K11a256

K11a255

K11a257

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a256 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X8394 X12,6,13,5 X20,8,21,7 X18,9,19,10 X16,11,17,12 X22,13,1,14 X4,16,5,15 X10,17,11,18 X2,19,3,20 X14,21,15,22
Gauss code	1, -10, 2, -8, 3, -1, 4, -2, 5, -9, 6, -3, 7, -11, 8, -6, 9, -5, 10, -4, 11, -7
Dowker-Thistlethwaite code	6 8 12 20 18 16 22 4 10 2 14

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

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a256/ThurstonBennequinNumber Hyperbolic Volume 16.7429 A-Polynomial See Data:K11a256/A-polynomial

[edit Notes for K11a256's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for K11a256's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+12t^{2}-31t+43-31t^{-1}+12t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 133, 0 }
Jones polynomial	$q^{4}-4q^{3}+9q^{2}-14q+19-21q^{-1}+21q^{-2}-18q^{-3}+13q^{-4}-8q^{-5}+4q^{-6}-q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{6}+2z^{4}a^{4}+2z^{2}a^{4}-z^{6}a^{2}-z^{4}a^{2}+a^{2}-z^{6}-2z^{4}-3z^{2}-1+z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}$
Kauffman polynomial (db, data sources)	$3a^{4}z^{10}+3a^{2}z^{10}+6a^{5}z^{9}+15a^{3}z^{9}+9az^{9}+4a^{6}z^{8}+2a^{4}z^{8}+11a^{2}z^{8}+13z^{8}+a^{7}z^{7}-20a^{5}z^{7}-43a^{3}z^{7}-9az^{7}+13z^{7}a^{-1}-14a^{6}z^{6}-30a^{4}z^{6}-43a^{2}z^{6}+9z^{6}a^{-2}-18z^{6}-3a^{7}z^{5}+19a^{5}z^{5}+34a^{3}z^{5}-9az^{5}-17z^{5}a^{-1}+4z^{5}a^{-3}+14a^{6}z^{4}+35a^{4}z^{4}+32a^{2}z^{4}-8z^{4}a^{-2}+z^{4}a^{-4}+2z^{4}+2a^{7}z^{3}-5a^{5}z^{3}-8a^{3}z^{3}+7az^{3}+7z^{3}a^{-1}-z^{3}a^{-3}-4a^{6}z^{2}-10a^{4}z^{2}-5a^{2}z^{2}+3z^{2}a^{-2}+4z^{2}-az-za^{-1}-a^{2}-a^{-2}-1$
The A2 invariant	Data:K11a256/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a256/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a256 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["K11a256"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+12t^{2}-31t+43-31t^{-1}+12t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 133, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3a^{4}z^{10}+3a^{2}z^{10}+6a^{5}z^{9}+15a^{3}z^{9}+9az^{9}+4a^{6}z^{8}+2a^{4}z^{8}+11a^{2}z^{8}+13z^{8}+a^{7}z^{7}-20a^{5}z^{7}-43a^{3}z^{7}-9az^{7}+13z^{7}a^{-1}-14a^{6}z^{6}-30a^{4}z^{6}-43a^{2}z^{6}+9z^{6}a^{-2}-18z^{6}-3a^{7}z^{5}+19a^{5}z^{5}+34a^{3}z^{5}-9az^{5}-17z^{5}a^{-1}+4z^{5}a^{-3}+14a^{6}z^{4}+35a^{4}z^{4}+32a^{2}z^{4}-8z^{4}a^{-2}+z^{4}a^{-4}+2z^{4}+2a^{7}z^{3}-5a^{5}z^{3}-8a^{3}z^{3}+7az^{3}+7z^{3}a^{-1}-z^{3}a^{-3}-4a^{6}z^{2}-10a^{4}z^{2}-5a^{2}z^{2}+3z^{2}a^{-2}+4z^{2}-az-za^{-1}-a^{2}-a^{-2}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["K11a256"];

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]=

{ $-2t^{3}+12t^{2}-31t+43-31t^{-1}+12t^{-2}-2t^{-3}$, $q^{4}-4q^{3}+9q^{2}-14q+19-21q^{-1}+21q^{-2}-18q^{-3}+13q^{-4}-8q^{-5}+4q^{-6}-q^{-7}$ }

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]=

{}

Vassiliev invariants

V2 and V3:

(-1, 0)

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 $-4$ $0$ $8$ ${\frac {34}{3}}$ ${\frac {14}{3}}$ $0$ $0$ $64$ $-64$ $-{\frac {32}{3}}$ $0$ $-{\frac {136}{3}}$ $-{\frac {56}{3}}$ $-{\frac {1231}{30}}$ $-{\frac {3218}{15}}$ ${\frac {4018}{45}}$ ${\frac {1807}{18}}$ ${\frac {689}{30}}$

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=$0 is the signature of K11a256. 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 9                       1 1 7                     3   -3 5                   6 1   5 3                 8 3     -5 1               11 6       5 -1             11 9         -2 -3           10 10           0 -5         8 11             3 -7       5 10               -5 -9     3 8                 5 -11   1 5                   -4 -13   3                     3 -15 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11a255 K11a257