K11a220

K11a219

K11a221

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a220 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a220/ThurstonBennequinNumber Hyperbolic Volume 12.6215 A-Polynomial See Data:K11a220/A-polynomial

[edit Notes for K11a220's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a220's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+10t^{2}-19t+23-19t^{-1}+10t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-2z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 85, 4 }
Jones polynomial	$-q^{11}+3q^{10}-6q^{9}+9q^{8}-12q^{7}+14q^{6}-13q^{5}+11q^{4}-8q^{3}+5q^{2}-2q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-2z^{4}a^{-6}+2z^{4}a^{-8}+3z^{2}a^{-2}-3z^{2}a^{-4}+4z^{2}a^{-8}-z^{2}a^{-10}+2a^{-2}-2a^{-4}+a^{-6}+a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-6}+z^{10}a^{-8}+2z^{9}a^{-5}+5z^{9}a^{-7}+3z^{9}a^{-9}+2z^{8}a^{-4}+z^{8}a^{-6}+4z^{8}a^{-8}+5z^{8}a^{-10}+2z^{7}a^{-3}-3z^{7}a^{-5}-13z^{7}a^{-7}-3z^{7}a^{-9}+5z^{7}a^{-11}+z^{6}a^{-2}-2z^{6}a^{-4}-3z^{6}a^{-6}-15z^{6}a^{-8}-12z^{6}a^{-10}+3z^{6}a^{-12}-6z^{5}a^{-3}+2z^{5}a^{-5}+18z^{5}a^{-7}-3z^{5}a^{-9}-12z^{5}a^{-11}+z^{5}a^{-13}-4z^{4}a^{-2}-7z^{4}a^{-4}+22z^{4}a^{-8}+13z^{4}a^{-10}-6z^{4}a^{-12}+4z^{3}a^{-3}-5z^{3}a^{-5}-11z^{3}a^{-7}+9z^{3}a^{-9}+9z^{3}a^{-11}-2z^{3}a^{-13}+5z^{2}a^{-2}+8z^{2}a^{-4}+z^{2}a^{-6}-9z^{2}a^{-8}-6z^{2}a^{-10}+z^{2}a^{-12}+za^{-5}+2za^{-7}-2za^{-9}-3za^{-11}-2a^{-2}-2a^{-4}-a^{-6}+a^{-8}+a^{-10}$
The A2 invariant	Data:K11a220/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a220/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a220 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["K11a220"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+10t^{2}-19t+23-19t^{-1}+10t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 85, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{11}+3q^{10}-6q^{9}+9q^{8}-12q^{7}+14q^{6}-13q^{5}+11q^{4}-8q^{3}+5q^{2}-2q+1$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-4}-z^{6}a^{-6}+z^{4}a^{-2}-3z^{4}a^{-4}-2z^{4}a^{-6}+2z^{4}a^{-8}+3z^{2}a^{-2}-3z^{2}a^{-4}+4z^{2}a^{-8}-z^{2}a^{-10}+2a^{-2}-2a^{-4}+a^{-6}+a^{-8}-a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-6}+z^{10}a^{-8}+2z^{9}a^{-5}+5z^{9}a^{-7}+3z^{9}a^{-9}+2z^{8}a^{-4}+z^{8}a^{-6}+4z^{8}a^{-8}+5z^{8}a^{-10}+2z^{7}a^{-3}-3z^{7}a^{-5}-13z^{7}a^{-7}-3z^{7}a^{-9}+5z^{7}a^{-11}+z^{6}a^{-2}-2z^{6}a^{-4}-3z^{6}a^{-6}-15z^{6}a^{-8}-12z^{6}a^{-10}+3z^{6}a^{-12}-6z^{5}a^{-3}+2z^{5}a^{-5}+18z^{5}a^{-7}-3z^{5}a^{-9}-12z^{5}a^{-11}+z^{5}a^{-13}-4z^{4}a^{-2}-7z^{4}a^{-4}+22z^{4}a^{-8}+13z^{4}a^{-10}-6z^{4}a^{-12}+4z^{3}a^{-3}-5z^{3}a^{-5}-11z^{3}a^{-7}+9z^{3}a^{-9}+9z^{3}a^{-11}-2z^{3}a^{-13}+5z^{2}a^{-2}+8z^{2}a^{-4}+z^{2}a^{-6}-9z^{2}a^{-8}-6z^{2}a^{-10}+z^{2}a^{-12}+za^{-5}+2za^{-7}-2za^{-9}-3za^{-11}-2a^{-2}-2a^{-4}-a^{-6}+a^{-8}+a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a60,}

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["K11a220"];

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]=

{ $-2t^{3}+10t^{2}-19t+23-19t^{-1}+10t^{-2}-2t^{-3}$, $-q^{11}+3q^{10}-6q^{9}+9q^{8}-12q^{7}+14q^{6}-13q^{5}+11q^{4}-8q^{3}+5q^{2}-2q+1$ }

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]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11a60,}

Vassiliev invariants

V2 and V3:

(3, 8)

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 $12$ $64$ $72$ $334$ $50$ $768$ ${\frac {5632}{3}}$ ${\frac {736}{3}}$ $352$ $288$ $2048$ $4008$ $600$ ${\frac {108991}{10}}$ $-{\frac {3902}{5}}$ ${\frac {26154}{5}}$ ${\frac {619}{2}}$ ${\frac {6911}{10}}$

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=$4 is the signature of K11a220. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 23                       1 -1 21                     2   2 19                   4 1   -3 17                 5 2     3 15               7 4       -3 13             7 5         2 11           6 7           1 9         5 7             -2 7       3 6               3 5     2 5                 -3 3   1 4                   3 1   1                     -1 -1 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a219 K11a221