K11a4

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K11a3

K11a5

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 K11a4's page at Knotilus!

Visit K11a4's page at the original Knot Atlas!

[edit K11a4 Quick Notes]

[edit K11a4 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₃₉₄ X_10,6,11,5 X_14,8,15,7 X_2,9,3,10 X_18,11,19,12 X_6,14,7,13 X_22,15,1,16 X_20,17,21,18 X_12,19,13,20 X_16,21,17,22
Gauss code	1, -5, 2, -1, 3, -7, 4, -2, 5, -3, 6, -10, 7, -4, 8, -11, 9, -6, 10, -9, 11, -8
Dowker-Thistlethwaite code	4 8 10 14 2 18 6 22 20 12 16

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	${1,2}$
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a4/ThurstonBennequinNumber
Hyperbolic Volume	14.1846
A-Polynomial	See Data:K11a4/A-polynomial

[edit Notes for K11a4's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$3$
Rasmussen s-Invariant	0

[edit Notes for K11a4's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$-2 t 3 + 10 t 2 -22 t + 29-22 t -1 + 10 t -2 -2 t -3$
Conway polynomial	$-2 z 6 -2 z 4 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 97, 0 }
Jones polynomial	$q 4 -3 q 3 + 6 q 2 -10 q + 14-15 q -1 + 15 q -2 -13 q -3 + 10 q -4 -6 q -5 + 3 q -6 - q -7$
HOMFLY-PT polynomial (db, data sources)	$- z 2 a 6 - a 6 + 2 z 4 a 4 + 4 z 2 a 4 + 2 a 4 - z 6 a 2 -2 z 4 a 2 - z 2 a 2 - z 6 -3 z 4 -4 z 2 -1 + z 4 a -2 + 2 z 2 a -2 + a -2$
Kauffman polynomial (db, data sources)	$a 4 z 10 + a 2 z 10 + 3 a 5 z 9 + 7 a 3 z 9 + 4 a z 9 + 3 a 6 z 8 + 7 a 4 z 8 + 10 a 2 z 8 + 6 z 8 + a 7 z 7 -7 a 5 z 7 -15 a 3 z 7 - a z 7 + 6 z 7 a -1 -12 a 6 z 6 -34 a 4 z 6 -33 a 2 z 6 + 5 z 6 a -2 -6 z 6 -4 a 7 z 5 -3 a 5 z 5 - a 3 z 5 -10 a z 5 -5 z 5 a -1 + 3 z 5 a -3 + 15 a 6 z 4 + 39 a 4 z 4 + 28 a 2 z 4 -5 z 4 a -2 + z 4 a -4 -2 z 4 + 5 a 7 z 3 + 11 a 5 z 3 + 10 a 3 z 3 + 7 a z 3 -3 z 3 a -3 -7 a 6 z 2 -16 a 4 z 2 -8 a 2 z 2 + 3 z 2 a -2 - z 2 a -4 + 5 z 2 -2 a 7 z -4 a 5 z -3 a 3 z - a z + z a -1 + z a -3 + a 6 + 2 a 4 - a -2 -1$
The A2 invariant	Data:K11a4/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a4/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a4 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["K11a4"];

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

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

Out[4]= $-2 t 3 + 10 t 2 -22 t + 29-22 t -1 + 10 t -2 -2 t -3$

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

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

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]= { 97, 0 }

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

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

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

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

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

Out[9]= $- z 2 a 6 - a 6 + 2 z 4 a 4 + 4 z 2 a 4 + 2 a 4 - z 6 a 2 -2 z 4 a 2 - z 2 a 2 - z 6 -3 z 4 -4 z 2 -1 + z 4 a -2 + 2 z 2 a -2 + a -2$

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

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

Out[10]= $a 4 z 10 + a 2 z 10 + 3 a 5 z 9 + 7 a 3 z 9 + 4 a z 9 + 3 a 6 z 8 + 7 a 4 z 8 + 10 a 2 z 8 + 6 z 8 + a 7 z 7 -7 a 5 z 7 -15 a 3 z 7 - a z 7 + 6 z 7 a -1 -12 a 6 z 6 -34 a 4 z 6 -33 a 2 z 6 + 5 z 6 a -2 -6 z 6 -4 a 7 z 5 -3 a 5 z 5 - a 3 z 5 -10 a z 5 -5 z 5 a -1 + 3 z 5 a -3 + 15 a 6 z 4 + 39 a 4 z 4 + 28 a 2 z 4 -5 z 4 a -2 + z 4 a -4 -2 z 4 + 5 a 7 z 3 + 11 a 5 z 3 + 10 a 3 z 3 + 7 a z 3 -3 z 3 a -3 -7 a 6 z 2 -16 a 4 z 2 -8 a 2 z 2 + 3 z 2 a -2 - z 2 a -4 + 5 z 2 -2 a 7 z -4 a 5 z -3 a 3 z - a z + z a -1 + z a -3 + a 6 + 2 a 4 - a -2 -1$

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

Same Alexander/Conway Polynomial: {K11a110,}

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["K11a4"];

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]= { $-2 t 3 + 10 t 2 -22 t + 29-22 t -1 + 10 t -2 -2 t -3$ , $q 4 -3 q 3 + 6 q 2 -10 q + 14-15 q -1 + 15 q -2 -13 q -3 + 10 q -4 -6 q -5 + 3 q -6 - q -7$ }

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]= {K11a110,}

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:

(0, -2)

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

0

-16

0

48

16

0

$-\frac{64}{3}$

$\frac{32}{3}$

16

0

128

0

0

216

$\frac{352}{3}$

$-\frac{352}{3}$

$\frac{104}{3}$

-40

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 =

0 is the signature of K11a4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

-2

-4

-1

-3

-5

-7

-3

-9

-11

-3

-13

-15

-1

Integral Khovanov Homology

(db, data source)

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