K11a169

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a169 at Knotilus!
	[edit K11a169 Quick Notes]

[edit K11a169 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,3,11,4 X_18,5,19,6 X_16,7,17,8 X_14,10,15,9 X_2,11,3,12 X_20,14,21,13 X_22,15,1,16 X_6,17,7,18 X_12,20,13,19 X_8,21,9,22
Gauss code	1, -6, 2, -1, 3, -9, 4, -11, 5, -2, 6, -10, 7, -5, 8, -4, 9, -3, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 10 18 16 14 2 20 22 6 12 8

A Braid Representative

A Morse Link Presentation

Three dimensional invariants

Symmetry type	Chiral
Unknotting number	$\{1,2\}$
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a169/ThurstonBennequinNumber
Hyperbolic Volume	16.1443
A-Polynomial	See Data:K11a169/A-polynomial

[edit Notes for K11a169's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a169's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+12t^{2}-28t+37-28t^{-1}+12t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 121, 0 }
Jones polynomial	$q^{4}-4q^{3}+8q^{2}-13q+18-19q^{-1}+19q^{-2}-16q^{-3}+12q^{-4}-7q^{-5}+3q^{-6}-q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{6}-a^{6}+2z^{4}a^{4}+3z^{2}a^{4}+a^{4}-z^{6}a^{2}-z^{4}a^{2}+z^{2}a^{2}+a^{2}-z^{6}-2z^{4}-2z^{2}+z^{4}a^{-2}+z^{2}a^{-2}$
Kauffman polynomial (db, data sources)	$2a^{4}z^{10}+2a^{2}z^{10}+4a^{5}z^{9}+11a^{3}z^{9}+7az^{9}+3a^{6}z^{8}+4a^{4}z^{8}+12a^{2}z^{8}+11z^{8}+a^{7}z^{7}-11a^{5}z^{7}-27a^{3}z^{7}-4az^{7}+11z^{7}a^{-1}-11a^{6}z^{6}-28a^{4}z^{6}-40a^{2}z^{6}+8z^{6}a^{-2}-15z^{6}-4a^{7}z^{5}+5a^{5}z^{5}+13a^{3}z^{5}-13az^{5}-13z^{5}a^{-1}+4z^{5}a^{-3}+13a^{6}z^{4}+31a^{4}z^{4}+29a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}+3z^{4}+5a^{7}z^{3}+4a^{5}z^{3}+a^{3}z^{3}+8az^{3}+4z^{3}a^{-1}-2z^{3}a^{-3}-6a^{6}z^{2}-11a^{4}z^{2}-5a^{2}z^{2}+2z^{2}a^{-2}+2z^{2}-2a^{7}z-2a^{5}z+a^{6}+a^{4}-a^{2}$
The A2 invariant	Data:K11a169/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a169/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a169 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["K11a169"];

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

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

Out[4]= $-2t^{3}+12t^{2}-28t+37-28t^{-1}+12t^{-2}-2t^{-3}$

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

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

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]= { 121, 0 }

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

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

Out[8]= $q^{4}-4q^{3}+8q^{2}-13q+18-19q^{-1}+19q^{-2}-16q^{-3}+12q^{-4}-7q^{-5}+3q^{-6}-q^{-7}$

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

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

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

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

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

Out[10]= $2a^{4}z^{10}+2a^{2}z^{10}+4a^{5}z^{9}+11a^{3}z^{9}+7az^{9}+3a^{6}z^{8}+4a^{4}z^{8}+12a^{2}z^{8}+11z^{8}+a^{7}z^{7}-11a^{5}z^{7}-27a^{3}z^{7}-4az^{7}+11z^{7}a^{-1}-11a^{6}z^{6}-28a^{4}z^{6}-40a^{2}z^{6}+8z^{6}a^{-2}-15z^{6}-4a^{7}z^{5}+5a^{5}z^{5}+13a^{3}z^{5}-13az^{5}-13z^{5}a^{-1}+4z^{5}a^{-3}+13a^{6}z^{4}+31a^{4}z^{4}+29a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}+3z^{4}+5a^{7}z^{3}+4a^{5}z^{3}+a^{3}z^{3}+8az^{3}+4z^{3}a^{-1}-2z^{3}a^{-3}-6a^{6}z^{2}-11a^{4}z^{2}-5a^{2}z^{2}+2z^{2}a^{-2}+2z^{2}-2a^{7}z-2a^{5}z+a^{6}+a^{4}-a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a36,}

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["K11a169"];

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]= { $-2t^{3}+12t^{2}-28t+37-28t^{-1}+12t^{-2}-2t^{-3}$ , $q^{4}-4q^{3}+8q^{2}-13q+18-19q^{-1}+19q^{-2}-16q^{-3}+12q^{-4}-7q^{-5}+3q^{-6}-q^{-7}$ }

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]= {K11a36,}

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, -3)

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

-24

32

{\frac {268}{3}}

{\frac {44}{3}}

-192

-368

0

-120

{\frac {256}{3}}

288

{\frac {2144}{3}}

{\frac {352}{3}}

{\frac {22951}{15}}

-{\frac {1148}{5}}

{\frac {35524}{45}}

{\frac {1001}{9}}

{\frac {1831}{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 K11a169. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

9

1

7

3

-3

5

1

4

3

8

3

-5

1

10

5

-1

10

9

-1

-3

9

0

-5

7

10

3

-7

5

9

-4

-9

2

7

5

-11

1

5

-4

-13

2

-15

1

-1

Integral Khovanov Homology

(db, data source)

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