K11a65

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a65 at Knotilus!
	[edit K11a65 Quick Notes]

[edit K11a65 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₃₉₄ X_16,5,17,6 X_10,8,11,7 X_2,9,3,10 X_22,11,1,12 X_20,13,21,14 X_18,15,19,16 X_6,17,7,18 X_14,19,15,20 X_12,21,13,22
Gauss code	1, -5, 2, -1, 3, -9, 4, -2, 5, -4, 6, -11, 7, -10, 8, -3, 9, -8, 10, -7, 11, -6
Dowker-Thistlethwaite code	4 8 16 10 2 22 20 18 6 14 12

A Braid Representative

A Morse Link Presentation

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	2
Bridge index	2
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a65/ThurstonBennequinNumber
Hyperbolic Volume	10.6865
A-Polynomial	See Data:K11a65/A-polynomial

[edit Notes for K11a65's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a65's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{2}+15t-23+15t^{-1}-3t^{-2}$
Conway polynomial	$-3z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 59, -2 }
Jones polynomial	$q-2+4q^{-1}-6q^{-2}+8q^{-3}-9q^{-4}+9q^{-5}-7q^{-6}+6q^{-7}-4q^{-8}+2q^{-9}-q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$-a^{10}+2z^{2}a^{8}+a^{8}-z^{4}a^{6}-z^{4}a^{4}+z^{2}a^{4}+a^{4}-z^{4}a^{2}-z^{2}a^{2}-a^{2}+z^{2}+1$
Kauffman polynomial (db, data sources)	$z^{7}a^{11}-5z^{5}a^{11}+7z^{3}a^{11}-3za^{11}+2z^{8}a^{10}-9z^{6}a^{10}+11z^{4}a^{10}-5z^{2}a^{10}+a^{10}+2z^{9}a^{9}-7z^{7}a^{9}+3z^{5}a^{9}+4z^{3}a^{9}-2za^{9}+z^{10}a^{8}-z^{8}a^{8}-7z^{6}a^{8}+9z^{4}a^{8}-3z^{2}a^{8}+a^{8}+4z^{9}a^{7}-15z^{7}a^{7}+17z^{5}a^{7}-8z^{3}a^{7}+za^{7}+z^{10}a^{6}-z^{8}a^{6}-2z^{6}a^{6}+z^{4}a^{6}+z^{2}a^{6}+2z^{9}a^{5}-5z^{7}a^{5}+7z^{5}a^{5}-4z^{3}a^{5}+2z^{8}a^{4}-2z^{6}a^{4}+2z^{4}a^{4}-2z^{2}a^{4}+a^{4}+2z^{7}a^{3}-2z^{3}a^{3}+za^{3}+2z^{6}a^{2}-3z^{2}a^{2}+a^{2}+2z^{5}a-3z^{3}a+za+z^{4}-2z^{2}+1$
The A2 invariant	Data:K11a65/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a65/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a65 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["K11a65"];

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

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

Out[4]= $-3t^{2}+15t-23+15t^{-1}-3t^{-2}$

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

Out[5]= $-3z^{4}+3z^{2}+1$

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]= { 59, -2 }

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

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

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

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

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

Out[9]= $-a^{10}+2z^{2}a^{8}+a^{8}-z^{4}a^{6}-z^{4}a^{4}+z^{2}a^{4}+a^{4}-z^{4}a^{2}-z^{2}a^{2}-a^{2}+z^{2}+1$

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

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

Out[10]= $z^{7}a^{11}-5z^{5}a^{11}+7z^{3}a^{11}-3za^{11}+2z^{8}a^{10}-9z^{6}a^{10}+11z^{4}a^{10}-5z^{2}a^{10}+a^{10}+2z^{9}a^{9}-7z^{7}a^{9}+3z^{5}a^{9}+4z^{3}a^{9}-2za^{9}+z^{10}a^{8}-z^{8}a^{8}-7z^{6}a^{8}+9z^{4}a^{8}-3z^{2}a^{8}+a^{8}+4z^{9}a^{7}-15z^{7}a^{7}+17z^{5}a^{7}-8z^{3}a^{7}+za^{7}+z^{10}a^{6}-z^{8}a^{6}-2z^{6}a^{6}+z^{4}a^{6}+z^{2}a^{6}+2z^{9}a^{5}-5z^{7}a^{5}+7z^{5}a^{5}-4z^{3}a^{5}+2z^{8}a^{4}-2z^{6}a^{4}+2z^{4}a^{4}-2z^{2}a^{4}+a^{4}+2z^{7}a^{3}-2z^{3}a^{3}+za^{3}+2z^{6}a^{2}-3z^{2}a^{2}+a^{2}+2z^{5}a-3z^{3}a+za+z^{4}-2z^{2}+1$

"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["K11a65"];

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]= { $-3t^{2}+15t-23+15t^{-1}-3t^{-2}$ , $q-2+4q^{-1}-6q^{-2}+8q^{-3}-9q^{-4}+9q^{-5}-7q^{-6}+6q^{-7}-4q^{-8}+2q^{-9}-q^{-10}$ }

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]= {}

Vassiliev invariants

V₂ and V₃:

(3, -9)

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

12

-72

72

366

58

-864

-2128

-320

-392

288

2592

4392

696

{\frac {127231}{10}}

-{\frac {10786}{15}}

{\frac {91862}{15}}

{\frac {1793}{6}}

{\frac {8351}{10}}

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.

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j-2r=s+1

or

j-2r=s-1

, where

s=

-2 is the signature of K11a65. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

1

-1

3

1

2

-3

4

2

-2

-5

4

2

-7

5

4

-1

-9

4

0

-11

3

5

2

-13

3

4

-1

-15

1

3

2

-17

1

3

-2

-19

1

-21

1

-1

Integral Khovanov Homology

(db, data source)

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