K11a266

K11a265

K11a267

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a266 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,3,11,4 X12,6,13,5 X14,7,15,8 X16,10,17,9 X18,11,19,12 X22,13,1,14 X20,16,21,15 X4,18,5,17 X2,19,3,20 X8,21,9,22
Gauss code	1, -10, 2, -9, 3, -1, 4, -11, 5, -2, 6, -3, 7, -4, 8, -5, 9, -6, 10, -8, 11, -7
Dowker-Thistlethwaite code	6 10 12 14 16 18 22 20 4 2 8

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a266/ThurstonBennequinNumber Hyperbolic Volume 20.2286 A-Polynomial See Data:K11a266/A-polynomial

[edit Notes for K11a266's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a266's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-7t^{3}+23t^{2}-45t+57-45t^{-1}+23t^{-2}-7t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+z^{6}+z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 209, 0 }
Jones polynomial	$-q^{5}+5q^{4}-12q^{3}+21q^{2}-29q+34-34q^{-1}+30q^{-2}-22q^{-3}+14q^{-4}-6q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+4z^{6}+a^{4}z^{4}-4a^{2}z^{4}-2z^{4}a^{-2}+6z^{4}-z^{2}a^{-2}+z^{2}-a^{4}+3a^{2}+a^{-2}-2$
Kauffman polynomial (db, data sources)	$6a^{2}z^{10}+6z^{10}+15a^{3}z^{9}+31az^{9}+16z^{9}a^{-1}+14a^{4}z^{8}+20a^{2}z^{8}+18z^{8}a^{-2}+24z^{8}+6a^{5}z^{7}-23a^{3}z^{7}-57az^{7}-16z^{7}a^{-1}+12z^{7}a^{-3}+a^{6}z^{6}-26a^{4}z^{6}-65a^{2}z^{6}-26z^{6}a^{-2}+5z^{6}a^{-4}-69z^{6}-7a^{5}z^{5}+2a^{3}z^{5}+19az^{5}-4z^{5}a^{-1}-13z^{5}a^{-3}+z^{5}a^{-5}+10a^{4}z^{4}+37a^{2}z^{4}+14z^{4}a^{-2}-3z^{4}a^{-4}+44z^{4}+2a^{3}z^{3}+5az^{3}+7z^{3}a^{-1}+4z^{3}a^{-3}+2a^{4}z^{2}+a^{2}z^{2}-2z^{2}a^{-2}-3z^{2}+a^{5}z+a^{3}z-az-za^{-1}-a^{4}-3a^{2}-a^{-2}-2$
The A2 invariant	$q^{18}-3q^{16}+3q^{12}-5q^{10}+7q^{8}-q^{6}+4q^{2}-7+6q^{-2}-6q^{-4}+2q^{-6}+4q^{-8}-4q^{-10}+3q^{-12}-q^{-14}$
The G2 invariant	Data:K11a266/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a266 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["K11a266"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-7t^{3}+23t^{2}-45t+57-45t^{-1}+23t^{-2}-7t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+z^{6}+z^{4}+1$

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]=

{ 209, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+5q^{4}-12q^{3}+21q^{2}-29q+34-34q^{-1}+30q^{-2}-22q^{-3}+14q^{-4}-6q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$6a^{2}z^{10}+6z^{10}+15a^{3}z^{9}+31az^{9}+16z^{9}a^{-1}+14a^{4}z^{8}+20a^{2}z^{8}+18z^{8}a^{-2}+24z^{8}+6a^{5}z^{7}-23a^{3}z^{7}-57az^{7}-16z^{7}a^{-1}+12z^{7}a^{-3}+a^{6}z^{6}-26a^{4}z^{6}-65a^{2}z^{6}-26z^{6}a^{-2}+5z^{6}a^{-4}-69z^{6}-7a^{5}z^{5}+2a^{3}z^{5}+19az^{5}-4z^{5}a^{-1}-13z^{5}a^{-3}+z^{5}a^{-5}+10a^{4}z^{4}+37a^{2}z^{4}+14z^{4}a^{-2}-3z^{4}a^{-4}+44z^{4}+2a^{3}z^{3}+5az^{3}+7z^{3}a^{-1}+4z^{3}a^{-3}+2a^{4}z^{2}+a^{2}z^{2}-2z^{2}a^{-2}-3z^{2}+a^{5}z+a^{3}z-az-za^{-1}-a^{4}-3a^{2}-a^{-2}-2$

"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["K11a266"];

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]=

{ $t^{4}-7t^{3}+23t^{2}-45t+57-45t^{-1}+23t^{-2}-7t^{-3}+t^{-4}$, $-q^{5}+5q^{4}-12q^{3}+21q^{2}-29q+34-34q^{-1}+30q^{-2}-22q^{-3}+14q^{-4}-6q^{-5}+q^{-6}$ }

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:

(0, -1)

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 $0$ $-8$ $0$ $0$ $-8$ $0$ ${\frac {16}{3}}$ $-{\frac {32}{3}}$ $24$ $0$ $32$ $0$ $0$ $48$ ${\frac {40}{3}}$ ${\frac {104}{3}}$ $-{\frac {112}{3}}$ $0$

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 K11a266. 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 11                       1 -1 9                     4   4 7                   8 1   -7 5                 13 4     9 3               16 8       -8 1             18 13         5 -1           17 17           0 -3         13 17             -4 -5       9 17               8 -7     5 13                 -8 -9   1 9                   8 -11   5                     -5 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{17}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=-1$ ${\mathbb {Z} }^{17}\oplus {\mathbb {Z} }_{2}^{17}$ ${\mathbb {Z} }^{17}$ $r=0$ ${\mathbb {Z} }^{17}\oplus {\mathbb {Z} }_{2}^{17}$ ${\mathbb {Z} }^{18}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{16}$ ${\mathbb {Z} }^{16}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a265 K11a267