K11a69

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K11a68

K11a70

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

(Knotscape image)

See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a69's page at Knotilus!

Visit K11a69's page at the original Knot Atlas!

[edit K11a69 Quick Notes]

[edit K11a69 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₃₉₄ X_16,5,17,6 X_14,8,15,7 X_2,9,3,10 X_18,12,19,11 X_20,14,21,13 X_22,15,1,16 X_12,18,13,17 X_10,20,11,19 X_6,21,7,22
Gauss code	1, -5, 2, -1, 3, -11, 4, -2, 5, -10, 6, -9, 7, -4, 8, -3, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 8 16 14 2 18 20 22 12 10 6

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Chiral
Unknotting number	1
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a69/ThurstonBennequinNumber
Hyperbolic Volume	16.7403
A-Polynomial	See Data:K11a69/A-polynomial

[edit Notes for K11a69's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$3$
Rasmussen s-Invariant	0

[edit Notes for K11a69's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$-2 t 3 + 13 t 2 -33 t + 45-33 t -1 + 13 t -2 -2 t -3$
Conway polynomial	$-2 z 6 + z 4 + z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 141, 0 }
Jones polynomial	$q 6 -4 q 5 + 8 q 4 -14 q 3 + 20 q 2 -22 q + 23-20 q -1 + 15 q -2 -9 q -3 + 4 q -4 - q -5$
HOMFLY-PT polynomial (db, data sources)	$- z 6 a -2 - z 6 + 2 a 2 z 4 -2 z 4 a -2 + z 4 a -4 - a 4 z 2 + a 2 z 2 -3 z 2 a -2 + z 2 a -4 + 3 z 2 - a 2 - a -2 + 3$
Kauffman polynomial (db, data sources)	$2 z 10 a -2 + 2 z 10 + 6 a z 9 + 12 z 9 a -1 + 6 z 9 a -3 + 9 a 2 z 8 + 12 z 8 a -2 + 7 z 8 a -4 + 14 z 8 + 8 a 3 z 7 + 3 a z 7 -17 z 7 a -1 -8 z 7 a -3 + 4 z 7 a -5 + 4 a 4 z 6 -10 a 2 z 6 -37 z 6 a -2 -18 z 6 a -4 + z 6 a -6 -32 z 6 + a 5 z 5 -12 a 3 z 5 -18 a z 5 + z 5 a -1 -4 z 5 a -3 -10 z 5 a -5 -5 a 4 z 4 + 2 a 2 z 4 + 34 z 4 a -2 + 15 z 4 a -4 -2 z 4 a -6 + 24 z 4 - a 5 z 3 + 7 a 3 z 3 + 11 a z 3 + 5 z 3 a -1 + 8 z 3 a -3 + 6 z 3 a -5 + 2 a 4 z 2 - a 2 z 2 -13 z 2 a -2 -4 z 2 a -4 -12 z 2 - a 3 z -2 a z -2 z a -1 - z a -3 + a 2 + a -2 + 3$
The A2 invariant	Data:K11a69/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a69/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a69 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["K11a69"];

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

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

Out[4]= $-2 t 3 + 13 t 2 -33 t + 45-33 t -1 + 13 t -2 -2 t -3$

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

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

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]= { 141, 0 }

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

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

Out[8]= $q 6 -4 q 5 + 8 q 4 -14 q 3 + 20 q 2 -22 q + 23-20 q -1 + 15 q -2 -9 q -3 + 4 q -4 - q -5$

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

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

Out[9]= $- z 6 a -2 - z 6 + 2 a 2 z 4 -2 z 4 a -2 + z 4 a -4 - a 4 z 2 + a 2 z 2 -3 z 2 a -2 + z 2 a -4 + 3 z 2 - a 2 - a -2 + 3$

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

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

Out[10]= $2 z 10 a -2 + 2 z 10 + 6 a z 9 + 12 z 9 a -1 + 6 z 9 a -3 + 9 a 2 z 8 + 12 z 8 a -2 + 7 z 8 a -4 + 14 z 8 + 8 a 3 z 7 + 3 a z 7 -17 z 7 a -1 -8 z 7 a -3 + 4 z 7 a -5 + 4 a 4 z 6 -10 a 2 z 6 -37 z 6 a -2 -18 z 6 a -4 + z 6 a -6 -32 z 6 + a 5 z 5 -12 a 3 z 5 -18 a z 5 + z 5 a -1 -4 z 5 a -3 -10 z 5 a -5 -5 a 4 z 4 + 2 a 2 z 4 + 34 z 4 a -2 + 15 z 4 a -4 -2 z 4 a -6 + 24 z 4 - a 5 z 3 + 7 a 3 z 3 + 11 a z 3 + 5 z 3 a -1 + 8 z 3 a -3 + 6 z 3 a -5 + 2 a 4 z 2 - a 2 z 2 -13 z 2 a -2 -4 z 2 a -4 -12 z 2 - a 3 z -2 a z -2 z a -1 - z a -3 + a 2 + a -2 + 3$

[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["K11a69"];

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]= { $-2 t 3 + 13 t 2 -33 t + 45-33 t -1 + 13 t -2 -2 t -3$ , $q 6 -4 q 5 + 8 q 4 -14 q 3 + 20 q 2 -22 q + 23-20 q -1 + 15 q -2 -9 q -3 + 4 q -4 - q -5$ }

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:

(1, 0)

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

4

0

8

$\frac{14}{3}$

$-\frac{14}{3}$

0

0

64

-64

$\frac{32}{3}$

0

$\frac{56}{3}$

$-\frac{56}{3}$

$\frac{511}{30}$

$\frac{1858}{15}$

$-\frac{7618}{45}$

$\frac{641}{18}$

$-\frac{1409}{30}$

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 =

0 is the signature of K11a69. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

-3

-6

-2

-1

-3

-5

-7

-5

-9

-11

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$