K11n166

K11n165

K11n167

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n166 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,4,11,3 X5,15,6,14 X18,7,19,8 X2,10,3,9 X20,11,21,12 X22,14,1,13 X15,5,16,4 X12,17,13,18 X8,19,9,20 X16,22,17,21
Gauss code	1, -5, 2, 8, -3, -1, 4, -10, 5, -2, 6, -9, 7, 3, -8, -11, 9, -4, 10, -6, 11, -7
Dowker-Thistlethwaite code	6 10 -14 18 2 20 22 -4 12 8 16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n166/ThurstonBennequinNumber Hyperbolic Volume 14.1954 A-Polynomial See Data:K11n166/A-polynomial

[edit Notes for K11n166's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n166's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+4t^{3}-8t^{2}+11t-11+11t^{-1}-8t^{-2}+4t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-4z^{6}-4z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 59, 2 }
Jones polynomial	$-q^{6}+4q^{5}-7q^{4}+9q^{3}-10q^{2}+10q-8+6q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-13z^{4}a^{-2}+5z^{4}a^{-4}+4z^{4}-13z^{2}a^{-2}+8z^{2}a^{-4}-z^{2}a^{-6}+5z^{2}-5a^{-2}+4a^{-4}-a^{-6}+3$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-1}+2z^{9}a^{-3}+8z^{8}a^{-2}+4z^{8}a^{-4}+4z^{8}+3az^{7}-z^{7}a^{-1}-2z^{7}a^{-3}+2z^{7}a^{-5}+a^{2}z^{6}-27z^{6}a^{-2}-14z^{6}a^{-4}-12z^{6}-9az^{5}-9z^{5}a^{-1}-3z^{5}a^{-3}-3z^{5}a^{-5}-3a^{2}z^{4}+35z^{4}a^{-2}+26z^{4}a^{-4}+4z^{4}a^{-6}+10z^{4}+6az^{3}+7z^{3}a^{-1}+7z^{3}a^{-3}+7z^{3}a^{-5}+z^{3}a^{-7}+2a^{2}z^{2}-23z^{2}a^{-2}-17z^{2}a^{-4}-3z^{2}a^{-6}-7z^{2}-az-2za^{-1}-3za^{-3}-3za^{-5}-za^{-7}+5a^{-2}+4a^{-4}+a^{-6}+3$
The A2 invariant	$q^{8}-q^{6}+2q^{4}+1+q^{-2}-3q^{-4}+2q^{-6}-3q^{-8}+2q^{-10}+2q^{-16}-q^{-18}+q^{-20}-q^{-22}$
The G2 invariant	Data:K11n166/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n166 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["K11n166"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+4t^{3}-8t^{2}+11t-11+11t^{-1}-8t^{-2}+4t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-4z^{6}-4z^{4}-z^{2}+1$

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]=

{ 59, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-13z^{4}a^{-2}+5z^{4}a^{-4}+4z^{4}-13z^{2}a^{-2}+8z^{2}a^{-4}-z^{2}a^{-6}+5z^{2}-5a^{-2}+4a^{-4}-a^{-6}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-1}+2z^{9}a^{-3}+8z^{8}a^{-2}+4z^{8}a^{-4}+4z^{8}+3az^{7}-z^{7}a^{-1}-2z^{7}a^{-3}+2z^{7}a^{-5}+a^{2}z^{6}-27z^{6}a^{-2}-14z^{6}a^{-4}-12z^{6}-9az^{5}-9z^{5}a^{-1}-3z^{5}a^{-3}-3z^{5}a^{-5}-3a^{2}z^{4}+35z^{4}a^{-2}+26z^{4}a^{-4}+4z^{4}a^{-6}+10z^{4}+6az^{3}+7z^{3}a^{-1}+7z^{3}a^{-3}+7z^{3}a^{-5}+z^{3}a^{-7}+2a^{2}z^{2}-23z^{2}a^{-2}-17z^{2}a^{-4}-3z^{2}a^{-6}-7z^{2}-az-2za^{-1}-3za^{-3}-3za^{-5}-za^{-7}+5a^{-2}+4a^{-4}+a^{-6}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n124,}

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["K11n166"];

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]=

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

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]=

{K11n124,}

Vassiliev invariants

V2 and V3:

(-1, 0)

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 $-4$ $0$ $8$ ${\frac {178}{3}}$ ${\frac {110}{3}}$ $0$ $96$ $32$ $32$ $-{\frac {32}{3}}$ $0$ $-{\frac {712}{3}}$ $-{\frac {440}{3}}$ $-{\frac {5311}{30}}$ ${\frac {5342}{15}}$ $-{\frac {27422}{45}}$ ${\frac {2047}{18}}$ $-{\frac {5311}{30}}$

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=$2 is the signature of K11n166. 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 χ 13                   1 -1 11                 3   3 9               4 1   -3 7             5 3     2 5           5 4       -1 3         5 5         0 1       4 6           2 -1     2 4             -2 -3   1 4               3 -5   2                 -2 -7 1                   1

Integral Khovanov Homology (db, data source)

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

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.

K11n165 K11n167