K11a165

K11a164

K11a166

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a165 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X18,6,19,5 X16,8,17,7 X2,10,3,9 X20,11,21,12 X22,13,1,14 X8,16,9,15 X6,18,7,17 X14,19,15,20 X12,21,13,22
Gauss code	1, -5, 2, -1, 3, -9, 4, -8, 5, -2, 6, -11, 7, -10, 8, -4, 9, -3, 10, -6, 11, -7
Dowker-Thistlethwaite code	4 10 18 16 2 20 22 8 6 14 12

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a165/ThurstonBennequinNumber Hyperbolic Volume 12.5729 A-Polynomial See Data:K11a165/A-polynomial

[edit Notes for K11a165'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 K11a165's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+9t^{2}-18t+23-18t^{-1}+9t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-3z^{4}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{2}+t+1\right\}$
Determinant and Signature	{ 81, 0 }
Jones polynomial	$-q^{7}+2q^{6}-4q^{5}+8q^{4}-10q^{3}+12q^{2}-13q+12-9q^{-1}+6q^{-2}-3q^{-3}+q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-3z^{4}a^{-2}+2z^{4}a^{-4}-3z^{4}+2a^{2}z^{2}-4z^{2}a^{-2}+6z^{2}a^{-4}-z^{2}a^{-6}-3z^{2}+a^{2}-3a^{-2}+5a^{-4}-2a^{-6}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+5z^{9}a^{-3}+2z^{9}a^{-5}+4z^{8}a^{-2}+z^{8}a^{-4}+2z^{8}a^{-6}+5z^{8}+6az^{7}-z^{7}a^{-1}-14z^{7}a^{-3}-6z^{7}a^{-5}+z^{7}a^{-7}+5a^{2}z^{6}-15z^{6}a^{-2}-13z^{6}a^{-4}-9z^{6}a^{-6}-6z^{6}+3a^{3}z^{5}-9az^{5}-8z^{5}a^{-1}+11z^{5}a^{-3}+2z^{5}a^{-5}-5z^{5}a^{-7}+a^{4}z^{4}-6a^{2}z^{4}+15z^{4}a^{-2}+21z^{4}a^{-4}+13z^{4}a^{-6}-3a^{3}z^{3}+7az^{3}+9z^{3}a^{-1}-4z^{3}a^{-3}+4z^{3}a^{-5}+7z^{3}a^{-7}-a^{4}z^{2}+3a^{2}z^{2}-10z^{2}a^{-2}-18z^{2}a^{-4}-8z^{2}a^{-6}+4z^{2}-2az-2za^{-1}-2za^{-5}-2za^{-7}-a^{2}+3a^{-2}+5a^{-4}+2a^{-6}$
The A2 invariant	Data:K11a165/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a165/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a165 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["K11a165"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+9t^{2}-18t+23-18t^{-1}+9t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-2z^{6}-3z^{4}+1$

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]=

$\left\{2,t^{2}+t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 81, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{7}+2q^{6}-4q^{5}+8q^{4}-10q^{3}+12q^{2}-13q+12-9q^{-1}+6q^{-2}-3q^{-3}+q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-3z^{4}a^{-2}+2z^{4}a^{-4}-3z^{4}+2a^{2}z^{2}-4z^{2}a^{-2}+6z^{2}a^{-4}-z^{2}a^{-6}-3z^{2}+a^{2}-3a^{-2}+5a^{-4}-2a^{-6}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+5z^{9}a^{-3}+2z^{9}a^{-5}+4z^{8}a^{-2}+z^{8}a^{-4}+2z^{8}a^{-6}+5z^{8}+6az^{7}-z^{7}a^{-1}-14z^{7}a^{-3}-6z^{7}a^{-5}+z^{7}a^{-7}+5a^{2}z^{6}-15z^{6}a^{-2}-13z^{6}a^{-4}-9z^{6}a^{-6}-6z^{6}+3a^{3}z^{5}-9az^{5}-8z^{5}a^{-1}+11z^{5}a^{-3}+2z^{5}a^{-5}-5z^{5}a^{-7}+a^{4}z^{4}-6a^{2}z^{4}+15z^{4}a^{-2}+21z^{4}a^{-4}+13z^{4}a^{-6}-3a^{3}z^{3}+7az^{3}+9z^{3}a^{-1}-4z^{3}a^{-3}+4z^{3}a^{-5}+7z^{3}a^{-7}-a^{4}z^{2}+3a^{2}z^{2}-10z^{2}a^{-2}-18z^{2}a^{-4}-8z^{2}a^{-6}+4z^{2}-2az-2za^{-1}-2za^{-5}-2za^{-7}-a^{2}+3a^{-2}+5a^{-4}+2a^{-6}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_87, 10_98, K11a58, K11n72,}

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["K11a165"];

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]=

{ $-2t^{3}+9t^{2}-18t+23-18t^{-1}+9t^{-2}-2t^{-3}$, $-q^{7}+2q^{6}-4q^{5}+8q^{4}-10q^{3}+12q^{2}-13q+12-9q^{-1}+6q^{-2}-3q^{-3}+q^{-4}$ }

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]=

{10_87, 10_98, K11a58, K11n72,}

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:

(0, 3)

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 $0$ $24$ $0$ $80$ $24$ $0$ $144$ $-64$ $120$ $0$ $288$ $0$ $0$ $728$ $-{\frac {856}{3}}$ $488$ $136$ $56$

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 K11a165. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a164 K11a166