K11n65

K11n64

K11n66

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n65 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,5,15,6 X2837 X18,9,19,10 X16,11,17,12 X13,20,14,21 X6,15,7,16 X10,17,11,18 X19,1,20,22 X21,12,22,13
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, 5, -9, 6, 11, -7, -3, 8, -6, 9, -5, -10, 7, -11, 10
Dowker-Thistlethwaite code	4 8 14 2 18 16 -20 6 10 -22 -12

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2,3\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n65/ThurstonBennequinNumber Hyperbolic Volume 11.4167 A-Polynomial See Data:K11n65/A-polynomial

[edit Notes for K11n65's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for K11n65's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3 t^2-8 t+11-8 t^{-1} +3 t^{-2}$
Conway polynomial	$3 z^4+4 z^2+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 33, 0 }
Jones polynomial	$-2 q+4-4 q^{-1} +6 q^{-2} -5 q^{-3} +5 q^{-4} -4 q^{-5} +2 q^{-6} - q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-z^2 a^6-a^6+z^4 a^4+z^2 a^4-a^4+2 z^4 a^2+6 z^2 a^2+5 a^2-2 z^2-2$
Kauffman polynomial (db, data sources)	$a^5 z^9+a^3 z^9+2 a^6 z^8+4 a^4 z^8+2 a^2 z^8+a^7 z^7-a^5 z^7-a^3 z^7+a z^7-9 a^6 z^6-15 a^4 z^6-6 a^2 z^6-5 a^7 z^5-9 a^5 z^5-5 a^3 z^5-a z^5+11 a^6 z^4+12 a^4 z^4+2 a^2 z^4+z^4+8 a^7 z^3+13 a^5 z^3+2 a^3 z^3-3 a z^3-4 a^6 z^2-a^4 z^2+5 a^2 z^2+2 z^2-4 a^7 z-4 a^5 z+2 a^3 z+4 a z+2 z a^{-1} +a^6-a^4-5 a^2-2$
The A2 invariant	Data:K11n65/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n65/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n65 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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11n65"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3 t^2-8 t+11-8 t^{-1} +3 t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$3 z^4+4 z^2+1$

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]=

{ 33, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-2 q+4-4 q^{-1} +6 q^{-2} -5 q^{-3} +5 q^{-4} -4 q^{-5} +2 q^{-6} - q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^2 a^6-a^6+z^4 a^4+z^2 a^4-a^4+2 z^4 a^2+6 z^2 a^2+5 a^2-2 z^2-2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^5 z^9+a^3 z^9+2 a^6 z^8+4 a^4 z^8+2 a^2 z^8+a^7 z^7-a^5 z^7-a^3 z^7+a z^7-9 a^6 z^6-15 a^4 z^6-6 a^2 z^6-5 a^7 z^5-9 a^5 z^5-5 a^3 z^5-a z^5+11 a^6 z^4+12 a^4 z^4+2 a^2 z^4+z^4+8 a^7 z^3+13 a^5 z^3+2 a^3 z^3-3 a z^3-4 a^6 z^2-a^4 z^2+5 a^2 z^2+2 z^2-4 a^7 z-4 a^5 z+2 a^3 z+4 a z+2 z a^{-1} +a^6-a^4-5 a^2-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_15,}

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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11n65"];

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]=

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

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]=

{8_15,}

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:

(4, -5)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $16$ $-40$ $128$ $\frac{680}{3}$ $\frac{112}{3}$ $-640$ $-\frac{3184}{3}$ $-\frac{448}{3}$ $-200$ $\frac{2048}{3}$ $800$ $\frac{10880}{3}$ $\frac{1792}{3}$ $\frac{85862}{15}$ $-\frac{1808}{15}$ $\frac{114848}{45}$ $\frac{682}{9}$ $\frac{5222}{15}$

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 $t^rq^j$ 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 K11n65. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 χ 3                 2 -2 1               2   2 -1             3 3   0 -3           3 1     2 -5         2 3       1 -7       3 3         0 -9     1 2           1 -11   1 3             -2 -13   1               1 -15 1                 -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11n64 K11n66