K11n40

K11n39

K11n41

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n40 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,13,6,12 X2837 X18,10,19,9 X16,11,17,12 X13,7,14,6 X20,15,21,16 X22,18,1,17 X14,19,15,20 X10,22,11,21
Gauss code	1, -4, 2, -1, -3, 7, 4, -2, 5, -11, 6, 3, -7, -10, 8, -6, 9, -5, 10, -8, 11, -9
Dowker-Thistlethwaite code	4 8 -12 2 18 16 -6 20 22 14 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n40/ThurstonBennequinNumber Hyperbolic Volume 15.4047 A-Polynomial See Data:K11n40/A-polynomial

[edit Notes for K11n40'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 K11n40's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-8t^{2}+18t-23+18t^{-1}-8t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+4z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 79, 2 }
Jones polynomial	$q^{9}-4q^{8}+7q^{7}-11q^{6}+13q^{5}-13q^{4}+13q^{3}-9q^{2}+6q-2$
HOMFLY-PT polynomial (db, data sources)	$2z^{6}a^{-4}-2z^{4}a^{-2}+9z^{4}a^{-4}-3z^{4}a^{-6}-4z^{2}a^{-2}+15z^{2}a^{-4}-8z^{2}a^{-6}+z^{2}a^{-8}-2a^{-2}+8a^{-4}-6a^{-6}+a^{-8}$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-5}+2z^{9}a^{-7}+6z^{8}a^{-4}+11z^{8}a^{-6}+5z^{8}a^{-8}+5z^{7}a^{-3}+8z^{7}a^{-5}+7z^{7}a^{-7}+4z^{7}a^{-9}+z^{6}a^{-2}-14z^{6}a^{-4}-27z^{6}a^{-6}-11z^{6}a^{-8}+z^{6}a^{-10}-9z^{5}a^{-3}-30z^{5}a^{-5}-32z^{5}a^{-7}-11z^{5}a^{-9}+6z^{4}a^{-2}+23z^{4}a^{-4}+22z^{4}a^{-6}+3z^{4}a^{-8}-2z^{4}a^{-10}+3z^{3}a^{-1}+15z^{3}a^{-3}+32z^{3}a^{-5}+28z^{3}a^{-7}+8z^{3}a^{-9}-6z^{2}a^{-2}-20z^{2}a^{-4}-15z^{2}a^{-6}+z^{2}a^{-10}-2za^{-1}-7za^{-3}-13za^{-5}-9za^{-7}-za^{-9}+2a^{-2}+8a^{-4}+6a^{-6}+a^{-8}$
The A2 invariant	$-2+2q^{-2}-2q^{-4}+q^{-6}+4q^{-8}+5q^{-12}-q^{-14}+q^{-16}-q^{-18}-4q^{-20}+q^{-22}-2q^{-24}+q^{-28}$
The G2 invariant	Data:K11n40/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n40 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["K11n40"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-8t^{2}+18t-23+18t^{-1}-8t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 79, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$2z^{6}a^{-4}-2z^{4}a^{-2}+9z^{4}a^{-4}-3z^{4}a^{-6}-4z^{2}a^{-2}+15z^{2}a^{-4}-8z^{2}a^{-6}+z^{2}a^{-8}-2a^{-2}+8a^{-4}-6a^{-6}+a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-5}+2z^{9}a^{-7}+6z^{8}a^{-4}+11z^{8}a^{-6}+5z^{8}a^{-8}+5z^{7}a^{-3}+8z^{7}a^{-5}+7z^{7}a^{-7}+4z^{7}a^{-9}+z^{6}a^{-2}-14z^{6}a^{-4}-27z^{6}a^{-6}-11z^{6}a^{-8}+z^{6}a^{-10}-9z^{5}a^{-3}-30z^{5}a^{-5}-32z^{5}a^{-7}-11z^{5}a^{-9}+6z^{4}a^{-2}+23z^{4}a^{-4}+22z^{4}a^{-6}+3z^{4}a^{-8}-2z^{4}a^{-10}+3z^{3}a^{-1}+15z^{3}a^{-3}+32z^{3}a^{-5}+28z^{3}a^{-7}+8z^{3}a^{-9}-6z^{2}a^{-2}-20z^{2}a^{-4}-15z^{2}a^{-6}+z^{2}a^{-10}-2za^{-1}-7za^{-3}-13za^{-5}-9za^{-7}-za^{-9}+2a^{-2}+8a^{-4}+6a^{-6}+a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_57, K11n46,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n46,}

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["K11n40"];

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]=

{ $2t^{3}-8t^{2}+18t-23+18t^{-1}-8t^{-2}+2t^{-3}$, $q^{9}-4q^{8}+7q^{7}-11q^{6}+13q^{5}-13q^{4}+13q^{3}-9q^{2}+6q-2$ }

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]=

{10_57, K11n46,}

In[6]:=

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

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

Out[6]=

{K11n46,}

Vassiliev invariants

V2 and V3:

(4, 6)

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 $16$ $48$ $128$ ${\frac {680}{3}}$ ${\frac {88}{3}}$ $768$ $1152$ $192$ $144$ ${\frac {2048}{3}}$ $1152$ ${\frac {10880}{3}}$ ${\frac {1408}{3}}$ ${\frac {90422}{15}}$ ${\frac {1544}{5}}$ ${\frac {93368}{45}}$ ${\frac {346}{9}}$ ${\frac {3782}{15}}$

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 K11n40. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n39 K11n41