K11n91

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K11n90

K11n92

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 K11n91's page at Knotilus!

Visit K11n91's page at the original Knot Atlas!

[edit K11n91 Quick Notes]

[edit K11n91 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,3,11,4 X_12,6,13,5 X_7,16,8,17 X_9,18,10,19 X_2,11,3,12 X_20,13,21,14 X_15,8,16,9 X_17,6,18,7 X_22,19,1,20 X_14,21,15,22
Gauss code	1, -6, 2, -1, 3, 9, -4, 8, -5, -2, 6, -3, 7, -11, -8, 4, -9, 5, 10, -7, 11, -10
Dowker-Thistlethwaite code	4 10 12 -16 -18 2 20 -8 -6 22 14

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	${1,2}$
3-genus	2
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11n91/ThurstonBennequinNumber
Hyperbolic Volume	9.90352
A-Polynomial	See Data:K11n91/A-polynomial

[edit Notes for K11n91's three dimensional invariants]

[edit] Four dimensional invariants

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

[edit Notes for K11n91's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$- t 2 + 8 t -13 + 8 t -1 - t -2$
Conway polynomial	$- z 4 + 4 z 2 + 1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^2+t+1\right\}$
Determinant and Signature	{ 31, -2 }
Jones polynomial	$q -1 -2 q -2 + 4 q -3 -4 q -4 + 5 q -5 -5 q -6 + 4 q -7 -3 q -8 + 2 q -9 - q -10$
HOMFLY-PT polynomial (db, data sources)	$- a 10 + 2 z 2 a 8 + 2 a 8 - z 4 a 6 -2 z 2 a 6 -3 a 6 + 3 z 2 a 4 + 3 a 4 + z 2 a 2$
Kauffman polynomial (db, data sources)	$z 7 a 11 -5 z 5 a 11 + 7 z 3 a 11 -3 z a 11 + 2 z 8 a 10 -10 z 6 a 10 + 14 z 4 a 10 -6 z 2 a 10 + a 10 + z 9 a 9 -2 z 7 a 9 -6 z 5 a 9 + 11 z 3 a 9 -3 z a 9 + 4 z 8 a 8 -18 z 6 a 8 + 23 z 4 a 8 -11 z 2 a 8 + 2 a 8 + z 9 a 7 -2 z 7 a 7 -2 z 5 a 7 + 2 z 3 a 7 - z a 7 + 2 z 8 a 6 -8 z 6 a 6 + 13 z 4 a 6 -12 z 2 a 6 + 3 a 6 + z 7 a 5 - z 5 a 5 - z a 5 + 4 z 4 a 4 -6 z 2 a 4 + 3 a 4 + 2 z 3 a 3 + z 2 a 2$
The A2 invariant	Data:K11n91/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n91/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n91 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["K11n91"];

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

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

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

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

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

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]= $\left\{2,t^2+t+1\right\}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 31, -2 }

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

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

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

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

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

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

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

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

Out[10]= $z 7 a 11 -5 z 5 a 11 + 7 z 3 a 11 -3 z a 11 + 2 z 8 a 10 -10 z 6 a 10 + 14 z 4 a 10 -6 z 2 a 10 + a 10 + z 9 a 9 -2 z 7 a 9 -6 z 5 a 9 + 11 z 3 a 9 -3 z a 9 + 4 z 8 a 8 -18 z 6 a 8 + 23 z 4 a 8 -11 z 2 a 8 + 2 a 8 + z 9 a 7 -2 z 7 a 7 -2 z 5 a 7 + 2 z 3 a 7 - z a 7 + 2 z 8 a 6 -8 z 6 a 6 + 13 z 4 a 6 -12 z 2 a 6 + 3 a 6 + z 7 a 5 - z 5 a 5 - z a 5 + 4 z 4 a 4 -6 z 2 a 4 + 3 a 4 + 2 z 3 a 3 + z 2 a 2$

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

Same Alexander/Conway Polynomial: {}

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["K11n91"];

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

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]= {}

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:

(4, -9)

[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 =

-2 is the signature of K11n91. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

-3

-1

-5

-7

-9

-11

-13

-1

-15

-17

-1

-19

-21

-1

Integral Khovanov Homology

(db, data source)

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