K11a326

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K11a325

K11a327

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 K11a326's page at Knotilus!

Visit K11a326's page at the original Knot Atlas!

[edit K11a326 Quick Notes]

[edit K11a326 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₆₂₇₁ X_12,4,13,3 X_18,5,19,6 X_14,8,15,7 X_22,10,1,9 X_4,12,5,11 X_20,13,21,14 X_8,16,9,15 X_10,17,11,18 X_2,19,3,20 X_16,21,17,22
Gauss code	1, -10, 2, -6, 3, -1, 4, -8, 5, -9, 6, -2, 7, -4, 8, -11, 9, -3, 10, -7, 11, -5
Dowker-Thistlethwaite code	6 12 18 14 22 4 20 8 10 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:K11a326/ThurstonBennequinNumber
Hyperbolic Volume	18.6893
A-Polynomial	See Data:K11a326/A-polynomial

[edit Notes for K11a326's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$[0,4]$
Rasmussen s-Invariant	0

[edit Notes for K11a326's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$t 4 -6 t 3 + 19 t 2 -36 t + 45-36 t -1 + 19 t -2 -6 t -3 + t -4$
Conway polynomial	$z 8 + 2 z 6 + 3 z 4 + 2 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 169, 0 }
Jones polynomial	$q 6 -5 q 5 + 11 q 4 -18 q 3 + 24 q 2 -27 q + 28-23 q -1 + 17 q -2 -10 q -3 + 4 q -4 - q -5$
HOMFLY-PT polynomial (db, data sources)	$z 8 - a 2 z 6 -2 z 6 a -2 + 5 z 6 -3 a 2 z 4 -6 z 4 a -2 + z 4 a -4 + 11 z 4 -4 a 2 z 2 -6 z 2 a -2 + z 2 a -4 + 11 z 2 -2 a 2 -2 a -2 + 5$
Kauffman polynomial (db, data sources)	$4 z 10 a -2 + 4 z 10 + 11 a z 9 + 21 z 9 a -1 + 10 z 9 a -3 + 13 a 2 z 8 + 13 z 8 a -2 + 10 z 8 a -4 + 16 z 8 + 9 a 3 z 7 -14 a z 7 -44 z 7 a -1 -16 z 7 a -3 + 5 z 7 a -5 + 4 a 4 z 6 -24 a 2 z 6 -47 z 6 a -2 -21 z 6 a -4 + z 6 a -6 -53 z 6 + a 5 z 5 -13 a 3 z 5 + 3 a z 5 + 28 z 5 a -1 + 2 z 5 a -3 -9 z 5 a -5 -4 a 4 z 4 + 22 a 2 z 4 + 39 z 4 a -2 + 11 z 4 a -4 - z 4 a -6 + 53 z 4 - a 5 z 3 + 7 a 3 z 3 + 6 a z 3 -5 z 3 a -1 + 3 z 3 a -5 -10 a 2 z 2 -14 z 2 a -2 -2 z 2 a -4 -22 z 2 -2 a 3 z -3 a z - z a -1 + z a -3 + z a -5 + 2 a 2 + 2 a -2 + 5$
The A2 invariant	$- q 14 + 2 q 12 -4 q 10 + 2 q 8 + q 6 -4 q 4 + 7 q 2 -3 + 5 q -2 -2 q -6 + 4 q -8 -5 q -10 + 2 q -12 -2 q -16 + q -18$
The G2 invariant	Data:K11a326/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a326 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["K11a326"];

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

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

Out[4]= $t 4 -6 t 3 + 19 t 2 -36 t + 45-36 t -1 + 19 t -2 -6 t -3 + t -4$

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

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

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]= { 169, 0 }

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

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

Out[8]= $q 6 -5 q 5 + 11 q 4 -18 q 3 + 24 q 2 -27 q + 28-23 q -1 + 17 q -2 -10 q -3 + 4 q -4 - q -5$

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

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

Out[9]= $z 8 - a 2 z 6 -2 z 6 a -2 + 5 z 6 -3 a 2 z 4 -6 z 4 a -2 + z 4 a -4 + 11 z 4 -4 a 2 z 2 -6 z 2 a -2 + z 2 a -4 + 11 z 2 -2 a 2 -2 a -2 + 5$

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

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

Out[10]= $4 z 10 a -2 + 4 z 10 + 11 a z 9 + 21 z 9 a -1 + 10 z 9 a -3 + 13 a 2 z 8 + 13 z 8 a -2 + 10 z 8 a -4 + 16 z 8 + 9 a 3 z 7 -14 a z 7 -44 z 7 a -1 -16 z 7 a -3 + 5 z 7 a -5 + 4 a 4 z 6 -24 a 2 z 6 -47 z 6 a -2 -21 z 6 a -4 + z 6 a -6 -53 z 6 + a 5 z 5 -13 a 3 z 5 + 3 a z 5 + 28 z 5 a -1 + 2 z 5 a -3 -9 z 5 a -5 -4 a 4 z 4 + 22 a 2 z 4 + 39 z 4 a -2 + 11 z 4 a -4 - z 4 a -6 + 53 z 4 - a 5 z 3 + 7 a 3 z 3 + 6 a z 3 -5 z 3 a -1 + 3 z 3 a -5 -10 a 2 z 2 -14 z 2 a -2 -2 z 2 a -4 -22 z 2 -2 a 3 z -3 a z - z a -1 + z a -3 + z a -5 + 2 a 2 + 2 a -2 + 5$

[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["K11a326"];

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]= { $t 4 -6 t 3 + 19 t 2 -36 t + 45-36 t -1 + 19 t -2 -6 t -3 + t -4$ , $q 6 -5 q 5 + 11 q 4 -18 q 3 + 24 q 2 -27 q + 28-23 q -1 + 17 q -2 -10 q -3 + 4 q -4 - q -5$ }

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:

(2, 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 =

0 is the signature of K11a326. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

-4

-7

-3

-1

-3

-6

-5

-7

-6

-9

-11

-1

Integral Khovanov Homology

(db, data source)

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