K11a205

K11a204

K11a206

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a205 at Knotilus!

Knot presentations

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

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:K11a205/ThurstonBennequinNumber Hyperbolic Volume 12.8798 A-Polynomial See Data:K11a205/A-polynomial

[edit Notes for K11a205'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 K11a205's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-10t^{2}+21t-25+21t^{-1}-10t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 91, -2 }
Jones polynomial	$-q^{4}+3q^{3}-5q^{2}+9q-12+14q^{-1}-14q^{-2}+13q^{-3}-10q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}+a^{6}-2z^{4}a^{4}-4z^{2}a^{4}-2a^{4}+z^{6}a^{2}+2z^{4}a^{2}+z^{2}a^{2}+a^{2}+z^{6}+3z^{4}+3z^{2}+1-z^{4}a^{-2}-2z^{2}a^{-2}$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+6az^{9}+3z^{9}a^{-1}+5a^{4}z^{8}+6a^{2}z^{8}+3z^{8}a^{-2}+4z^{8}+6a^{5}z^{7}+2a^{3}z^{7}-15az^{7}-10z^{7}a^{-1}+z^{7}a^{-3}+5a^{6}z^{6}-2a^{4}z^{6}-16a^{2}z^{6}-13z^{6}a^{-2}-22z^{6}+3a^{7}z^{5}-6a^{5}z^{5}-11a^{3}z^{5}+11az^{5}+9z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-5a^{6}z^{4}-8a^{4}z^{4}+6a^{2}z^{4}+17z^{4}a^{-2}+25z^{4}-3a^{7}z^{3}+2a^{5}z^{3}+4a^{3}z^{3}-8az^{3}-3z^{3}a^{-1}+4z^{3}a^{-3}-a^{8}z^{2}+3a^{6}z^{2}+9a^{4}z^{2}+a^{2}z^{2}-6z^{2}a^{-2}-10z^{2}+a^{7}z+a^{5}z+a^{3}z+2az+za^{-1}-a^{6}-2a^{4}-a^{2}+1$
The A2 invariant	Data:K11a205/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a205/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a205 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["K11a205"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-10t^{2}+21t-25+21t^{-1}-10t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 91, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+6az^{9}+3z^{9}a^{-1}+5a^{4}z^{8}+6a^{2}z^{8}+3z^{8}a^{-2}+4z^{8}+6a^{5}z^{7}+2a^{3}z^{7}-15az^{7}-10z^{7}a^{-1}+z^{7}a^{-3}+5a^{6}z^{6}-2a^{4}z^{6}-16a^{2}z^{6}-13z^{6}a^{-2}-22z^{6}+3a^{7}z^{5}-6a^{5}z^{5}-11a^{3}z^{5}+11az^{5}+9z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-5a^{6}z^{4}-8a^{4}z^{4}+6a^{2}z^{4}+17z^{4}a^{-2}+25z^{4}-3a^{7}z^{3}+2a^{5}z^{3}+4a^{3}z^{3}-8az^{3}-3z^{3}a^{-1}+4z^{3}a^{-3}-a^{8}z^{2}+3a^{6}z^{2}+9a^{4}z^{2}+a^{2}z^{2}-6z^{2}a^{-2}-10z^{2}+a^{7}z+a^{5}z+a^{3}z+2az+za^{-1}-a^{6}-2a^{4}-a^{2}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a86,}

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["K11a205"];

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]=

{ $2t^{3}-10t^{2}+21t-25+21t^{-1}-10t^{-2}+2t^{-3}$, $-q^{4}+3q^{3}-5q^{2}+9q-12+14q^{-1}-14q^{-2}+13q^{-3}-10q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$ }

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]=

{K11a86,}

Vassiliev invariants

V2 and V3:

(-1, 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 $-4$ $16$ $8$ $-{\frac {110}{3}}$ $-{\frac {34}{3}}$ $-64$ $-{\frac {128}{3}}$ $-{\frac {224}{3}}$ $48$ $-{\frac {32}{3}}$ $128$ ${\frac {440}{3}}$ ${\frac {136}{3}}$ ${\frac {13409}{30}}$ $-{\frac {178}{15}}$ ${\frac {15058}{45}}$ $-{\frac {1121}{18}}$ ${\frac {1889}{30}}$

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 K11a205. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a204 K11a206