K11a258

K11a257

K11a259

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a258 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X8493 X14,5,15,6 X2837 X16,10,17,9 X20,11,21,12 X18,13,19,14 X4,15,5,16 X22,18,1,17 X12,19,13,20 X10,21,11,22
Gauss code	1, -4, 2, -8, 3, -1, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -7, 10, -6, 11, -9
Dowker-Thistlethwaite code	6 8 14 2 16 20 18 4 22 12 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a258/ThurstonBennequinNumber Hyperbolic Volume 11.9877 A-Polynomial See Data:K11a258/A-polynomial

[edit Notes for K11a258'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 K11a258's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-9t^{2}+17t-19+17t^{-1}-9t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+3z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 75, 2 }
Jones polynomial	$-q^{6}+3q^{5}-6q^{4}+9q^{3}-10q^{2}+12q-11+9q^{-1}-7q^{-2}+4q^{-3}-2q^{-4}+q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}-2a^{2}z^{4}+3z^{4}a^{-2}-z^{4}a^{-4}+3z^{4}+a^{4}z^{2}-6a^{2}z^{2}+3z^{2}a^{-2}-2z^{2}a^{-4}+3z^{2}+2a^{4}-4a^{2}+2a^{-2}-a^{-4}+2$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+2a^{3}z^{9}+6az^{9}+4z^{9}a^{-1}+a^{4}z^{8}+7z^{8}a^{-2}+6z^{8}-11a^{3}z^{7}-26az^{7}-6z^{7}a^{-1}+9z^{7}a^{-3}-6a^{4}z^{6}-18a^{2}z^{6}-13z^{6}a^{-2}+9z^{6}a^{-4}-34z^{6}+20a^{3}z^{5}+35az^{5}-7z^{5}a^{-1}-16z^{5}a^{-3}+6z^{5}a^{-5}+12a^{4}z^{4}+39a^{2}z^{4}-z^{4}a^{-2}-14z^{4}a^{-4}+3z^{4}a^{-6}+43z^{4}-14a^{3}z^{3}-19az^{3}+4z^{3}a^{-1}+4z^{3}a^{-3}-4z^{3}a^{-5}+z^{3}a^{-7}-9a^{4}z^{2}-25a^{2}z^{2}+4z^{2}a^{-2}+7z^{2}a^{-4}-19z^{2}+4a^{3}z+6az+2za^{-1}+za^{-3}+za^{-5}+2a^{4}+4a^{2}-2a^{-2}-a^{-4}+2$
The A2 invariant	Data:K11a258/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a258/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a258 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["K11a258"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-9t^{2}+17t-19+17t^{-1}-9t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 75, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+2a^{3}z^{9}+6az^{9}+4z^{9}a^{-1}+a^{4}z^{8}+7z^{8}a^{-2}+6z^{8}-11a^{3}z^{7}-26az^{7}-6z^{7}a^{-1}+9z^{7}a^{-3}-6a^{4}z^{6}-18a^{2}z^{6}-13z^{6}a^{-2}+9z^{6}a^{-4}-34z^{6}+20a^{3}z^{5}+35az^{5}-7z^{5}a^{-1}-16z^{5}a^{-3}+6z^{5}a^{-5}+12a^{4}z^{4}+39a^{2}z^{4}-z^{4}a^{-2}-14z^{4}a^{-4}+3z^{4}a^{-6}+43z^{4}-14a^{3}z^{3}-19az^{3}+4z^{3}a^{-1}+4z^{3}a^{-3}-4z^{3}a^{-5}+z^{3}a^{-7}-9a^{4}z^{2}-25a^{2}z^{2}+4z^{2}a^{-2}+7z^{2}a^{-4}-19z^{2}+4a^{3}z+6az+2za^{-1}+za^{-3}+za^{-5}+2a^{4}+4a^{2}-2a^{-2}-a^{-4}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n150,}

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["K11a258"];

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]=

{ $2t^{3}-9t^{2}+17t-19+17t^{-1}-9t^{-2}+2t^{-3}$, $-q^{6}+3q^{5}-6q^{4}+9q^{3}-10q^{2}+12q-11+9q^{-1}-7q^{-2}+4q^{-3}-2q^{-4}+q^{-5}$ }

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]=

{K11n150,}

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, 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 $-4$ $24$ $8$ $-{\frac {62}{3}}$ $-{\frac {58}{3}}$ $-96$ $-144$ $-128$ $56$ $-{\frac {32}{3}}$ $288$ ${\frac {248}{3}}$ ${\frac {232}{3}}$ ${\frac {18929}{30}}$ $-{\frac {846}{5}}$ ${\frac {25978}{45}}$ $-{\frac {1361}{18}}$ ${\frac {2609}{30}}$

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 K11a258. 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 13                       1 -1 11                     2   2 9                   4 1   -3 7                 5 2     3 5               5 4       -1 3             7 5         2 1           5 6           1 -1         4 6             -2 -3       3 5               2 -5     1 4                 -3 -7   1 3                   2 -9   1                     -1 -11 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a257 K11a259