K11a312

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K11a311

K11a313

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 K11a312's page at Knotilus!

Visit K11a312's page at the original Knot Atlas!

[edit K11a312 Quick Notes]

[edit K11a312 Further Notes and Views]

[edit] Knot presentations

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

A Braid Representative

A Morse Link Presentation

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	${1,2}$
3-genus	3
Bridge index	Missing
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a312/ThurstonBennequinNumber
Hyperbolic Volume	16.0409
A-Polynomial	See Data:K11a312/A-polynomial

[edit Notes for K11a312's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$3$
Rasmussen s-Invariant	2

[edit Notes for K11a312's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$3 t 3 -13 t 2 + 27 t -33 + 27 t -1 -13 t -2 + 3 t -3$
Conway polynomial	$3 z 6 + 5 z 4 + 2 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 119, -2 }
Jones polynomial	$- q 2 + 3 q -6 + 12 q -1 -16 q -2 + 19 q -3 -19 q -4 + 17 q -5 -13 q -6 + 8 q -7 -4 q -8 + q -9$
HOMFLY-PT polynomial (db, data sources)	$z 2 a 8 + a 8 -3 z 4 a 6 -7 z 2 a 6 -4 a 6 + 2 z 6 a 4 + 7 z 4 a 4 + 9 z 2 a 4 + 4 a 4 + z 6 a 2 + 2 z 4 a 2 + z 2 a 2 - z 4 -2 z 2$
Kauffman polynomial (db, data sources)	$z 6 a 10 -2 z 4 a 10 + z 2 a 10 + 4 z 7 a 9 -10 z 5 a 9 + 6 z 3 a 9 + 6 z 8 a 8 -13 z 6 a 8 + 5 z 4 a 8 - z 2 a 8 + a 8 + 5 z 9 a 7 -6 z 7 a 7 -5 z 5 a 7 + 2 z 3 a 7 + 2 z 10 a 6 + 7 z 8 a 6 -27 z 6 a 6 + 29 z 4 a 6 -18 z 2 a 6 + 4 a 6 + 11 z 9 a 5 -28 z 7 a 5 + 31 z 5 a 5 -16 z 3 a 5 + 2 z a 5 + 2 z 10 a 4 + 8 z 8 a 4 -33 z 6 a 4 + 48 z 4 a 4 -25 z 2 a 4 + 4 a 4 + 6 z 9 a 3 -13 z 7 a 3 + 16 z 5 a 3 -6 z 3 a 3 + 2 z a 3 + 7 z 8 a 2 -17 z 6 a 2 + 20 z 4 a 2 -7 z 2 a 2 + 5 z 7 a -9 z 5 a + 4 z 3 a + 3 z 6 -6 z 4 + 2 z 2 + z 5 a -1 -2 z 3 a -1$
The A2 invariant	Data:K11a312/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a312/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a312 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["K11a312"];

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

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

Out[4]= $3 t 3 -13 t 2 + 27 t -33 + 27 t -1 -13 t -2 + 3 t -3$

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

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

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]= { 119, -2 }

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

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

Out[8]= $- q 2 + 3 q -6 + 12 q -1 -16 q -2 + 19 q -3 -19 q -4 + 17 q -5 -13 q -6 + 8 q -7 -4 q -8 + q -9$

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

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

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

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

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

Out[10]= $z 6 a 10 -2 z 4 a 10 + z 2 a 10 + 4 z 7 a 9 -10 z 5 a 9 + 6 z 3 a 9 + 6 z 8 a 8 -13 z 6 a 8 + 5 z 4 a 8 - z 2 a 8 + a 8 + 5 z 9 a 7 -6 z 7 a 7 -5 z 5 a 7 + 2 z 3 a 7 + 2 z 10 a 6 + 7 z 8 a 6 -27 z 6 a 6 + 29 z 4 a 6 -18 z 2 a 6 + 4 a 6 + 11 z 9 a 5 -28 z 7 a 5 + 31 z 5 a 5 -16 z 3 a 5 + 2 z a 5 + 2 z 10 a 4 + 8 z 8 a 4 -33 z 6 a 4 + 48 z 4 a 4 -25 z 2 a 4 + 4 a 4 + 6 z 9 a 3 -13 z 7 a 3 + 16 z 5 a 3 -6 z 3 a 3 + 2 z a 3 + 7 z 8 a 2 -17 z 6 a 2 + 20 z 4 a 2 -7 z 2 a 2 + 5 z 7 a -9 z 5 a + 4 z 3 a + 3 z 6 -6 z 4 + 2 z 2 + z 5 a -1 -2 z 3 a -1$

[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["K11a312"];

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]= { $3 t 3 -13 t 2 + 27 t -33 + 27 t -1 -13 t -2 + 3 t -3$ , $- q 2 + 3 q -6 + 12 q -1 -16 q -2 + 19 q -3 -19 q -4 + 17 q -5 -13 q -6 + 8 q -7 -4 q -8 + q -9$ }

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:

(2, -2)

[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 K11a312. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-8

-7

-6

-5

-4

-3

-2

-1

-3

-1

-3

-4

-5

-7

-9

-2

-11

-13

-5

-15

-17

-3

-19

Integral Khovanov Homology

(db, data source)

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