K11a202

K11a201

K11a203

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a202 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2,3\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a202/ThurstonBennequinNumber Hyperbolic Volume 14.8711 A-Polynomial See Data:K11a202/A-polynomial

[edit Notes for K11a202'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 K11a202's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-12t^{2}+26t-31+26t^{-1}-12t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}-4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 111, 2 }
Jones polynomial	$-q^{8}+4q^{7}-8q^{6}+13q^{5}-16q^{4}+18q^{3}-18q^{2}+14q-10+6q^{-1}-2q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}+2z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}-2z^{2}a^{-2}+2z^{2}a^{-4}-z^{2}a^{-6}-4z^{2}+2a^{2}-2a^{-2}+2a^{-4}-1$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+7z^{9}a^{-3}+4z^{9}a^{-5}+7z^{8}a^{-2}+11z^{8}a^{-4}+7z^{8}a^{-6}+3z^{8}+2az^{7}-5z^{7}a^{-1}-11z^{7}a^{-3}+3z^{7}a^{-5}+7z^{7}a^{-7}+a^{2}z^{6}-22z^{6}a^{-2}-27z^{6}a^{-4}-8z^{6}a^{-6}+4z^{6}a^{-8}-6z^{6}-5az^{5}+6z^{5}a^{-1}+8z^{5}a^{-3}-15z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+28z^{4}a^{-2}+27z^{4}a^{-4}-z^{4}a^{-6}-6z^{4}a^{-8}+2z^{4}+2az^{3}-10z^{3}a^{-1}-6z^{3}a^{-3}+11z^{3}a^{-5}+4z^{3}a^{-7}-z^{3}a^{-9}+5a^{2}z^{2}-17z^{2}a^{-2}-13z^{2}a^{-4}+2z^{2}a^{-6}+2z^{2}a^{-8}+z^{2}+az+6za^{-1}+4za^{-3}-2za^{-5}-za^{-7}-2a^{2}+2a^{-2}+2a^{-4}-1$
The A2 invariant	Data:K11a202/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a202/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a202 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["K11a202"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-12t^{2}+26t-31+26t^{-1}-12t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 111, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+4q^{7}-8q^{6}+13q^{5}-16q^{4}+18q^{3}-18q^{2}+14q-10+6q^{-1}-2q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+7z^{9}a^{-3}+4z^{9}a^{-5}+7z^{8}a^{-2}+11z^{8}a^{-4}+7z^{8}a^{-6}+3z^{8}+2az^{7}-5z^{7}a^{-1}-11z^{7}a^{-3}+3z^{7}a^{-5}+7z^{7}a^{-7}+a^{2}z^{6}-22z^{6}a^{-2}-27z^{6}a^{-4}-8z^{6}a^{-6}+4z^{6}a^{-8}-6z^{6}-5az^{5}+6z^{5}a^{-1}+8z^{5}a^{-3}-15z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+28z^{4}a^{-2}+27z^{4}a^{-4}-z^{4}a^{-6}-6z^{4}a^{-8}+2z^{4}+2az^{3}-10z^{3}a^{-1}-6z^{3}a^{-3}+11z^{3}a^{-5}+4z^{3}a^{-7}-z^{3}a^{-9}+5a^{2}z^{2}-17z^{2}a^{-2}-13z^{2}a^{-4}+2z^{2}a^{-6}+2z^{2}a^{-8}+z^{2}+az+6za^{-1}+4za^{-3}-2za^{-5}-za^{-7}-2a^{2}+2a^{-2}+2a^{-4}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a137,}

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["K11a202"];

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]=

{ $2t^{3}-12t^{2}+26t-31+26t^{-1}-12t^{-2}+2t^{-3}$, $-q^{8}+4q^{7}-8q^{6}+13q^{5}-16q^{4}+18q^{3}-18q^{2}+14q-10+6q^{-1}-2q^{-2}+q^{-3}$ }

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]=

{K11a137,}

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, -2)

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 $-16$ $-16$ $128$ ${\frac {520}{3}}$ ${\frac {200}{3}}$ $256$ ${\frac {1088}{3}}$ ${\frac {224}{3}}$ $48$ $-{\frac {2048}{3}}$ $128$ $-{\frac {8320}{3}}$ $-{\frac {3200}{3}}$ $-{\frac {36062}{15}}$ ${\frac {5048}{15}}$ $-{\frac {92648}{45}}$ ${\frac {2030}{9}}$ $-{\frac {6542}{15}}$

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 K11a202. 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 17                       1 -1 15                     3   3 13                   5 1   -4 11                 8 3     5 9               8 5       -3 7             10 8         2 5           8 8           0 3         6 10             -4 1       5 9               4 -1     1 5                 -4 -3   1 5                   4 -5   1                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a201 K11a203