K11a227

K11a226

K11a228

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a227 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X12,4,13,3 X18,6,19,5 X22,8,1,7 X16,10,17,9 X2,12,3,11 X8,14,9,13 X20,16,21,15 X10,18,11,17 X6,20,7,19 X14,22,15,21
Gauss code	1, -6, 2, -1, 3, -10, 4, -7, 5, -9, 6, -2, 7, -11, 8, -5, 9, -3, 10, -8, 11, -4
Dowker-Thistlethwaite code	4 12 18 22 16 2 8 20 10 6 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a227/ThurstonBennequinNumber Hyperbolic Volume 17.3495 A-Polynomial See Data:K11a227/A-polynomial

[edit Notes for K11a227's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -6

[edit Notes for K11a227's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$5t^{3}-17t^{2}+31t-37+31t^{-1}-17t^{-2}+5t^{-3}$
Conway polynomial	$5z^{6}+13z^{4}+8z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 143, 6 }
Jones polynomial	$-q^{14}+4q^{13}-9q^{12}+15q^{11}-21q^{10}+23q^{9}-23q^{8}+20q^{7}-14q^{6}+9q^{5}-3q^{4}+q^{3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-6}+3z^{6}a^{-8}+z^{6}a^{-10}+3z^{4}a^{-6}+11z^{4}a^{-8}-z^{4}a^{-12}+3z^{2}a^{-6}+12z^{2}a^{-8}-6z^{2}a^{-10}-z^{2}a^{-12}+a^{-6}+4a^{-8}-5a^{-10}+a^{-12}$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-10}+2z^{10}a^{-12}+5z^{9}a^{-9}+12z^{9}a^{-11}+7z^{9}a^{-13}+6z^{8}a^{-8}+12z^{8}a^{-10}+16z^{8}a^{-12}+10z^{8}a^{-14}+3z^{7}a^{-7}-z^{7}a^{-9}-13z^{7}a^{-11}-z^{7}a^{-13}+8z^{7}a^{-15}+z^{6}a^{-6}-14z^{6}a^{-8}-34z^{6}a^{-10}-38z^{6}a^{-12}-15z^{6}a^{-14}+4z^{6}a^{-16}-6z^{5}a^{-7}-18z^{5}a^{-9}-14z^{5}a^{-11}-15z^{5}a^{-13}-12z^{5}a^{-15}+z^{5}a^{-17}-3z^{4}a^{-6}+15z^{4}a^{-8}+29z^{4}a^{-10}+25z^{4}a^{-12}+9z^{4}a^{-14}-5z^{4}a^{-16}+3z^{3}a^{-7}+20z^{3}a^{-9}+24z^{3}a^{-11}+15z^{3}a^{-13}+7z^{3}a^{-15}-z^{3}a^{-17}+3z^{2}a^{-6}-12z^{2}a^{-8}-16z^{2}a^{-10}-4z^{2}a^{-12}-2z^{2}a^{-14}+z^{2}a^{-16}-8za^{-9}-10za^{-11}-4za^{-13}-2za^{-15}-a^{-6}+4a^{-8}+5a^{-10}+a^{-12}$
The A2 invariant	Data:K11a227/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a227/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a227 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["K11a227"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$5t^{3}-17t^{2}+31t-37+31t^{-1}-17t^{-2}+5t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$5z^{6}+13z^{4}+8z^{2}+1$

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]=

{ 143, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{14}+4q^{13}-9q^{12}+15q^{11}-21q^{10}+23q^{9}-23q^{8}+20q^{7}-14q^{6}+9q^{5}-3q^{4}+q^{3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-6}+3z^{6}a^{-8}+z^{6}a^{-10}+3z^{4}a^{-6}+11z^{4}a^{-8}-z^{4}a^{-12}+3z^{2}a^{-6}+12z^{2}a^{-8}-6z^{2}a^{-10}-z^{2}a^{-12}+a^{-6}+4a^{-8}-5a^{-10}+a^{-12}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-10}+2z^{10}a^{-12}+5z^{9}a^{-9}+12z^{9}a^{-11}+7z^{9}a^{-13}+6z^{8}a^{-8}+12z^{8}a^{-10}+16z^{8}a^{-12}+10z^{8}a^{-14}+3z^{7}a^{-7}-z^{7}a^{-9}-13z^{7}a^{-11}-z^{7}a^{-13}+8z^{7}a^{-15}+z^{6}a^{-6}-14z^{6}a^{-8}-34z^{6}a^{-10}-38z^{6}a^{-12}-15z^{6}a^{-14}+4z^{6}a^{-16}-6z^{5}a^{-7}-18z^{5}a^{-9}-14z^{5}a^{-11}-15z^{5}a^{-13}-12z^{5}a^{-15}+z^{5}a^{-17}-3z^{4}a^{-6}+15z^{4}a^{-8}+29z^{4}a^{-10}+25z^{4}a^{-12}+9z^{4}a^{-14}-5z^{4}a^{-16}+3z^{3}a^{-7}+20z^{3}a^{-9}+24z^{3}a^{-11}+15z^{3}a^{-13}+7z^{3}a^{-15}-z^{3}a^{-17}+3z^{2}a^{-6}-12z^{2}a^{-8}-16z^{2}a^{-10}-4z^{2}a^{-12}-2z^{2}a^{-14}+z^{2}a^{-16}-8za^{-9}-10za^{-11}-4za^{-13}-2za^{-15}-a^{-6}+4a^{-8}+5a^{-10}+a^{-12}$

"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["K11a227"];

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]=

{ $5t^{3}-17t^{2}+31t-37+31t^{-1}-17t^{-2}+5t^{-3}$, $-q^{14}+4q^{13}-9q^{12}+15q^{11}-21q^{10}+23q^{9}-23q^{8}+20q^{7}-14q^{6}+9q^{5}-3q^{4}+q^{3}$ }

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:

(8, 21)

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 $32$ $168$ $512$ ${\frac {3424}{3}}$ ${\frac {440}{3}}$ $5376$ $8688$ $1440$ $936$ ${\frac {16384}{3}}$ $14112$ ${\frac {109568}{3}}$ ${\frac {14080}{3}}$ ${\frac {1024804}{15}}$ ${\frac {61144}{15}}$ ${\frac {992536}{45}}$ ${\frac {3884}{9}}$ ${\frac {39364}{15}}$

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=$6 is the signature of K11a227. 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 29                       1 -1 27                     3   3 25                   6 1   -5 23                 9 3     6 21               12 6       -6 19             11 9         2 17           12 12           0 15         8 11             -3 13       6 12               6 11     3 8                 -5 9     6                   6 7 1 3                     -2 5 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=5$ $i=7$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=5$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=6$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=7$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=8$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=9$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=10$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=11$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a226 K11a228