q 15$

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

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

Out[9]= Data:T(11,4)/HOMFLYPT Polynomial

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

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

Out[10]= Data:T(11,4)/Kauffman Polynomial

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

Same Alexander/Conway Polynomial: {}

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

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(11,4)"];

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]= { $t 15 - t 14 + t 11 - t 10 + t 7 - t 6 + t 4 - t 2 + 1- t -2 + t -4 - t -6 + t -7 - t -10 + t -11 - t -14 + t -15$ , $- q 28 - q 26 + q 25 - q 24 + q 23 - q 22 + q 21 - q 20 + q 19 + q 17 + q 15$ }

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]= {}

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

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

Out[6]= {}

[edit] Vassiliev invariants

V₂ and V₃:

(75, 550)

[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 =

22 is the signature of T(11,4). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 19$	$i = 21$	$i = 23$	$i = 25$	$i = 27$	$i = 29$	$i = 31$
$r = 0$						${\mathbb Z}$	${\mathbb Z}$
$r = 1$
$r = 2$						${\mathbb Z}$
$r = 3$						${\mathbb Z}_2$	${\mathbb Z}$
$r = 4$					${\mathbb Z}$	${\mathbb Z}$
$r = 5$						${\mathbb Z}$	${\mathbb Z}$
$r = 6$				${\mathbb Z}$	${\mathbb Z}$
$r = 7$				${\mathbb Z}_2$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 8$				${\mathbb Z}^{2}$
$r = 9$				${\mathbb Z}_2\oplus{\mathbb Z}_4$	${\mathbb Z}^{2}$
$r = 10$			${\mathbb Z}$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}_2$
$r = 11$			${\mathbb Z}_2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 12$		${\mathbb Z}$	${\mathbb Z}^{2}$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 13$		${\mathbb Z}_2$	${\mathbb Z}\oplus{\mathbb Z}_2\oplus{\mathbb Z}_4$	${\mathbb Z}^{2}$
$r = 14$		${\mathbb Z}^{2}$	${\mathbb Z}_2$	${\mathbb Z}_2$
$r = 15$		${\mathbb Z}_2^{2}\oplus{\mathbb Z}_4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$
$r = 16$	${\mathbb Z}$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}_2\oplus{\mathbb Z}_4$	${\mathbb Z}$
$r = 17$	${\mathbb Z}_2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2\oplus{\mathbb Z}_4$	${\mathbb Z}^{2}$
$r = 18$	${\mathbb Z}$	${\mathbb Z}_2^{2}$	${\mathbb Z}\oplus{\mathbb Z}_2$
$r = 19$	${\mathbb Z}_2\oplus{\mathbb Z}_4$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$
$r = 20$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}_2\oplus{\mathbb Z}_4$	${\mathbb Z}$
$r = 21$	${\mathbb Z}_4$	${\mathbb Z}$
$r = 22$	${\mathbb Z}_2$	${\mathbb Z}_2$