K11n126

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n126 at Knotilus!
	[edit K11n126 Quick Notes]

[edit K11n126 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₄₂₅₁ X_10,4,11,3 X_5,19,6,18 X_7,13,8,12 X_2,10,3,9 X_11,17,12,16 X_13,21,14,20 X_15,9,16,8 X_17,1,18,22 X_19,15,20,14 X_21,7,22,6
Gauss code	1, -5, 2, -1, -3, 11, -4, 8, 5, -2, -6, 4, -7, 10, -8, 6, -9, 3, -10, 7, -11, 9
Dowker-Thistlethwaite code	4 10 -18 -12 2 -16 -20 -8 -22 -14 -6

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	3
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11n126/ThurstonBennequinNumber
Hyperbolic Volume	10.7639
A-Polynomial	See Data:K11n126/A-polynomial

[edit Notes for K11n126's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$3$
Rasmussen s-Invariant	-6

[edit Notes for K11n126's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{3}-6t^{2}+4t-1+4t^{-1}-6t^{-2}+3t^{-3}$
Conway polynomial	$3z^{6}+12z^{4}+7z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 27, 6 }
Jones polynomial	$q^{11}-3q^{10}+3q^{9}-5q^{8}+5q^{7}-3q^{6}+4q^{5}-2q^{4}+q^{3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-6}+2z^{6}a^{-8}+4z^{4}a^{-6}+10z^{4}a^{-8}-2z^{4}a^{-10}+3z^{2}a^{-6}+13z^{2}a^{-8}-9z^{2}a^{-10}+a^{-6}+5a^{-8}-7a^{-10}+2a^{-12}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-9}+z^{9}a^{-11}+3z^{8}a^{-8}+4z^{8}a^{-10}+z^{8}a^{-12}+2z^{7}a^{-7}-z^{7}a^{-9}-3z^{7}a^{-11}+z^{6}a^{-6}-14z^{6}a^{-8}-20z^{6}a^{-10}-5z^{6}a^{-12}-7z^{5}a^{-7}-11z^{5}a^{-9}-4z^{5}a^{-11}-4z^{4}a^{-6}+19z^{4}a^{-8}+31z^{4}a^{-10}+8z^{4}a^{-12}+3z^{3}a^{-7}+20z^{3}a^{-9}+19z^{3}a^{-11}+2z^{3}a^{-13}+3z^{2}a^{-6}-13z^{2}a^{-8}-19z^{2}a^{-10}-3z^{2}a^{-12}-10za^{-9}-13za^{-11}-3za^{-13}-a^{-6}+5a^{-8}+7a^{-10}+2a^{-12}$
The A2 invariant	Data:K11n126/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n126/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n126 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["K11n126"];

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

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

Out[4]= $3t^{3}-6t^{2}+4t-1+4t^{-1}-6t^{-2}+3t^{-3}$

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

Out[5]= $3z^{6}+12z^{4}+7z^{2}+1$

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]= $\{3,t+1\}$

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

Out[7]= { 27, 6 }

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

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

Out[8]= $q^{11}-3q^{10}+3q^{9}-5q^{8}+5q^{7}-3q^{6}+4q^{5}-2q^{4}+q^{3}$

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

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

Out[9]= $z^{6}a^{-6}+2z^{6}a^{-8}+4z^{4}a^{-6}+10z^{4}a^{-8}-2z^{4}a^{-10}+3z^{2}a^{-6}+13z^{2}a^{-8}-9z^{2}a^{-10}+a^{-6}+5a^{-8}-7a^{-10}+2a^{-12}$

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

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

Out[10]= $z^{9}a^{-9}+z^{9}a^{-11}+3z^{8}a^{-8}+4z^{8}a^{-10}+z^{8}a^{-12}+2z^{7}a^{-7}-z^{7}a^{-9}-3z^{7}a^{-11}+z^{6}a^{-6}-14z^{6}a^{-8}-20z^{6}a^{-10}-5z^{6}a^{-12}-7z^{5}a^{-7}-11z^{5}a^{-9}-4z^{5}a^{-11}-4z^{4}a^{-6}+19z^{4}a^{-8}+31z^{4}a^{-10}+8z^{4}a^{-12}+3z^{3}a^{-7}+20z^{3}a^{-9}+19z^{3}a^{-11}+2z^{3}a^{-13}+3z^{2}a^{-6}-13z^{2}a^{-8}-19z^{2}a^{-10}-3z^{2}a^{-12}-10za^{-9}-13za^{-11}-3za^{-13}-a^{-6}+5a^{-8}+7a^{-10}+2a^{-12}$

"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["K11n126"];

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]= { $3t^{3}-6t^{2}+4t-1+4t^{-1}-6t^{-2}+3t^{-3}$ , $q^{11}-3q^{10}+3q^{9}-5q^{8}+5q^{7}-3q^{6}+4q^{5}-2q^{4}+q^{3}$ }

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:

(7, 16)

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

28

128

392

{\frac {2306}{3}}

{\frac {286}{3}}

3584

{\frac {15584}{3}}

{\frac {2528}{3}}

512

{\frac {10976}{3}}

8192

{\frac {64568}{3}}

{\frac {8008}{3}}

{\frac {1096537}{30}}

{\frac {40606}{15}}

{\frac {480434}{45}}

{\frac {4295}{18}}

{\frac {37177}{30}}

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=

6 is the signature of K11n126. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

0

1

2

3

4

5

6

7

8

χ

23

1

21

2

-2

19

1

0

17

4

2

-2

15

2

0

13

2

4

2

11

2

1

9

2

7

1

2

-1

5

1

Integral Khovanov Homology

(db, data source)

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