K11n153

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n153 at Knotilus!
	[edit K11n153 Quick Notes]

[edit K11n153 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₆₂₇₁ X₈₃₉₄ X_12,6,13,5 X_20,8,21,7 X_16,9,17,10 X_18,11,19,12 X_13,1,14,22 X_4,16,5,15 X_10,17,11,18 X_2,19,3,20 X_21,15,22,14
Gauss code	1, -10, 2, -8, 3, -1, 4, -2, 5, -9, 6, -3, -7, 11, 8, -5, 9, -6, 10, -4, -11, 7
Dowker-Thistlethwaite code	6 8 12 20 16 18 -22 4 10 2 -14

A Braid Representative

A Morse Link Presentation

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11n153/ThurstonBennequinNumber
Hyperbolic Volume	13.0745
A-Polynomial	See Data:K11n153/A-polynomial

[edit Notes for K11n153's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$4$
Rasmussen s-Invariant	0

[edit Notes for K11n153's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-4t^{3}+7t^{2}-10t+13-10t^{-1}+7t^{-2}-4t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+4z^{6}+3z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 57, 0 }
Jones polynomial	$2q^{4}-4q^{3}+6q^{2}-9q+10-9q^{-1}+8q^{-2}-5q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-5z^{4}a^{-2}+12z^{4}-4a^{2}z^{2}-8z^{2}a^{-2}+z^{2}a^{-4}+9z^{2}-4a^{-2}+2a^{-4}+3$
Kauffman polynomial (db, data sources)	$2az^{9}+2z^{9}a^{-1}+4a^{2}z^{8}+3z^{8}a^{-2}+7z^{8}+4a^{3}z^{7}-3az^{7}-6z^{7}a^{-1}+z^{7}a^{-3}+3a^{4}z^{6}-11a^{2}z^{6}-12z^{6}a^{-2}-26z^{6}+a^{5}z^{5}-9a^{3}z^{5}+11z^{5}a^{-1}+z^{5}a^{-3}-7a^{4}z^{4}+12a^{2}z^{4}+25z^{4}a^{-2}+3z^{4}a^{-4}+41z^{4}-2a^{5}z^{3}+4a^{3}z^{3}+5az^{3}-7z^{3}a^{-1}-6z^{3}a^{-3}+2a^{4}z^{2}-5a^{2}z^{2}-21z^{2}a^{-2}-7z^{2}a^{-4}-21z^{2}-a^{3}z-az+3za^{-1}+3za^{-3}+4a^{-2}+2a^{-4}+3$
The A2 invariant	$-q^{14}+q^{12}-q^{10}+2q^{8}+q^{6}+3q^{2}-2+2q^{-2}-3q^{-4}-q^{-6}-q^{-10}+2q^{-12}+q^{-16}$
The G2 invariant	Data:K11n153/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n153 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["K11n153"];

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

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

Out[4]= $t^{4}-4t^{3}+7t^{2}-10t+13-10t^{-1}+7t^{-2}-4t^{-3}+t^{-4}$

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

Out[5]= $z^{8}+4z^{6}+3z^{4}-2z^{2}+1$

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, 0 }

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

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

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

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

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

Out[9]= $z^{8}-a^{2}z^{6}-z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-5z^{4}a^{-2}+12z^{4}-4a^{2}z^{2}-8z^{2}a^{-2}+z^{2}a^{-4}+9z^{2}-4a^{-2}+2a^{-4}+3$

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

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

Out[10]= $2az^{9}+2z^{9}a^{-1}+4a^{2}z^{8}+3z^{8}a^{-2}+7z^{8}+4a^{3}z^{7}-3az^{7}-6z^{7}a^{-1}+z^{7}a^{-3}+3a^{4}z^{6}-11a^{2}z^{6}-12z^{6}a^{-2}-26z^{6}+a^{5}z^{5}-9a^{3}z^{5}+11z^{5}a^{-1}+z^{5}a^{-3}-7a^{4}z^{4}+12a^{2}z^{4}+25z^{4}a^{-2}+3z^{4}a^{-4}+41z^{4}-2a^{5}z^{3}+4a^{3}z^{3}+5az^{3}-7z^{3}a^{-1}-6z^{3}a^{-3}+2a^{4}z^{2}-5a^{2}z^{2}-21z^{2}a^{-2}-7z^{2}a^{-4}-21z^{2}-a^{3}z-az+3za^{-1}+3za^{-3}+4a^{-2}+2a^{-4}+3$

"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["K11n153"];

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

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:

(-2, -2)

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

-16

32

{\frac {20}{3}}

-{\frac {20}{3}}

128

{\frac {608}{3}}

{\frac {224}{3}}

16

-{\frac {256}{3}}

128

-{\frac {160}{3}}

{\frac {160}{3}}

{\frac {4409}{15}}

-{\frac {4196}{15}}

{\frac {26276}{45}}

-{\frac {953}{9}}

{\frac {2009}{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.

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=

0 is the signature of K11n153. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

9

2

7

2

-2

5

4

2

3

5

2

-3

1

5

4

1

-1

5

6

1

-3

3

4

-1

-5

2

5

3

-7

1

3

-2

-9

2

-11

1

-1

Integral Khovanov Homology

(db, data source)

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