K11n65

From Knot Atlas

Jump to: navigation, search

K11n64

K11n66

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 K11n65's page at Knotilus!

Visit K11n65's page at the original Knot Atlas!

[edit K11n65 Quick Notes]

[edit K11n65 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_14,5,15,6 X₂₈₃₇ X_18,9,19,10 X_16,11,17,12 X_13,20,14,21 X_6,15,7,16 X_10,17,11,18 X_19,1,20,22 X_21,12,22,13
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, 5, -9, 6, 11, -7, -3, 8, -6, 9, -5, -10, 7, -11, 10
Dowker-Thistlethwaite code	4 8 14 2 18 16 -20 6 10 -22 -12

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	${1,2,3}$
3-genus	2
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11n65/ThurstonBennequinNumber
Hyperbolic Volume	11.4167
A-Polynomial	See Data:K11n65/A-polynomial

[edit Notes for K11n65's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$2$
Rasmussen s-Invariant	0

[edit Notes for K11n65's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$3 t 2 -8 t + 11-8 t -1 + 3 t -2$
Conway polynomial	$3 z 4 + 4 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 33, 0 }
Jones polynomial	$-2 q + 4-4 q -1 + 6 q -2 -5 q -3 + 5 q -4 -4 q -5 + 2 q -6 - q -7$
HOMFLY-PT polynomial (db, data sources)	$- z 2 a 6 - a 6 + z 4 a 4 + z 2 a 4 - a 4 + 2 z 4 a 2 + 6 z 2 a 2 + 5 a 2 -2 z 2 -2$
Kauffman polynomial (db, data sources)	$a 5 z 9 + a 3 z 9 + 2 a 6 z 8 + 4 a 4 z 8 + 2 a 2 z 8 + a 7 z 7 - a 5 z 7 - a 3 z 7 + a z 7 -9 a 6 z 6 -15 a 4 z 6 -6 a 2 z 6 -5 a 7 z 5 -9 a 5 z 5 -5 a 3 z 5 - a z 5 + 11 a 6 z 4 + 12 a 4 z 4 + 2 a 2 z 4 + z 4 + 8 a 7 z 3 + 13 a 5 z 3 + 2 a 3 z 3 -3 a z 3 -4 a 6 z 2 - a 4 z 2 + 5 a 2 z 2 + 2 z 2 -4 a 7 z -4 a 5 z + 2 a 3 z + 4 a z + 2 z a -1 + a 6 - a 4 -5 a 2 -2$
The A2 invariant	Data:K11n65/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n65/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n65 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["K11n65"];

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

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

Out[4]= $3 t 2 -8 t + 11-8 t -1 + 3 t -2$

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

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

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]= ${1}$

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

Out[7]= { 33, 0 }

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

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

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

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

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

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

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

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

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

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

Same Alexander/Conway Polynomial: {8_15,}

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["K11n65"];

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]= { $3 t 2 -8 t + 11-8 t -1 + 3 t -2$ , $-2 q + 4-4 q -1 + 6 q -2 -5 q -3 + 5 q -4 -4 q -5 + 2 q -6 - q -7$ }

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_15,}

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, -5)

[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 K11n65. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

-2

-1

-3

-5

-7

-9

-11

-2

-13

-15

-1

Integral Khovanov Homology

(db, data source)

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