K11a321

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K11a320

K11a322

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 K11a321's page at Knotilus!

Visit K11a321's page at the original Knot Atlas!

[edit K11a321 Quick Notes]

[edit K11a321 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₆₂₇₁ X_12,3,13,4 X_16,5,17,6 X_22,8,1,7 X_18,9,19,10 X_20,11,21,12 X_2,13,3,14 X_8,15,9,16 X_4,17,5,18 X_10,19,11,20 X_14,21,15,22
Gauss code	1, -7, 2, -9, 3, -1, 4, -8, 5, -10, 6, -2, 7, -11, 8, -3, 9, -5, 10, -6, 11, -4
Dowker-Thistlethwaite code	6 12 16 22 18 20 2 8 4 10 14

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:K11a321/ThurstonBennequinNumber
Hyperbolic Volume	16.2583
A-Polynomial	See Data:K11a321/A-polynomial

[edit Notes for K11a321'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 K11a321's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$-3 t 3 + 15 t 2 -27 t + 31-27 t -1 + 15 t -2 -3 t -3$
Conway polynomial	$-3 z 6 -3 z 4 + 6 z 2 + 1$
2nd Alexander ideal (db, data sources)	${11, t + 1}$
Determinant and Signature	{ 121, -4 }
Jones polynomial	$1-3 q -1 + 7 q -2 -12 q -3 + 17 q -4 -19 q -5 + 20 q -6 -17 q -7 + 13 q -8 -8 q -9 + 3 q -10 - q -11$
HOMFLY-PT polynomial (db, data sources)	$- z 2 a 10 -2 a 10 + 3 z 4 a 8 + 7 z 2 a 8 + 3 a 8 -2 z 6 a 6 -6 z 4 a 6 -5 z 2 a 6 -2 a 6 - z 6 a 4 - z 4 a 4 + 3 z 2 a 4 + 2 a 4 + z 4 a 2 + 2 z 2 a 2$
Kauffman polynomial (db, data sources)	$z 5 a 13 -2 z 3 a 13 + z a 13 + 3 z 6 a 12 -4 z 4 a 12 + z 2 a 12 + 6 z 7 a 11 -10 z 5 a 11 + 8 z 3 a 11 -4 z a 11 + 7 z 8 a 10 -10 z 6 a 10 + 7 z 4 a 10 -5 z 2 a 10 + 2 a 10 + 5 z 9 a 9 - z 7 a 9 -10 z 5 a 9 + 12 z 3 a 9 -4 z a 9 + 2 z 10 a 8 + 7 z 8 a 8 -18 z 6 a 8 + 16 z 4 a 8 -8 z 2 a 8 + 3 a 8 + 10 z 9 a 7 -21 z 7 a 7 + 20 z 5 a 7 -11 z 3 a 7 + 2 z a 7 + 2 z 10 a 6 + 5 z 8 a 6 -19 z 6 a 6 + 20 z 4 a 6 -11 z 2 a 6 + 2 a 6 + 5 z 9 a 5 -11 z 7 a 5 + 11 z 5 a 5 -8 z 3 a 5 + z a 5 + 5 z 8 a 4 -13 z 6 a 4 + 12 z 4 a 4 -7 z 2 a 4 + 2 a 4 + 3 z 7 a 3 -8 z 5 a 3 + 5 z 3 a 3 + z 6 a 2 -3 z 4 a 2 + 2 z 2 a 2$
The A2 invariant	Data:K11a321/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a321/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a321 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["K11a321"];

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

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

Out[4]= $-3 t 3 + 15 t 2 -27 t + 31-27 t -1 + 15 t -2 -3 t -3$

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

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

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]= ${11, t + 1}$

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

Out[7]= { 121, -4 }

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

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

Out[8]= $1-3 q -1 + 7 q -2 -12 q -3 + 17 q -4 -19 q -5 + 20 q -6 -17 q -7 + 13 q -8 -8 q -9 + 3 q -10 - q -11$

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

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

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

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

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

Out[10]= $z 5 a 13 -2 z 3 a 13 + z a 13 + 3 z 6 a 12 -4 z 4 a 12 + z 2 a 12 + 6 z 7 a 11 -10 z 5 a 11 + 8 z 3 a 11 -4 z a 11 + 7 z 8 a 10 -10 z 6 a 10 + 7 z 4 a 10 -5 z 2 a 10 + 2 a 10 + 5 z 9 a 9 - z 7 a 9 -10 z 5 a 9 + 12 z 3 a 9 -4 z a 9 + 2 z 10 a 8 + 7 z 8 a 8 -18 z 6 a 8 + 16 z 4 a 8 -8 z 2 a 8 + 3 a 8 + 10 z 9 a 7 -21 z 7 a 7 + 20 z 5 a 7 -11 z 3 a 7 + 2 z a 7 + 2 z 10 a 6 + 5 z 8 a 6 -19 z 6 a 6 + 20 z 4 a 6 -11 z 2 a 6 + 2 a 6 + 5 z 9 a 5 -11 z 7 a 5 + 11 z 5 a 5 -8 z 3 a 5 + z a 5 + 5 z 8 a 4 -13 z 6 a 4 + 12 z 4 a 4 -7 z 2 a 4 + 2 a 4 + 3 z 7 a 3 -8 z 5 a 3 + 5 z 3 a 3 + z 6 a 2 -3 z 4 a 2 + 2 z 2 a 2$

[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["K11a321"];

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]= { $-3 t 3 + 15 t 2 -27 t + 31-27 t -1 + 15 t -2 -3 t -3$ , $1-3 q -1 + 7 q -2 -12 q -3 + 17 q -4 -19 q -5 + 20 q -6 -17 q -7 + 13 q -8 -8 q -9 + 3 q -10 - q -11$ }

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:

(6, -16)

[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 K11a321. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

-2

-3

-5

-7

-9

-2

-11

-13

-15

-4

-17

-19

-5

-21

-23

-1

Integral Khovanov Homology

(db, data source)

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