K11a230

K11a229

K11a231

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a230 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a230/ThurstonBennequinNumber Hyperbolic Volume 9.27442 A-Polynomial See Data:K11a230/A-polynomial

[edit Notes for K11a230's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a230's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{2}+13t-19+13t^{-1}-3t^{-2}$
Conway polynomial	$-3z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 51, 2 }
Jones polynomial	$-q^{10}+2q^{9}-3q^{8}+5q^{7}-6q^{6}+7q^{5}-8q^{4}+7q^{3}-5q^{2}+4q-2+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{-2}-z^{4}a^{-4}-z^{4}a^{-6}-z^{2}a^{-2}-z^{2}a^{-6}+2z^{2}a^{-8}+z^{2}-a^{-6}+2a^{-8}-a^{-10}+1$
Kauffman polynomial (db, data sources)	$z^{10}a^{-6}+z^{10}a^{-8}+2z^{9}a^{-5}+4z^{9}a^{-7}+2z^{9}a^{-9}+2z^{8}a^{-4}-2z^{8}a^{-6}-2z^{8}a^{-8}+2z^{8}a^{-10}+2z^{7}a^{-3}-7z^{7}a^{-5}-19z^{7}a^{-7}-9z^{7}a^{-9}+z^{7}a^{-11}+2z^{6}a^{-2}-4z^{6}a^{-4}-2z^{6}a^{-6}-6z^{6}a^{-8}-10z^{6}a^{-10}+2z^{5}a^{-1}-2z^{5}a^{-3}+11z^{5}a^{-5}+31z^{5}a^{-7}+11z^{5}a^{-9}-5z^{5}a^{-11}-z^{4}a^{-2}+4z^{4}a^{-4}+7z^{4}a^{-6}+15z^{4}a^{-8}+14z^{4}a^{-10}+z^{4}-3z^{3}a^{-1}+z^{3}a^{-3}-7z^{3}a^{-5}-23z^{3}a^{-7}-6z^{3}a^{-9}+6z^{3}a^{-11}-2z^{2}a^{-2}-5z^{2}a^{-6}-12z^{2}a^{-8}-7z^{2}a^{-10}-2z^{2}+3za^{-5}+6za^{-7}+2za^{-9}-za^{-11}+a^{-6}+2a^{-8}+a^{-10}+1$
The A2 invariant	Data:K11a230/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a230/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a230 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["K11a230"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{2}+13t-19+13t^{-1}-3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-3z^{4}+z^{2}+1$

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]=

{ 51, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{10}+2q^{9}-3q^{8}+5q^{7}-6q^{6}+7q^{5}-8q^{4}+7q^{3}-5q^{2}+4q-2+q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}a^{-2}-z^{4}a^{-4}-z^{4}a^{-6}-z^{2}a^{-2}-z^{2}a^{-6}+2z^{2}a^{-8}+z^{2}-a^{-6}+2a^{-8}-a^{-10}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-6}+z^{10}a^{-8}+2z^{9}a^{-5}+4z^{9}a^{-7}+2z^{9}a^{-9}+2z^{8}a^{-4}-2z^{8}a^{-6}-2z^{8}a^{-8}+2z^{8}a^{-10}+2z^{7}a^{-3}-7z^{7}a^{-5}-19z^{7}a^{-7}-9z^{7}a^{-9}+z^{7}a^{-11}+2z^{6}a^{-2}-4z^{6}a^{-4}-2z^{6}a^{-6}-6z^{6}a^{-8}-10z^{6}a^{-10}+2z^{5}a^{-1}-2z^{5}a^{-3}+11z^{5}a^{-5}+31z^{5}a^{-7}+11z^{5}a^{-9}-5z^{5}a^{-11}-z^{4}a^{-2}+4z^{4}a^{-4}+7z^{4}a^{-6}+15z^{4}a^{-8}+14z^{4}a^{-10}+z^{4}-3z^{3}a^{-1}+z^{3}a^{-3}-7z^{3}a^{-5}-23z^{3}a^{-7}-6z^{3}a^{-9}+6z^{3}a^{-11}-2z^{2}a^{-2}-5z^{2}a^{-6}-12z^{2}a^{-8}-7z^{2}a^{-10}-2z^{2}+3za^{-5}+6za^{-7}+2za^{-9}-za^{-11}+a^{-6}+2a^{-8}+a^{-10}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_36, K11n29,}

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["K11a230"];

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]=

{ $-3t^{2}+13t-19+13t^{-1}-3t^{-2}$, $-q^{10}+2q^{9}-3q^{8}+5q^{7}-6q^{6}+7q^{5}-8q^{4}+7q^{3}-5q^{2}+4q-2+q^{-1}$ }

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_36, K11n29,}

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:

(1, 4)

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 $4$ $32$ $8$ ${\frac {494}{3}}$ ${\frac {82}{3}}$ $128$ ${\frac {2240}{3}}$ ${\frac {128}{3}}$ $224$ ${\frac {32}{3}}$ $512$ ${\frac {1976}{3}}$ ${\frac {328}{3}}$ ${\frac {102751}{30}}$ $-{\frac {15422}{15}}$ ${\frac {108542}{45}}$ ${\frac {4769}{18}}$ ${\frac {10591}{30}}$

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 K11a230. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a229 K11a231