K11n45

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n45 at Knotilus!
	[edit K11n45 Quick Notes]

[edit K11n45 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_12,5,13,6 X₂₈₃₇ X_9,21,10,20 X_11,18,12,19 X_6,13,7,14 X_15,10,16,11 X_17,1,18,22 X_19,14,20,15 X_21,17,22,16
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, -5, 8, -6, -3, 7, 10, -8, 11, -9, 6, -10, 5, -11, 9
Dowker-Thistlethwaite code	4 8 12 2 -20 -18 6 -10 -22 -14 -16

A Braid Representative

A Morse Link Presentation

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11n45/ThurstonBennequinNumber
Hyperbolic Volume	12.7511
A-Polynomial	See Data:K11n45/A-polynomial

[edit Notes for K11n45's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$[0,3]$
Rasmussen s-Invariant	0

[edit Notes for K11n45's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{2}-6t+9-6t^{-1}+2t^{-2}$
Conway polynomial	$2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 25, 0 }
Jones polynomial	$q^{6}-3q^{5}+4q^{4}-5q^{3}+5q^{2}-4q+4-q^{-1}+q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}+z^{6}-a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}+7z^{4}-4a^{2}z^{2}-9z^{2}a^{-2}+2z^{2}a^{-4}+13z^{2}-3a^{2}-5a^{-2}+a^{-4}+8$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+a^{2}z^{8}+4z^{8}a^{-2}+3z^{8}a^{-4}+2z^{8}+a^{3}z^{7}+az^{7}-3z^{7}a^{-1}+3z^{7}a^{-5}-7a^{2}z^{6}-20z^{6}a^{-2}-11z^{6}a^{-4}+z^{6}a^{-6}-15z^{6}-6a^{3}z^{5}-10az^{5}-7z^{5}a^{-1}-14z^{5}a^{-3}-11z^{5}a^{-5}+13a^{2}z^{4}+30z^{4}a^{-2}+8z^{4}a^{-4}-3z^{4}a^{-6}+32z^{4}+9a^{3}z^{3}+20az^{3}+23z^{3}a^{-1}+20z^{3}a^{-3}+8z^{3}a^{-5}-10a^{2}z^{2}-18z^{2}a^{-2}-3z^{2}a^{-4}+z^{2}a^{-6}-24z^{2}-4a^{3}z-10az-12za^{-1}-8za^{-3}-2za^{-5}+3a^{2}+5a^{-2}+a^{-4}+8$
The A2 invariant	$-q^{12}-q^{10}-q^{8}+q^{6}+3q^{2}+3+q^{-2}+2q^{-4}-2q^{-6}-2q^{-10}-q^{-12}+q^{-14}-q^{-16}+q^{-18}$
The G2 invariant	Data:K11n45/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n45 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["K11n45"];

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

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

Out[4]= $2t^{2}-6t+9-6t^{-1}+2t^{-2}$

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

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

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]= { 25, 0 }

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

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

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

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

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

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

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

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

Out[10]= $z^{9}a^{-1}+z^{9}a^{-3}+a^{2}z^{8}+4z^{8}a^{-2}+3z^{8}a^{-4}+2z^{8}+a^{3}z^{7}+az^{7}-3z^{7}a^{-1}+3z^{7}a^{-5}-7a^{2}z^{6}-20z^{6}a^{-2}-11z^{6}a^{-4}+z^{6}a^{-6}-15z^{6}-6a^{3}z^{5}-10az^{5}-7z^{5}a^{-1}-14z^{5}a^{-3}-11z^{5}a^{-5}+13a^{2}z^{4}+30z^{4}a^{-2}+8z^{4}a^{-4}-3z^{4}a^{-6}+32z^{4}+9a^{3}z^{3}+20az^{3}+23z^{3}a^{-1}+20z^{3}a^{-3}+8z^{3}a^{-5}-10a^{2}z^{2}-18z^{2}a^{-2}-3z^{2}a^{-4}+z^{2}a^{-6}-24z^{2}-4a^{3}z-10az-12za^{-1}-8za^{-3}-2za^{-5}+3a^{2}+5a^{-2}+a^{-4}+8$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_8, 10_129, K11n39, K11n50, K11n132,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11n39,}

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["K11n45"];

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

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]= {8_8, 10_129, K11n39, K11n50, K11n132,}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {K11n39,}

Vassiliev invariants

V₂ and V₃:

(2, -1)

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

-8

32

{\frac {76}{3}}

-{\frac {4}{3}}

-64

-{\frac {272}{3}}

{\frac {64}{3}}

-40

{\frac {256}{3}}

32

{\frac {608}{3}}

-{\frac {32}{3}}

{\frac {3751}{15}}

{\frac {476}{15}}

{\frac {1204}{45}}

{\frac {329}{9}}

-{\frac {89}{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 K11n45. 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

5

6

χ

13

1

11

2

-2

9

2

1

7

3

2

-1

5

1

3

2

0

3

2

3

1

4

3

0

-1

1

3

-3

1

2

-1

-5

1

-7

0

-9

1

-1

Integral Khovanov Homology

(db, data source)

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