K11n66

K11n65

K11n67

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n66 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,5,15,6 X2837 X9,19,10,18 X11,17,12,16 X20,14,21,13 X6,15,7,16 X17,11,18,10 X22,19,1,20 X12,22,13,21
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, -5, 9, -6, -11, 7, -3, 8, 6, -9, 5, 10, -7, 11, -10
Dowker-Thistlethwaite code	4 8 14 2 -18 -16 20 6 -10 22 12

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:K11n66/ThurstonBennequinNumber Hyperbolic Volume 14.4341 A-Polynomial See Data:K11n66/A-polynomial

[edit Notes for K11n66's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $[1,3]$ Rasmussen s-Invariant -2

[edit Notes for K11n66's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-7t^{2}+18t-23+18t^{-1}-7t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 75, 2 }
Jones polynomial	$2q^{7}-5q^{6}+8q^{5}-12q^{4}+13q^{3}-12q^{2}+11q-7+4q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+3z^{4}a^{-2}-3z^{4}a^{-4}-z^{4}+5z^{2}a^{-2}-7z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}+4a^{-2}-5a^{-4}+2a^{-6}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+4z^{8}a^{-2}+6z^{8}a^{-4}+2z^{8}a^{-6}+6z^{7}a^{-1}+10z^{7}a^{-3}+5z^{7}a^{-5}+z^{7}a^{-7}+z^{6}a^{-2}-3z^{6}a^{-4}+4z^{6}+az^{5}-10z^{5}a^{-1}-21z^{5}a^{-3}-7z^{5}a^{-5}+3z^{5}a^{-7}-15z^{4}a^{-2}-11z^{4}a^{-4}+3z^{4}a^{-8}-7z^{4}-az^{3}+3z^{3}a^{-1}+10z^{3}a^{-3}-6z^{3}a^{-7}+12z^{2}a^{-2}+14z^{2}a^{-4}+z^{2}a^{-6}-4z^{2}a^{-8}+3z^{2}-za^{-1}+4za^{-5}+3za^{-7}-4a^{-2}-5a^{-4}-2a^{-6}$
The A2 invariant	Data:K11n66/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n66/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n66 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["K11n66"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-7t^{2}+18t-23+18t^{-1}-7t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$2q^{7}-5q^{6}+8q^{5}-12q^{4}+13q^{3}-12q^{2}+11q-7+4q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+3z^{4}a^{-2}-3z^{4}a^{-4}-z^{4}+5z^{2}a^{-2}-7z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}+4a^{-2}-5a^{-4}+2a^{-6}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-3}+z^{9}a^{-5}+4z^{8}a^{-2}+6z^{8}a^{-4}+2z^{8}a^{-6}+6z^{7}a^{-1}+10z^{7}a^{-3}+5z^{7}a^{-5}+z^{7}a^{-7}+z^{6}a^{-2}-3z^{6}a^{-4}+4z^{6}+az^{5}-10z^{5}a^{-1}-21z^{5}a^{-3}-7z^{5}a^{-5}+3z^{5}a^{-7}-15z^{4}a^{-2}-11z^{4}a^{-4}+3z^{4}a^{-8}-7z^{4}-az^{3}+3z^{3}a^{-1}+10z^{3}a^{-3}-6z^{3}a^{-7}+12z^{2}a^{-2}+14z^{2}a^{-4}+z^{2}a^{-6}-4z^{2}a^{-8}+3z^{2}-za^{-1}+4za^{-5}+3za^{-7}-4a^{-2}-5a^{-4}-2a^{-6}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_40, 10_59,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n150,}

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["K11n66"];

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]=

{ $t^{3}-7t^{2}+18t-23+18t^{-1}-7t^{-2}+t^{-3}$, $2q^{7}-5q^{6}+8q^{5}-12q^{4}+13q^{3}-12q^{2}+11q-7+4q^{-1}-q^{-2}$ }

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]=

{9_40, 10_59,}

In[6]:=

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

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

Out[6]=

{K11n150,}

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 $-4$ $-24$ $8$ $-{\frac {110}{3}}$ ${\frac {38}{3}}$ $96$ $144$ $64$ $72$ $-{\frac {32}{3}}$ $288$ ${\frac {440}{3}}$ $-{\frac {152}{3}}$ ${\frac {29249}{30}}$ $-{\frac {1018}{15}}$ ${\frac {26218}{45}}$ ${\frac {1183}{18}}$ ${\frac {1409}{30}}$

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 K11n66. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n65 K11n67