K11a366

K11a365

K11a367

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a366 at Knotilus!

Knot presentations

Planar diagram presentation	X10,2,11,1 X14,4,15,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X22,12,1,11 X4,14,5,13 X2,16,3,15 X8,18,9,17 X6,20,7,19 X12,22,13,21
Gauss code	1, -8, 2, -7, 3, -10, 4, -9, 5, -1, 6, -11, 7, -2, 8, -3, 9, -4, 10, -5, 11, -6
Dowker-Thistlethwaite code	10 14 16 18 20 22 4 2 8 6 12

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3,4\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a366/ThurstonBennequinNumber Hyperbolic Volume 12.7681 A-Polynomial See Data:K11a366/A-polynomial

[edit Notes for K11a366's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant -4

[edit Notes for K11a366's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$8t^{2}-20t+25-20t^{-1}+8t^{-2}$
Conway polynomial	$8z^{4}+12z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 81, 4 }
Jones polynomial	$-q^{13}+q^{12}-4q^{11}+7q^{10}-9q^{9}+13q^{8}-13q^{7}+12q^{6}-10q^{5}+7q^{4}-3q^{3}+q^{2}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-4}+3z^{4}a^{-6}+3z^{4}a^{-8}+z^{4}a^{-10}+z^{2}a^{-4}+5z^{2}a^{-6}+6z^{2}a^{-8}+z^{2}a^{-10}-z^{2}a^{-12}+3a^{-8}-2a^{-12}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-10}+z^{10}a^{-12}+3z^{9}a^{-9}+4z^{9}a^{-11}+z^{9}a^{-13}+6z^{8}a^{-8}+2z^{8}a^{-10}-3z^{8}a^{-12}+z^{8}a^{-14}+7z^{7}a^{-7}-2z^{7}a^{-9}-11z^{7}a^{-11}-z^{7}a^{-13}+z^{7}a^{-15}+6z^{6}a^{-6}-11z^{6}a^{-8}-3z^{6}a^{-10}+11z^{6}a^{-12}-3z^{6}a^{-14}+3z^{5}a^{-5}-10z^{5}a^{-7}+2z^{5}a^{-9}+15z^{5}a^{-11}-6z^{5}a^{-13}-6z^{5}a^{-15}+z^{4}a^{-4}-8z^{4}a^{-6}+15z^{4}a^{-8}-4z^{4}a^{-10}-28z^{4}a^{-12}-2z^{3}a^{-5}+6z^{3}a^{-7}-6z^{3}a^{-9}-16z^{3}a^{-11}+10z^{3}a^{-13}+12z^{3}a^{-15}-z^{2}a^{-4}+5z^{2}a^{-6}-12z^{2}a^{-8}+z^{2}a^{-10}+23z^{2}a^{-12}+4z^{2}a^{-14}+4za^{-11}-4za^{-13}-8za^{-15}+3a^{-8}-2a^{-12}$
The A2 invariant	Data:K11a366/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a366/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a366 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["K11a366"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$8t^{2}-20t+25-20t^{-1}+8t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$8z^{4}+12z^{2}+1$

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]=

$\{3,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 81, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{13}+q^{12}-4q^{11}+7q^{10}-9q^{9}+13q^{8}-13q^{7}+12q^{6}-10q^{5}+7q^{4}-3q^{3}+q^{2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{4}a^{-4}+3z^{4}a^{-6}+3z^{4}a^{-8}+z^{4}a^{-10}+z^{2}a^{-4}+5z^{2}a^{-6}+6z^{2}a^{-8}+z^{2}a^{-10}-z^{2}a^{-12}+3a^{-8}-2a^{-12}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-10}+z^{10}a^{-12}+3z^{9}a^{-9}+4z^{9}a^{-11}+z^{9}a^{-13}+6z^{8}a^{-8}+2z^{8}a^{-10}-3z^{8}a^{-12}+z^{8}a^{-14}+7z^{7}a^{-7}-2z^{7}a^{-9}-11z^{7}a^{-11}-z^{7}a^{-13}+z^{7}a^{-15}+6z^{6}a^{-6}-11z^{6}a^{-8}-3z^{6}a^{-10}+11z^{6}a^{-12}-3z^{6}a^{-14}+3z^{5}a^{-5}-10z^{5}a^{-7}+2z^{5}a^{-9}+15z^{5}a^{-11}-6z^{5}a^{-13}-6z^{5}a^{-15}+z^{4}a^{-4}-8z^{4}a^{-6}+15z^{4}a^{-8}-4z^{4}a^{-10}-28z^{4}a^{-12}-2z^{3}a^{-5}+6z^{3}a^{-7}-6z^{3}a^{-9}-16z^{3}a^{-11}+10z^{3}a^{-13}+12z^{3}a^{-15}-z^{2}a^{-4}+5z^{2}a^{-6}-12z^{2}a^{-8}+z^{2}a^{-10}+23z^{2}a^{-12}+4z^{2}a^{-14}+4za^{-11}-4za^{-13}-8za^{-15}+3a^{-8}-2a^{-12}$

"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["K11a366"];

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]=

{ $8t^{2}-20t+25-20t^{-1}+8t^{-2}$, $-q^{13}+q^{12}-4q^{11}+7q^{10}-9q^{9}+13q^{8}-13q^{7}+12q^{6}-10q^{5}+7q^{4}-3q^{3}+q^{2}$ }

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]=

{}

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:

(12, 40)

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 $48$ $320$ $1152$ $2872$ $504$ $15360$ ${\frac {84032}{3}}$ ${\frac {15104}{3}}$ $4288$ $18432$ $51200$ $137856$ $24192$ ${\frac {1380382}{5}}$ $-{\frac {8344}{15}}$ ${\frac {1816648}{15}}$ ${\frac {5666}{3}}$ ${\frac {84542}{5}}$

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=$4 is the signature of K11a366. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 11 χ 27                       1 -1 25                         0 23                   4 1   -3 21                 3       3 19               6 4       -2 17             7 3         4 15           6 6           0 13         6 7             -1 11       4 6               2 9     3 6                 -3 7     4                   4 5 1 3                     -2 3 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a365 K11a367