K11n146

K11n145

K11n147

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n146 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X14,4,15,3 X10,5,11,6 X20,8,21,7 X22,9,1,10 X11,19,12,18 X2,14,3,13 X8,15,9,16 X6,18,7,17 X19,13,20,12 X16,22,17,21
Gauss code	1, -7, 2, -1, 3, -9, 4, -8, 5, -3, -6, 10, 7, -2, 8, -11, 9, 6, -10, -4, 11, -5
Dowker-Thistlethwaite code	4 14 10 20 22 -18 2 8 6 -12 16

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:K11n146/ThurstonBennequinNumber Hyperbolic Volume 14.497 A-Polynomial See Data:K11n146/A-polynomial

[edit Notes for K11n146's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for K11n146's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-5t^{2}+15t-21+15t^{-1}-5t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 63, 2 }
Jones polynomial	$q^{9}-4q^{8}+6q^{7}-9q^{6}+11q^{5}-10q^{4}+10q^{3}-7q^{2}+4q-1$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-4}-z^{4}a^{-2}+4z^{4}a^{-4}-2z^{4}a^{-6}-z^{2}a^{-2}+8z^{2}a^{-4}-4z^{2}a^{-6}+z^{2}a^{-8}-a^{-2}+5a^{-4}-3a^{-6}$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-5}+2z^{9}a^{-7}+4z^{8}a^{-4}+9z^{8}a^{-6}+5z^{8}a^{-8}+2z^{7}a^{-3}+2z^{7}a^{-7}+4z^{7}a^{-9}-13z^{6}a^{-4}-29z^{6}a^{-6}-15z^{6}a^{-8}+z^{6}a^{-10}-3z^{5}a^{-3}-11z^{5}a^{-5}-20z^{5}a^{-7}-12z^{5}a^{-9}+4z^{4}a^{-2}+24z^{4}a^{-4}+33z^{4}a^{-6}+11z^{4}a^{-8}-2z^{4}a^{-10}+z^{3}a^{-1}+8z^{3}a^{-3}+18z^{3}a^{-5}+18z^{3}a^{-7}+7z^{3}a^{-9}-3z^{2}a^{-2}-17z^{2}a^{-4}-18z^{2}a^{-6}-4z^{2}a^{-8}-za^{-1}-4za^{-3}-8za^{-5}-4za^{-7}+za^{-9}+a^{-2}+5a^{-4}+3a^{-6}$
The A2 invariant	$-1+2q^{-2}-2q^{-4}+3q^{-8}+4q^{-12}+q^{-16}-3q^{-20}+q^{-22}-2q^{-24}-q^{-26}+q^{-28}$
The G2 invariant	Data:K11n146/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n146 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["K11n146"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-5t^{2}+15t-21+15t^{-1}-5t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 63, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-5}+2z^{9}a^{-7}+4z^{8}a^{-4}+9z^{8}a^{-6}+5z^{8}a^{-8}+2z^{7}a^{-3}+2z^{7}a^{-7}+4z^{7}a^{-9}-13z^{6}a^{-4}-29z^{6}a^{-6}-15z^{6}a^{-8}+z^{6}a^{-10}-3z^{5}a^{-3}-11z^{5}a^{-5}-20z^{5}a^{-7}-12z^{5}a^{-9}+4z^{4}a^{-2}+24z^{4}a^{-4}+33z^{4}a^{-6}+11z^{4}a^{-8}-2z^{4}a^{-10}+z^{3}a^{-1}+8z^{3}a^{-3}+18z^{3}a^{-5}+18z^{3}a^{-7}+7z^{3}a^{-9}-3z^{2}a^{-2}-17z^{2}a^{-4}-18z^{2}a^{-6}-4z^{2}a^{-8}-za^{-1}-4za^{-3}-8za^{-5}-4za^{-7}+za^{-9}+a^{-2}+5a^{-4}+3a^{-6}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n167,}

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["K11n146"];

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]=

{ $t^{3}-5t^{2}+15t-21+15t^{-1}-5t^{-2}+t^{-3}$, $q^{9}-4q^{8}+6q^{7}-9q^{6}+11q^{5}-10q^{4}+10q^{3}-7q^{2}+4q-1$ }

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]=

{K11n167,}

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, 7)

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 $16$ $56$ $128$ ${\frac {872}{3}}$ ${\frac {160}{3}}$ $896$ ${\frac {4784}{3}}$ ${\frac {896}{3}}$ $248$ ${\frac {2048}{3}}$ $1568$ ${\frac {13952}{3}}$ ${\frac {2560}{3}}$ ${\frac {132422}{15}}$ $-{\frac {688}{15}}$ ${\frac {179408}{45}}$ ${\frac {346}{9}}$ ${\frac {8822}{15}}$

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=$2 is the signature of K11n146. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -1 0 1 2 3 4 5 6 7 8 χ 19                   1 1 17                 3   -3 15               3 1   2 13             6 3     -3 11           5 3       2 9         5 6         1 7       5 5           0 5     2 5             3 3   2 5               -3 1   3                 3 -1 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

K11n145 K11n147