K11a325

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K11a324

K11a326

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 K11a325's page at Knotilus!

Visit K11a325's page at the original Knot Atlas!

[edit K11a325 Quick Notes]

[edit K11a325 Further Notes and Views]

[edit] Knot presentations

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

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	${1,2}$
3-genus	2
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a325/ThurstonBennequinNumber
Hyperbolic Volume	14.4126
A-Polynomial	See Data:K11a325/A-polynomial

[edit Notes for K11a325's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for K11a325's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$-6 t 2 + 24 t -35 + 24 t -1 -6 t -2$
Conway polynomial	$1-6 z 4$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 95, -2 }
Jones polynomial	$q 3 -3 q 2 + 6 q -9 + 13 q -1 -15 q -2 + 15 q -3 -13 q -4 + 10 q -5 -6 q -6 + 3 q -7 - q -8$
HOMFLY-PT polynomial (db, data sources)	$- a 8 + 3 z 2 a 6 + 2 a 6 -2 z 4 a 4 - z 2 a 4 -3 z 4 a 2 -4 z 2 a 2 -2 a 2 - z 4 + z 2 + 2 + z 2 a -2$
Kauffman polynomial (db, data sources)	$2 a 4 z 10 + 2 a 2 z 10 + 4 a 5 z 9 + 9 a 3 z 9 + 5 a z 9 + 4 a 6 z 8 + a 2 z 8 + 5 z 8 + 4 a 7 z 7 -8 a 5 z 7 -32 a 3 z 7 -17 a z 7 + 3 z 7 a -1 + 3 a 8 z 6 -3 a 6 z 6 -9 a 4 z 6 -21 a 2 z 6 + z 6 a -2 -17 z 6 + a 9 z 5 -5 a 7 z 5 + 10 a 5 z 5 + 45 a 3 z 5 + 20 a z 5 -9 z 5 a -1 -6 a 8 z 4 -6 a 6 z 4 + 17 a 4 z 4 + 37 a 2 z 4 -3 z 4 a -2 + 17 z 4 -2 a 9 z 3 -2 a 7 z 3 -6 a 5 z 3 -21 a 3 z 3 -11 a z 3 + 4 z 3 a -1 + 3 a 8 z 2 + 6 a 6 z 2 -6 a 4 z 2 -19 a 2 z 2 + z 2 a -2 -9 z 2 + a 9 z + a 7 z + a 5 z + 3 a 3 z + 2 a z - a 8 -2 a 6 + 2 a 2 + 2$
The A2 invariant	Data:K11a325/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a325/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a325 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["K11a325"];

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

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

Out[4]= $-6 t 2 + 24 t -35 + 24 t -1 -6 t -2$

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

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

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]= { 95, -2 }

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

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

Out[8]= $q 3 -3 q 2 + 6 q -9 + 13 q -1 -15 q -2 + 15 q -3 -13 q -4 + 10 q -5 -6 q -6 + 3 q -7 - q -8$

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

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

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

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

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

Out[10]= $2 a 4 z 10 + 2 a 2 z 10 + 4 a 5 z 9 + 9 a 3 z 9 + 5 a z 9 + 4 a 6 z 8 + a 2 z 8 + 5 z 8 + 4 a 7 z 7 -8 a 5 z 7 -32 a 3 z 7 -17 a z 7 + 3 z 7 a -1 + 3 a 8 z 6 -3 a 6 z 6 -9 a 4 z 6 -21 a 2 z 6 + z 6 a -2 -17 z 6 + a 9 z 5 -5 a 7 z 5 + 10 a 5 z 5 + 45 a 3 z 5 + 20 a z 5 -9 z 5 a -1 -6 a 8 z 4 -6 a 6 z 4 + 17 a 4 z 4 + 37 a 2 z 4 -3 z 4 a -2 + 17 z 4 -2 a 9 z 3 -2 a 7 z 3 -6 a 5 z 3 -21 a 3 z 3 -11 a z 3 + 4 z 3 a -1 + 3 a 8 z 2 + 6 a 6 z 2 -6 a 4 z 2 -19 a 2 z 2 + z 2 a -2 -9 z 2 + a 9 z + a 7 z + a 5 z + 3 a 3 z + 2 a z - a 8 -2 a 6 + 2 a 2 + 2$

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a7,}

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["K11a325"];

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]= { $-6 t 2 + 24 t -35 + 24 t -1 -6 t -2$ , $q 3 -3 q 2 + 6 q -9 + 13 q -1 -15 q -2 + 15 q -3 -13 q -4 + 10 q -5 -6 q -6 + 3 q -7 - q -8$ }

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]= {K11a7,}

[edit] Vassiliev invariants

V₂ and V₃:

(0, -2)

[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 K11a325. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

-2

-3

-1

-3

-2

-5

-7

-9

-3

-11

-13

-3

-15

-17

-1

Integral Khovanov Homology

(db, data source)

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