T(11,3)

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T(21,2)

T(23,2)

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(11,3)'s page at Knotilus!

Visit T(11,3)'s page at the original Knot Atlas!

Edit T(11,3) Quick Notes

Edit T(11,3) Further Notes and Views

[edit] Knot presentations

Planar diagram presentation	X_7,37,8,36 X_22,38,23,37 X_23,9,24,8 X_38,10,39,9 X_39,25,40,24 X_10,26,11,25 X_11,41,12,40 X_26,42,27,41 X_27,13,28,12 X_42,14,43,13 X_43,29,44,28 X_14,30,15,29 X_15,1,16,44 X_30,2,31,1 X_31,17,32,16 X_2,18,3,17 X_3,33,4,32 X_18,34,19,33 X_19,5,20,4 X_34,6,35,5 X_35,21,36,20 X_6,22,7,21
Gauss code	14, -16, -17, 19, 20, -22, -1, 3, 4, -6, -7, 9, 10, -12, -13, 15, 16, -18, -19, 21, 22, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 17, 18, -20, -21, 1, 2, -4, -5, 7, 8, -10, -11, 13
Dowker-Thistlethwaite code	30 -32 34 -36 38 -40 42 -44 2 -4 6 -8 10 -12 14 -16 18 -20 22 -24 26 -28

Braid presentation

[edit] Polynomial invariants

Alexander polynomial	$t 10 - t 9 + t 7 - t 6 + t 4 - t 3 + t -1 + t -1 - t -3 + t -4 - t -6 + t -7 - t -9 + t -10$
Conway polynomial	$z 20 + 19 z 18 + 152 z 16 + 666 z 14 + 1742 z 12 + 2782 z 10 + 2665 z 8 + 1443 z 6 + 390 z 4 + 40 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 1, 16 }
Jones polynomial	$- q 22 + q 12 + q 10$
HOMFLY-PT polynomial (db, data sources)	$z 20 a -20 + 20 z 18 a -20 - z 18 a -22 + 171 z 16 a -20 -19 z 16 a -22 + 817 z 14 a -20 -152 z 14 a -22 + z 14 a -24 + 2394 z 12 a -20 -666 z 12 a -22 + 14 z 12 a -24 + 4446 z 10 a -20 -1742 z 10 a -22 + 78 z 10 a -24 + 5226 z 8 a -20 -2782 z 8 a -22 + 221 z 8 a -24 + 3770 z 6 a -20 -2665 z 6 a -22 + 338 z 6 a -24 + 1560 z 4 a -20 -1443 z 4 a -22 + 273 z 4 a -24 + 325 z 2 a -20 -390 z 2 a -22 + 105 z 2 a -24 + 26 a -20 -40 a -22 + 15 a -24$
Kauffman polynomial (db, data sources)	Data:T(11,3)/Kauffman Polynomial
The A2 invariant	Data:T(11,3)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(11,3)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(11,3) 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(11,3)"];

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

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

Out[4]= $t 10 - t 9 + t 7 - t 6 + t 4 - t 3 + t -1 + t -1 - t -3 + t -4 - t -6 + t -7 - t -9 + t -10$

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

Out[5]= $z 20 + 19 z 18 + 152 z 16 + 666 z 14 + 1742 z 12 + 2782 z 10 + 2665 z 8 + 1443 z 6 + 390 z 4 + 40 z 2 + 1$

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, 16 }

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

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

Out[8]= $- q 22 + q 12 + q 10$

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

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

Out[9]= $z 20 a -20 + 20 z 18 a -20 - z 18 a -22 + 171 z 16 a -20 -19 z 16 a -22 + 817 z 14 a -20 -152 z 14 a -22 + z 14 a -24 + 2394 z 12 a -20 -666 z 12 a -22 + 14 z 12 a -24 + 4446 z 10 a -20 -1742 z 10 a -22 + 78 z 10 a -24 + 5226 z 8 a -20 -2782 z 8 a -22 + 221 z 8 a -24 + 3770 z 6 a -20 -2665 z 6 a -22 + 338 z 6 a -24 + 1560 z 4 a -20 -1443 z 4 a -22 + 273 z 4 a -24 + 325 z 2 a -20 -390 z 2 a -22 + 105 z 2 a -24 + 26 a -20 -40 a -22 + 15 a -24$

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

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

Out[10]= Data:T(11,3)/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(11,3)"];

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]= { $t 10 - t 9 + t 7 - t 6 + t 4 - t 3 + t -1 + t -1 - t -3 + t -4 - t -6 + t -7 - t -9 + t -10$ , $- q 22 + q 12 + q 10$ }

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:

(40, 220)

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 =

16 is the signature of T(11,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 13$	$i = 15$	$i = 17$	$i = 19$	$i = 21$
$r = 0$				${\mathbb Z}$	${\mathbb Z}$
$r = 1$
$r = 2$				${\mathbb Z}$
$r = 3$				${\mathbb Z}_2$	${\mathbb Z}$
$r = 4$			${\mathbb Z}$	${\mathbb Z}$
$r = 5$				${\mathbb Z}$	${\mathbb Z}$
$r = 6$			${\mathbb Z}$
$r = 7$			${\mathbb Z}_2$	${\mathbb Z}$
$r = 8$		${\mathbb Z}$	${\mathbb Z}$
$r = 9$			${\mathbb Z}$	${\mathbb Z}$
$r = 10$		${\mathbb Z}$
$r = 11$		${\mathbb Z}_2$	${\mathbb Z}$
$r = 12$	${\mathbb Z}$	${\mathbb Z}$
$r = 13$		${\mathbb Z}$	${\mathbb Z}$
$r = 14$	${\mathbb Z}$
$r = 15$	${\mathbb Z}_2$	${\mathbb Z}$