K11a56

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K11a55

K11a57

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 K11a56's page at Knotilus!

Visit K11a56's page at the original Knot Atlas!

[edit K11a56 Quick Notes]

[edit K11a56 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_16,6,17,5 X₂₈₃₇ X_18,9,19,10 X_20,11,21,12 X_22,13,1,14 X_6,16,7,15 X_14,17,15,18 X_12,19,13,20 X_10,21,11,22
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -9, 8, -3, 9, -5, 10, -6, 11, -7
Dowker-Thistlethwaite code	4 8 16 2 18 20 22 6 14 12 10

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a56/ThurstonBennequinNumber
Hyperbolic Volume	14.5406
A-Polynomial	See Data:K11a56/A-polynomial

[edit Notes for K11a56'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 K11a56's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$-2 t 3 + 11 t 2 -25 t + 33-25 t -1 + 11 t -2 -2 t -3$
Conway polynomial	$-2 z 6 - z 4 + z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 109, 0 }
Jones polynomial	$- q 5 + 3 q 4 -7 q 3 + 12 q 2 -15 q + 18-17 q -1 + 15 q -2 -11 q -3 + 6 q -4 -3 q -5 + q -6$
HOMFLY-PT polynomial (db, data sources)	$- a 2 z 6 - z 6 + a 4 z 4 -3 a 2 z 4 + 2 z 4 a -2 - z 4 + 2 a 4 z 2 -5 a 2 z 2 + 3 z 2 a -2 - z 2 a -4 + 2 z 2 + a 4 -3 a 2 + a -2 - a -4 + 3$
Kauffman polynomial (db, data sources)	$a 2 z 10 + z 10 + 3 a 3 z 9 + 6 a z 9 + 3 z 9 a -1 + 4 a 4 z 8 + 7 a 2 z 8 + 5 z 8 a -2 + 8 z 8 + 3 a 5 z 7 - a 3 z 7 -4 a z 7 + 5 z 7 a -1 + 5 z 7 a -3 + a 6 z 6 -9 a 4 z 6 -17 a 2 z 6 -3 z 6 a -2 + 3 z 6 a -4 -13 z 6 -9 a 5 z 5 -10 a 3 z 5 -8 a z 5 -15 z 5 a -1 -7 z 5 a -3 + z 5 a -5 -3 a 6 z 4 + 5 a 4 z 4 + 13 a 2 z 4 -4 z 4 a -2 -5 z 4 a -4 + 6 z 4 + 8 a 5 z 3 + 12 a 3 z 3 + 10 a z 3 + 11 z 3 a -1 + 3 z 3 a -3 -2 z 3 a -5 + 2 a 6 z 2 - a 4 z 2 -8 a 2 z 2 + 4 z 2 a -2 + 3 z 2 a -4 -4 z 2 -2 a 5 z -5 a 3 z -5 a z -3 z a -1 + z a -5 + a 4 + 3 a 2 - a -2 - a -4 + 3$
The A2 invariant	Data:K11a56/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a56/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a56 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["K11a56"];

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

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

Out[4]= $-2 t 3 + 11 t 2 -25 t + 33-25 t -1 + 11 t -2 -2 t -3$

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

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

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]= { 109, 0 }

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

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

Out[8]= $- q 5 + 3 q 4 -7 q 3 + 12 q 2 -15 q + 18-17 q -1 + 15 q -2 -11 q -3 + 6 q -4 -3 q -5 + q -6$

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

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

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

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

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

Out[10]= $a 2 z 10 + z 10 + 3 a 3 z 9 + 6 a z 9 + 3 z 9 a -1 + 4 a 4 z 8 + 7 a 2 z 8 + 5 z 8 a -2 + 8 z 8 + 3 a 5 z 7 - a 3 z 7 -4 a z 7 + 5 z 7 a -1 + 5 z 7 a -3 + a 6 z 6 -9 a 4 z 6 -17 a 2 z 6 -3 z 6 a -2 + 3 z 6 a -4 -13 z 6 -9 a 5 z 5 -10 a 3 z 5 -8 a z 5 -15 z 5 a -1 -7 z 5 a -3 + z 5 a -5 -3 a 6 z 4 + 5 a 4 z 4 + 13 a 2 z 4 -4 z 4 a -2 -5 z 4 a -4 + 6 z 4 + 8 a 5 z 3 + 12 a 3 z 3 + 10 a z 3 + 11 z 3 a -1 + 3 z 3 a -3 -2 z 3 a -5 + 2 a 6 z 2 - a 4 z 2 -8 a 2 z 2 + 4 z 2 a -2 + 3 z 2 a -4 -4 z 2 -2 a 5 z -5 a 3 z -5 a z -3 z a -1 + z a -5 + a 4 + 3 a 2 - a -2 - a -4 + 3$

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

Same Alexander/Conway Polynomial: {K11a185, K11a265,}

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["K11a56"];

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]= { $-2 t 3 + 11 t 2 -25 t + 33-25 t -1 + 11 t -2 -2 t -3$ , $- q 5 + 3 q 4 -7 q 3 + 12 q 2 -15 q + 18-17 q -1 + 15 q -2 -11 q -3 + 6 q -4 -3 q -5 + q -6$ }

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]= {K11a185, K11a265,}

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:

(1, 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

4

16

8

$\frac{62}{3}$

$\frac{34}{3}$

64

$\frac{256}{3}$

$-\frac{128}{3}$

48

$\frac{32}{3}$

128

$\frac{248}{3}$

$\frac{136}{3}$

$\frac{11311}{30}$

$\frac{338}{15}$

$\frac{7262}{45}$

$\frac{689}{18}$

$\frac{271}{30}$

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 =

0 is the signature of K11a56. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

-4

-3

-1

-3

-2

-5

-7

-5

-9

-11

-2

-13

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$