K11a267

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K11a266

K11a268

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 K11a267's page at Knotilus!

Visit K11a267's page at the original Knot Atlas!

[edit K11a267 Quick Notes]

[edit K11a267 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₆₂₇₁ X_10,3,11,4 X_12,6,13,5 X_14,7,15,8 X_20,10,21,9 X_18,11,19,12 X_22,13,1,14 X_8,15,9,16 X_4,18,5,17 X_2,19,3,20 X_16,21,17,22
Gauss code	1, -10, 2, -9, 3, -1, 4, -8, 5, -2, 6, -3, 7, -4, 8, -11, 9, -6, 10, -5, 11, -7
Dowker-Thistlethwaite code	6 10 12 14 20 18 22 8 4 2 16

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Chiral
Unknotting number	1
3-genus	4
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a267/ThurstonBennequinNumber
Hyperbolic Volume	19.5158
A-Polynomial	See Data:K11a267/A-polynomial

[edit Notes for K11a267's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$4$
Rasmussen s-Invariant	2

[edit Notes for K11a267's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 4 + 7 t 3 -22 t 2 + 41 t -49 + 41 t -1 -22 t -2 + 7 t -3 - t -4$
Conway polynomial	$- z 8 - z 6 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 191, -2 }
Jones polynomial	$q 3 -5 q 2 + 12 q -19 + 27 q -1 -31 q -2 + 31 q -3 -27 q -4 + 20 q -5 -12 q -6 + 5 q -7 - q -8$
HOMFLY-PT polynomial (db, data sources)	$- a 2 z 8 + 2 a 4 z 6 -4 a 2 z 6 + z 6 - a 6 z 4 + 5 a 4 z 4 -6 a 2 z 4 + 2 z 4 - a 6 z 2 + 3 a 4 z 2 -3 a 2 z 2 + z 2 + 1$
Kauffman polynomial (db, data sources)	$5 a 4 z 10 + 5 a 2 z 10 + 14 a 5 z 9 + 26 a 3 z 9 + 12 a z 9 + 17 a 6 z 8 + 21 a 4 z 8 + 15 a 2 z 8 + 11 z 8 + 12 a 7 z 7 -13 a 5 z 7 -51 a 3 z 7 -21 a z 7 + 5 z 7 a -1 + 5 a 8 z 6 -25 a 6 z 6 -60 a 4 z 6 -54 a 2 z 6 + z 6 a -2 -23 z 6 + a 9 z 5 -14 a 7 z 5 -5 a 5 z 5 + 26 a 3 z 5 + 8 a z 5 -8 z 5 a -1 -3 a 8 z 4 + 12 a 6 z 4 + 42 a 4 z 4 + 42 a 2 z 4 - z 4 a -2 + 14 z 4 + 4 a 7 z 3 + 4 a 5 z 3 -2 a 3 z 3 + 2 z 3 a -1 -3 a 6 z 2 -9 a 4 z 2 -9 a 2 z 2 -3 z 2 + 1$
The A2 invariant	$- q 24 + 2 q 22 -3 q 18 + 5 q 16 -5 q 14 + 2 q 12 + q 10 -4 q 8 + 6 q 6 -6 q 4 + 6 q 2 -2 q -2 + 4 q -4 -3 q -6 + q -8$
The G2 invariant	Data:K11a267/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a267 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["K11a267"];

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

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

Out[4]= $- t 4 + 7 t 3 -22 t 2 + 41 t -49 + 41 t -1 -22 t -2 + 7 t -3 - t -4$

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

Out[5]= $- z 8 - z 6 + 1$

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]= { 191, -2 }

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

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

Out[8]= $q 3 -5 q 2 + 12 q -19 + 27 q -1 -31 q -2 + 31 q -3 -27 q -4 + 20 q -5 -12 q -6 + 5 q -7 - q -8$

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

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

Out[9]= $- a 2 z 8 + 2 a 4 z 6 -4 a 2 z 6 + z 6 - a 6 z 4 + 5 a 4 z 4 -6 a 2 z 4 + 2 z 4 - a 6 z 2 + 3 a 4 z 2 -3 a 2 z 2 + z 2 + 1$

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

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

Out[10]= $5 a 4 z 10 + 5 a 2 z 10 + 14 a 5 z 9 + 26 a 3 z 9 + 12 a z 9 + 17 a 6 z 8 + 21 a 4 z 8 + 15 a 2 z 8 + 11 z 8 + 12 a 7 z 7 -13 a 5 z 7 -51 a 3 z 7 -21 a z 7 + 5 z 7 a -1 + 5 a 8 z 6 -25 a 6 z 6 -60 a 4 z 6 -54 a 2 z 6 + z 6 a -2 -23 z 6 + a 9 z 5 -14 a 7 z 5 -5 a 5 z 5 + 26 a 3 z 5 + 8 a z 5 -8 z 5 a -1 -3 a 8 z 4 + 12 a 6 z 4 + 42 a 4 z 4 + 42 a 2 z 4 - z 4 a -2 + 14 z 4 + 4 a 7 z 3 + 4 a 5 z 3 -2 a 3 z 3 + 2 z 3 a -1 -3 a 6 z 2 -9 a 4 z 2 -9 a 2 z 2 -3 z 2 + 1$

[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["K11a267"];

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]= { $- t 4 + 7 t 3 -22 t 2 + 41 t -49 + 41 t -1 -22 t -2 + 7 t -3 - t -4$ , $q 3 -5 q 2 + 12 q -19 + 27 q -1 -31 q -2 + 31 q -3 -27 q -4 + 20 q -5 -12 q -6 + 5 q -7 - q -8$ }

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:

(0, 0)

[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 =

-2 is the signature of K11a267. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

-4

-7

-1

-3

-4

-5

-7

-9

-7

-11

-13

-7

-15

-17

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -4$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -3$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -2$	${\mathbb Z}^{16}\oplus{\mathbb Z}_2^{15}$	${\mathbb Z}^{15}$
$r = -1$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{16}$	${\mathbb Z}^{16}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{15}$	${\mathbb Z}^{16}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$