K11n49

From Knot Atlas

Jump to: navigation, search

K11n48

K11n50

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 K11n49's page at Knotilus!

Visit K11n49's page at the original Knot Atlas!

[edit K11n49 Quick Notes]

[edit K11n49 Further Notes and Views]

K11n49 is not $k$ -colourable for any $k$ . See The Determinant and the Signature.

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₃₉₄ X_5,12,6,13 X_7,17,8,16 X_2,9,3,10 X_11,19,12,18 X_13,22,14,1 X_15,20,16,21 X_17,11,18,10 X_19,7,20,6 X_21,14,22,15
Gauss code	1, -5, 2, -1, -3, 10, -4, -2, 5, 9, -6, 3, -7, 11, -8, 4, -9, 6, -10, 8, -11, 7
Dowker-Thistlethwaite code	4 8 -12 -16 2 -18 -22 -20 -10 -6 -14

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:K11n49/ThurstonBennequinNumber
Hyperbolic Volume	6.90911
A-Polynomial	See Data:K11n49/A-polynomial

[edit Notes for K11n49'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 K11n49's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 2 + 3- t -2$
Conway polynomial	$- z 4 -4 z 2 + 1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^2+t+1\right\}$
Determinant and Signature	{ 1, 0 }
Jones polynomial	$q 4 - q 3 + q 2 - q + 1- q -2 + q -3 - q -4 + q -5$
HOMFLY-PT polynomial (db, data sources)	$z 2 a 4 + 2 a 4 - z 4 a 2 -4 z 2 a 2 -3 a 2 + 2- z 2 a -2 - a -2 + a -4$
Kauffman polynomial (db, data sources)	$a 3 z 9 + a z 9 + a 4 z 8 + 2 a 2 z 8 + z 8 -7 a 3 z 7 -7 a z 7 -7 a 4 z 6 -14 a 2 z 6 -7 z 6 + 15 a 3 z 5 + 14 a z 5 + z 5 a -3 + 15 a 4 z 4 + 28 a 2 z 4 + z 4 a -2 + z 4 a -4 + 13 z 4 -12 a 3 z 3 -10 a z 3 - z 3 a -1 -3 z 3 a -3 -11 a 4 z 2 -18 a 2 z 2 -3 z 2 a -2 -3 z 2 a -4 -7 z 2 + 3 a 3 z + 3 a z + z a -1 + z a -3 + 2 a 4 + 3 a 2 + a -2 + a -4 + 2$
The A2 invariant	Data:K11n49/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n49/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n49 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["K11n49"];

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

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

Out[4]= $- t 2 + 3- t -2$

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

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

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]= $\left\{2,t^2+t+1\right\}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 1, 0 }

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

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

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

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

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

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

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

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

Out[10]= $a 3 z 9 + a z 9 + a 4 z 8 + 2 a 2 z 8 + z 8 -7 a 3 z 7 -7 a z 7 -7 a 4 z 6 -14 a 2 z 6 -7 z 6 + 15 a 3 z 5 + 14 a z 5 + z 5 a -3 + 15 a 4 z 4 + 28 a 2 z 4 + z 4 a -2 + z 4 a -4 + 13 z 4 -12 a 3 z 3 -10 a z 3 - z 3 a -1 -3 z 3 a -3 -11 a 4 z 2 -18 a 2 z 2 -3 z 2 a -2 -3 z 2 a -4 -7 z 2 + 3 a 3 z + 3 a z + z a -1 + z a -3 + 2 a 4 + 3 a 2 + a -2 + a -4 + 2$

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

Same Alexander/Conway Polynomial: {K11n116,}

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["K11n49"];

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

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]= {K11n116,}

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:

(-4, 1)

[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 K11n49. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

-3

-1

-5

-7

-9

-11

Integral Khovanov Homology

(db, data source)

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