K11n19

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K11n18

K11n20

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 K11n19's page at Knotilus!

Visit K11n19's page at the original Knot Atlas!

[edit K11n19 Quick Notes]

[edit K11n19 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₃₉₄ X_10,6,11,5 X_7,17,8,16 X_2,9,3,10 X_11,18,12,19 X_13,20,14,21 X_15,22,16,1 X_17,7,18,6 X_19,12,20,13 X_21,14,22,15
Gauss code	1, -5, 2, -1, 3, 9, -4, -2, 5, -3, -6, 10, -7, 11, -8, 4, -9, 6, -10, 7, -11, 8
Dowker-Thistlethwaite code	4 8 10 -16 2 -18 -20 -22 -6 -12 -14

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	3
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11n19/ThurstonBennequinNumber
Hyperbolic Volume	4.76989
A-Polynomial	See Data:K11n19/A-polynomial

[edit Notes for K11n19's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$3$
Rasmussen s-Invariant	2

[edit Notes for K11n19's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 3 + 2 t 2 -1 + 2 t -2 - t -3$
Conway polynomial	$- z 6 -4 z 4 - z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 5, -4 }
Jones polynomial	$q 2 - q + 1- q -1 + q -2$
HOMFLY-PT polynomial (db, data sources)	$- a 6 + z 4 a 4 + 5 z 2 a 4 + 4 a 4 - z 6 a 2 -6 z 4 a 2 -10 z 2 a 2 -5 a 2 + z 4 + 4 z 2 + 3$
Kauffman polynomial (db, data sources)	$a 3 z 9 + a z 9 + a 4 z 8 + 2 a 2 z 8 + z 8 -7 a 3 z 7 -7 a z 7 -7 a 4 z 6 -14 a 2 z 6 -7 z 6 + 15 a 3 z 5 + 15 a z 5 + 15 a 4 z 4 + 30 a 2 z 4 + 15 z 4 - a 5 z 3 -12 a 3 z 3 -11 a z 3 - a 6 z 2 -13 a 4 z 2 -23 a 2 z 2 -11 z 2 + a 5 z + 3 a 3 z + 2 a z + a 6 + 4 a 4 + 5 a 2 + 3$
The A2 invariant	$- q 20 + q 16 + q 14 + q 12 - q 6 - q 4 + q -2 + q -4 + q -6$
The G2 invariant	Data:K11n19/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n19 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["K11n19"];

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

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

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

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

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

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]= { 5, -4 }

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

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

Out[8]= $q 2 - q + 1- q -1 + q -2$

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

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

Out[9]= $- a 6 + z 4 a 4 + 5 z 2 a 4 + 4 a 4 - z 6 a 2 -6 z 4 a 2 -10 z 2 a 2 -5 a 2 + z 4 + 4 z 2 + 3$

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

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

Out[10]= $a 3 z 9 + a z 9 + a 4 z 8 + 2 a 2 z 8 + z 8 -7 a 3 z 7 -7 a z 7 -7 a 4 z 6 -14 a 2 z 6 -7 z 6 + 15 a 3 z 5 + 15 a z 5 + 15 a 4 z 4 + 30 a 2 z 4 + 15 z 4 - a 5 z 3 -12 a 3 z 3 -11 a z 3 - a 6 z 2 -13 a 4 z 2 -23 a 2 z 2 -11 z 2 + a 5 z + 3 a 3 z + 2 a z + a 6 + 4 a 4 + 5 a 2 + 3$

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

Same Alexander/Conway Polynomial: {K11n135,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {4_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["K11n19"];

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

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]= {K11n135,}

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

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

Out[6]= {4_1,}

[edit] Vassiliev invariants

V₂ and V₃:

(-1, 0)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

-4

0

8

$\frac{178}{3}$

$\frac{110}{3}$

0

-128

-32

-64

$-\frac{32}{3}$

0

$-\frac{712}{3}$

$-\frac{440}{3}$

$-\frac{2431}{30}$

$\frac{302}{15}$

$-\frac{8702}{45}$

$\frac{2335}{18}$

$-\frac{1951}{30}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

[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 =

-4 is the signature of K11n19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-3

-2

-1

-3

-5

-7

-9

Integral Khovanov Homology

(db, data source)

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