K11n2

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K11n1

K11n3

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 K11n2's page at Knotilus!

Visit K11n2's page at the original Knot Atlas!

[edit K11n2 Quick Notes]

[edit K11n2 Further Notes and Views]

Knot K11n2.

A graph, knot K11n2.

A part of a knot and a part of a graph.

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₃₉₄ X_10,6,11,5 X_7,15,8,14 X_2,9,3,10 X_16,12,17,11 X_20,14,21,13 X_15,7,16,6 X_22,18,1,17 X_12,20,13,19 X_18,22,19,21
Gauss code	1, -5, 2, -1, 3, 8, -4, -2, 5, -3, 6, -10, 7, 4, -8, -6, 9, -11, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 8 10 -14 2 16 20 -6 22 12 18

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:K11n2/ThurstonBennequinNumber
Hyperbolic Volume	12.5932
A-Polynomial	See Data:K11n2/A-polynomial

[edit Notes for K11n2'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 K11n2's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$-2 t 3 + 8 t 2 -12 t + 13-12 t -1 + 8 t -2 -2 t -3$
Conway polynomial	$-2 z 6 -4 z 4 + 2 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 57, 4 }
Jones polynomial	$-2 q 9 + 5 q 8 -7 q 7 + 9 q 6 -10 q 5 + 9 q 4 -7 q 3 + 5 q 2 -2 q + 1$
HOMFLY-PT polynomial (db, data sources)	$- z 6 a -4 - z 6 a -6 + z 4 a -2 -3 z 4 a -4 -3 z 4 a -6 + z 4 a -8 + 3 z 2 a -2 -2 z 2 a -4 -2 z 2 a -6 + 3 z 2 a -8 + 2 a -2 - a -4 - a -6 + 2 a -8 - a -10$
Kauffman polynomial (db, data sources)	$z 9 a -5 + z 9 a -7 + 2 z 8 a -4 + 5 z 8 a -6 + 3 z 8 a -8 + 2 z 7 a -3 + 2 z 7 a -5 + 3 z 7 a -7 + 3 z 7 a -9 + z 6 a -2 -3 z 6 a -4 -11 z 6 a -6 -6 z 6 a -8 + z 6 a -10 -6 z 5 a -3 -9 z 5 a -5 -7 z 5 a -7 -4 z 5 a -9 -4 z 4 a -2 -4 z 4 a -4 + 8 z 4 a -6 + 12 z 4 a -8 + 4 z 4 a -10 + 4 z 3 a -3 + 4 z 3 a -5 + 3 z 3 a -7 + 6 z 3 a -9 + 3 z 3 a -11 + 5 z 2 a -2 + 5 z 2 a -4 -5 z 2 a -6 -10 z 2 a -8 -5 z 2 a -10 -2 z a -9 -2 z a -11 -2 a -2 - a -4 + a -6 + 2 a -8 + a -10$
The A2 invariant	Data:K11n2/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n2/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n2 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["K11n2"];

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

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

Out[4]= $-2 t 3 + 8 t 2 -12 t + 13-12 t -1 + 8 t -2 -2 t -3$

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

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

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]= { 57, 4 }

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

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

Out[8]= $-2 q 9 + 5 q 8 -7 q 7 + 9 q 6 -10 q 5 + 9 q 4 -7 q 3 + 5 q 2 -2 q + 1$

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

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

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

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

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

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

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

Same Alexander/Conway Polynomial: {10_14, K11a161,}

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["K11n2"];

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]= { $-2 t 3 + 8 t 2 -12 t + 13-12 t -1 + 8 t -2 -2 t -3$ , $-2 q 9 + 5 q 8 -7 q 7 + 9 q 6 -10 q 5 + 9 q 4 -7 q 3 + 5 q 2 -2 q + 1$ }

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]= {10_14, K11a161,}

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:

(2, 5)

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

40

32

$\frac{652}{3}$

$\frac{140}{3}$

320

$\frac{3280}{3}$

$\frac{640}{3}$

200

$\frac{256}{3}$

800

$\frac{5216}{3}$

$\frac{1120}{3}$

$\frac{81511}{15}$

$-\frac{1684}{15}$

$\frac{118564}{45}$

$\frac{905}{9}$

$\frac{5191}{15}$

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 K11n2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-2

-1

-2

-1

-2

-1

Integral Khovanov Homology

(db, data source)

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