K11a218

K11a217

K11a219

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a218 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a218/ThurstonBennequinNumber Hyperbolic Volume 16.744 A-Polynomial See Data:K11a218/A-polynomial

[edit Notes for K11a218'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 K11a218's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-10t^{2}+32t-45+32t^{-1}-10t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-4z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 131, -2 }
Jones polynomial	$q^{3}-4q^{2}+9q-14+19q^{-1}-21q^{-2}+21q^{-3}-18q^{-4}+13q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{8}+3z^{2}a^{6}+2a^{6}-3z^{4}a^{4}-3z^{2}a^{4}-a^{4}+z^{6}a^{2}+z^{4}a^{2}+z^{2}a^{2}-2z^{4}-z^{2}+1+z^{2}a^{-2}$
Kauffman polynomial (db, data sources)	$2a^{4}z^{10}+2a^{2}z^{10}+5a^{5}z^{9}+11a^{3}z^{9}+6az^{9}+6a^{6}z^{8}+9a^{4}z^{8}+10a^{2}z^{8}+7z^{8}+5a^{7}z^{7}-a^{5}z^{7}-18a^{3}z^{7}-8az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-5a^{6}z^{6}-20a^{4}z^{6}-29a^{2}z^{6}+z^{6}a^{-2}-16z^{6}+a^{9}z^{5}-6a^{7}z^{5}-6a^{5}z^{5}+8a^{3}z^{5}-2az^{5}-9z^{5}a^{-1}-5a^{8}z^{4}-2a^{6}z^{4}+11a^{4}z^{4}+20a^{2}z^{4}-2z^{4}a^{-2}+10z^{4}-2a^{9}z^{3}+2a^{7}z^{3}+6a^{5}z^{3}-3a^{3}z^{3}+5z^{3}a^{-1}+3a^{8}z^{2}+6a^{6}z^{2}+a^{4}z^{2}-6a^{2}z^{2}+z^{2}a^{-2}-3z^{2}+a^{9}z-a^{5}z+a^{3}z+az-a^{8}-2a^{6}-a^{4}+1$
The A2 invariant	Data:K11a218/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a218/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a218 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["K11a218"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-10t^{2}+32t-45+32t^{-1}-10t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 131, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-4q^{2}+9q-14+19q^{-1}-21q^{-2}+21q^{-3}-18q^{-4}+13q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{8}+3z^{2}a^{6}+2a^{6}-3z^{4}a^{4}-3z^{2}a^{4}-a^{4}+z^{6}a^{2}+z^{4}a^{2}+z^{2}a^{2}-2z^{4}-z^{2}+1+z^{2}a^{-2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{4}z^{10}+2a^{2}z^{10}+5a^{5}z^{9}+11a^{3}z^{9}+6az^{9}+6a^{6}z^{8}+9a^{4}z^{8}+10a^{2}z^{8}+7z^{8}+5a^{7}z^{7}-a^{5}z^{7}-18a^{3}z^{7}-8az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-5a^{6}z^{6}-20a^{4}z^{6}-29a^{2}z^{6}+z^{6}a^{-2}-16z^{6}+a^{9}z^{5}-6a^{7}z^{5}-6a^{5}z^{5}+8a^{3}z^{5}-2az^{5}-9z^{5}a^{-1}-5a^{8}z^{4}-2a^{6}z^{4}+11a^{4}z^{4}+20a^{2}z^{4}-2z^{4}a^{-2}+10z^{4}-2a^{9}z^{3}+2a^{7}z^{3}+6a^{5}z^{3}-3a^{3}z^{3}+5z^{3}a^{-1}+3a^{8}z^{2}+6a^{6}z^{2}+a^{4}z^{2}-6a^{2}z^{2}+z^{2}a^{-2}-3z^{2}+a^{9}z-a^{5}z+a^{3}z+az-a^{8}-2a^{6}-a^{4}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a131,}

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["K11a218"];

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]=

{ $t^{3}-10t^{2}+32t-45+32t^{-1}-10t^{-2}+t^{-3}$, $q^{3}-4q^{2}+9q-14+19q^{-1}-21q^{-2}+21q^{-3}-18q^{-4}+13q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$ }

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]=

{K11a131,}

Vassiliev invariants

V2 and V3:

(1, -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 $4$ $-24$ $8$ ${\frac {350}{3}}$ ${\frac {106}{3}}$ $-96$ $-464$ $-32$ $-184$ ${\frac {32}{3}}$ $288$ ${\frac {1400}{3}}$ ${\frac {424}{3}}$ ${\frac {55471}{30}}$ $-{\frac {9622}{15}}$ ${\frac {68822}{45}}$ ${\frac {2129}{18}}$ ${\frac {7951}{30}}$

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 K11a218. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   6 1   5 1                 8 3     -5 -1               11 6       5 -3             11 9         -2 -5           10 10           0 -7         8 11             3 -9       5 10               -5 -11     2 8                 6 -13   1 5                   -4 -15   2                     2 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a217 K11a219