K11n132

From Knot Atlas

Jump to: navigation, search

K11n131

K11n133

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 K11n132's page at Knotilus!

Visit K11n132's page at the original Knot Atlas!

[edit K11n132 Quick Notes]

[edit K11n132 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,3,11,4 X_5,19,6,18 X_7,16,8,17 X_9,14,10,15 X_2,11,3,12 X_13,20,14,21 X_15,8,16,9 X_17,1,18,22 X_19,12,20,13 X_21,7,22,6
Gauss code	1, -6, 2, -1, -3, 11, -4, 8, -5, -2, 6, 10, -7, 5, -8, 4, -9, 3, -10, 7, -11, 9
Dowker-Thistlethwaite code	4 10 -18 -16 -14 2 -20 -8 -22 -12 -6

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Chiral
Unknotting number	${1,2}$
3-genus	2
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11n132/ThurstonBennequinNumber
Hyperbolic Volume	10.2668
A-Polynomial	See Data:K11n132/A-polynomial

[edit Notes for K11n132's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for K11n132's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$2 t 2 -6 t + 9-6 t -1 + 2 t -2$
Conway polynomial	$2 z 4 + 2 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 25, 0 }
Jones polynomial	$- q + 3-3 q -1 + 4 q -2 -4 q -3 + 4 q -4 -3 q -5 + 2 q -6 - q -7$
HOMFLY-PT polynomial (db, data sources)	$- z 2 a 6 - a 6 + z 4 a 4 + 2 z 2 a 4 + a 4 + z 4 a 2 + 2 z 2 a 2 + a 2 - z 2$
Kauffman polynomial (db, data sources)	$a 5 z 9 + a 3 z 9 + 2 a 6 z 8 + 3 a 4 z 8 + a 2 z 8 + a 7 z 7 -3 a 5 z 7 -4 a 3 z 7 -10 a 6 z 6 -13 a 4 z 6 -3 a 2 z 6 -5 a 7 z 5 - a 5 z 5 + 6 a 3 z 5 + 2 a z 5 + 14 a 6 z 4 + 15 a 4 z 4 + a 2 z 4 + 7 a 7 z 3 + 4 a 5 z 3 -8 a 3 z 3 -5 a z 3 -6 a 6 z 2 -7 a 4 z 2 + a 2 z 2 + 2 z 2 -2 a 7 z - a 5 z + 3 a 3 z + 3 a z + z a -1 + a 6 + a 4 - a 2$
The A2 invariant	Data:K11n132/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n132/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n132 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["K11n132"];

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

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

Out[4]= $2 t 2 -6 t + 9-6 t -1 + 2 t -2$

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

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

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]= { 25, 0 }

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

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

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

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

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

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

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

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

Out[10]= $a 5 z 9 + a 3 z 9 + 2 a 6 z 8 + 3 a 4 z 8 + a 2 z 8 + a 7 z 7 -3 a 5 z 7 -4 a 3 z 7 -10 a 6 z 6 -13 a 4 z 6 -3 a 2 z 6 -5 a 7 z 5 - a 5 z 5 + 6 a 3 z 5 + 2 a z 5 + 14 a 6 z 4 + 15 a 4 z 4 + a 2 z 4 + 7 a 7 z 3 + 4 a 5 z 3 -8 a 3 z 3 -5 a z 3 -6 a 6 z 2 -7 a 4 z 2 + a 2 z 2 + 2 z 2 -2 a 7 z - a 5 z + 3 a 3 z + 3 a z + z a -1 + a 6 + a 4 - a 2$

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

Same Alexander/Conway Polynomial: {8_8, 10_129, K11n39, K11n45, K11n50,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11n50, K11n133,}

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["K11n132"];

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]= { $2 t 2 -6 t + 9-6 t -1 + 2 t -2$ , $- q + 3-3 q -1 + 4 q -2 -4 q -3 + 4 q -4 -3 q -5 + 2 q -6 - q -7$ }

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]= {8_8, 10_129, K11n39, K11n45, K11n50,}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {K11n50, K11n133,}

[edit] Vassiliev invariants

V₂ and V₃:

(2, -3)

[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 K11n132. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

-3

-5

-7

-9

-11

-1

-13

-15

-1

Integral Khovanov Homology

(db, data source)

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