K11a2

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K11a1

K11a3

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 K11a2's page at Knotilus!

Visit K11a2's page at the original Knot Atlas!

[edit K11a2 Quick Notes]

[edit K11a2 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_18,12,19,11 X_6,14,7,13 X_20,16,21,15 X_12,18,13,17 X_22,20,1,19 X_16,22,17,21
Gauss code	1, -5, 2, -1, 3, -7, 4, -2, 5, -3, 6, -9, 7, -4, 8, -11, 9, -6, 10, -8, 11, -10
Dowker-Thistlethwaite code	4 8 10 14 2 18 6 20 12 22 16

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Chiral
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a2/ThurstonBennequinNumber
Hyperbolic Volume	16.5881
A-Polynomial	See Data:K11a2/A-polynomial

[edit Notes for K11a2's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for K11a2's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$-3 t 3 + 15 t 2 -31 t + 39-31 t -1 + 15 t -2 -3 t -3$
Conway polynomial	$-3 z 6 -3 z 4 + 2 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 137, 4 }
Jones polynomial	$- q 11 + 4 q 10 -9 q 9 + 15 q 8 -20 q 7 + 22 q 6 -22 q 5 + 19 q 4 -13 q 3 + 8 q 2 -3 q + 1$
HOMFLY-PT polynomial (db, data sources)	$- z 6 a -4 -2 z 6 a -6 + z 4 a -2 - z 4 a -4 -6 z 4 a -6 + 3 z 4 a -8 + 2 z 2 a -2 + 2 z 2 a -4 -7 z 2 a -6 + 6 z 2 a -8 - z 2 a -10 + a -2 + 2 a -4 -4 a -6 + 3 a -8 - a -10$
Kauffman polynomial (db, data sources)	$z 10 a -6 + z 10 a -8 + 4 z 9 a -5 + 9 z 9 a -7 + 5 z 9 a -9 + 5 z 8 a -4 + 16 z 8 a -6 + 20 z 8 a -8 + 9 z 8 a -10 + 3 z 7 a -3 + z 7 a -5 + 10 z 7 a -9 + 8 z 7 a -11 + z 6 a -2 -10 z 6 a -4 -42 z 6 a -6 -45 z 6 a -8 -10 z 6 a -10 + 4 z 6 a -12 -7 z 5 a -3 -17 z 5 a -5 -33 z 5 a -7 -36 z 5 a -9 -12 z 5 a -11 + z 5 a -13 -3 z 4 a -2 + 7 z 4 a -4 + 38 z 4 a -6 + 35 z 4 a -8 + 2 z 4 a -10 -5 z 4 a -12 + 5 z 3 a -3 + 17 z 3 a -5 + 33 z 3 a -7 + 30 z 3 a -9 + 8 z 3 a -11 - z 3 a -13 + 3 z 2 a -2 -4 z 2 a -4 -18 z 2 a -6 -14 z 2 a -8 - z 2 a -10 + 2 z 2 a -12 - z a -3 -6 z a -5 -10 z a -7 -8 z a -9 -3 z a -11 - a -2 + 2 a -4 + 4 a -6 + 3 a -8 + a -10$
The A2 invariant	Data:K11a2/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a2/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a2 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["K11a2"];

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

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

Out[4]= $-3 t 3 + 15 t 2 -31 t + 39-31 t -1 + 15 t -2 -3 t -3$

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

Out[5]= $-3 z 6 -3 z 4 + 2 z 2 + 1$

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]= { 137, 4 }

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

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

Out[8]= $- q 11 + 4 q 10 -9 q 9 + 15 q 8 -20 q 7 + 22 q 6 -22 q 5 + 19 q 4 -13 q 3 + 8 q 2 -3 q + 1$

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

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

Out[9]= $- z 6 a -4 -2 z 6 a -6 + z 4 a -2 - z 4 a -4 -6 z 4 a -6 + 3 z 4 a -8 + 2 z 2 a -2 + 2 z 2 a -4 -7 z 2 a -6 + 6 z 2 a -8 - z 2 a -10 + a -2 + 2 a -4 -4 a -6 + 3 a -8 - a -10$

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

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

Out[10]= $z 10 a -6 + z 10 a -8 + 4 z 9 a -5 + 9 z 9 a -7 + 5 z 9 a -9 + 5 z 8 a -4 + 16 z 8 a -6 + 20 z 8 a -8 + 9 z 8 a -10 + 3 z 7 a -3 + z 7 a -5 + 10 z 7 a -9 + 8 z 7 a -11 + z 6 a -2 -10 z 6 a -4 -42 z 6 a -6 -45 z 6 a -8 -10 z 6 a -10 + 4 z 6 a -12 -7 z 5 a -3 -17 z 5 a -5 -33 z 5 a -7 -36 z 5 a -9 -12 z 5 a -11 + z 5 a -13 -3 z 4 a -2 + 7 z 4 a -4 + 38 z 4 a -6 + 35 z 4 a -8 + 2 z 4 a -10 -5 z 4 a -12 + 5 z 3 a -3 + 17 z 3 a -5 + 33 z 3 a -7 + 30 z 3 a -9 + 8 z 3 a -11 - z 3 a -13 + 3 z 2 a -2 -4 z 2 a -4 -18 z 2 a -6 -14 z 2 a -8 - z 2 a -10 + 2 z 2 a -12 - z a -3 -6 z a -5 -10 z a -7 -8 z a -9 -3 z a -11 - a -2 + 2 a -4 + 4 a -6 + 3 a -8 + a -10$

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

Same Alexander/Conway Polynomial: {K11a116,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a116,}

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["K11a2"];

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]= { $-3 t 3 + 15 t 2 -31 t + 39-31 t -1 + 15 t -2 -3 t -3$ , $- q 11 + 4 q 10 -9 q 9 + 15 q 8 -20 q 7 + 22 q 6 -22 q 5 + 19 q 4 -13 q 3 + 8 q 2 -3 q + 1$ }

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]= {K11a116,}

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

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

Out[6]= {K11a116,}

[edit] Vassiliev invariants

V₂ and V₃:

(2, 4)

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

8

32

32

$\frac{508}{3}$

$\frac{116}{3}$

256

$\frac{2432}{3}$

$\frac{512}{3}$

128

$\frac{256}{3}$

512

$\frac{4064}{3}$

$\frac{928}{3}$

$\frac{56431}{15}$

$\frac{2116}{15}$

$\frac{70444}{45}$

$\frac{353}{9}$

$\frac{2911}{15}$

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 K11a2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-2

-1

-5

-3

-5

-2

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 3$	$i = 5$
$r = -2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 4$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 5$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 6$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 7$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 8$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 9$	${\mathbb Z}_2$	${\mathbb Z}$