K11a188

K11a187

K11a189

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a188 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a188/ThurstonBennequinNumber Hyperbolic Volume 10.6589 A-Polynomial See Data:K11a188/A-polynomial

[edit Notes for K11a188's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for K11a188's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-9t^{2}+15t-15+15t^{-1}-9t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+3z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 67, 2 }
Jones polynomial	$-q^{6}+3q^{5}-5q^{4}+8q^{3}-9q^{2}+10q-10+8q^{-1}-6q^{-2}+4q^{-3}-2q^{-4}+q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}-2a^{2}z^{4}+3z^{4}a^{-2}-z^{4}a^{-4}+3z^{4}+a^{4}z^{2}-6a^{2}z^{2}+2z^{2}a^{-2}-2z^{2}a^{-4}+2z^{2}+2a^{4}-3a^{2}+a^{-2}+1$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+2a^{3}z^{9}+5az^{9}+3z^{9}a^{-1}+a^{4}z^{8}-a^{2}z^{8}+5z^{8}a^{-2}+3z^{8}-11a^{3}z^{7}-21az^{7}-3z^{7}a^{-1}+7z^{7}a^{-3}-6a^{4}z^{6}-13a^{2}z^{6}-7z^{6}a^{-2}+7z^{6}a^{-4}-21z^{6}+19a^{3}z^{5}+24az^{5}-13z^{5}a^{-1}-13z^{5}a^{-3}+5z^{5}a^{-5}+12a^{4}z^{4}+30a^{2}z^{4}-8z^{4}a^{-2}-10z^{4}a^{-4}+3z^{4}a^{-6}+23z^{4}-11a^{3}z^{3}-6az^{3}+15z^{3}a^{-1}+6z^{3}a^{-3}-3z^{3}a^{-5}+z^{3}a^{-7}-9a^{4}z^{2}-19a^{2}z^{2}+8z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}a^{-6}-7z^{2}+2a^{3}z-az-5za^{-1}-2za^{-3}+2a^{4}+3a^{2}-a^{-2}+1$
The A2 invariant	Data:K11a188/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a188/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a188 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["K11a188"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-9t^{2}+15t-15+15t^{-1}-9t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{6}+3z^{4}-3z^{2}+1$

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]=

{ 67, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+z^{6}-2a^{2}z^{4}+3z^{4}a^{-2}-z^{4}a^{-4}+3z^{4}+a^{4}z^{2}-6a^{2}z^{2}+2z^{2}a^{-2}-2z^{2}a^{-4}+2z^{2}+2a^{4}-3a^{2}+a^{-2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+2a^{3}z^{9}+5az^{9}+3z^{9}a^{-1}+a^{4}z^{8}-a^{2}z^{8}+5z^{8}a^{-2}+3z^{8}-11a^{3}z^{7}-21az^{7}-3z^{7}a^{-1}+7z^{7}a^{-3}-6a^{4}z^{6}-13a^{2}z^{6}-7z^{6}a^{-2}+7z^{6}a^{-4}-21z^{6}+19a^{3}z^{5}+24az^{5}-13z^{5}a^{-1}-13z^{5}a^{-3}+5z^{5}a^{-5}+12a^{4}z^{4}+30a^{2}z^{4}-8z^{4}a^{-2}-10z^{4}a^{-4}+3z^{4}a^{-6}+23z^{4}-11a^{3}z^{3}-6az^{3}+15z^{3}a^{-1}+6z^{3}a^{-3}-3z^{3}a^{-5}+z^{3}a^{-7}-9a^{4}z^{2}-19a^{2}z^{2}+8z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}a^{-6}-7z^{2}+2a^{3}z-az-5za^{-1}-2za^{-3}+2a^{4}+3a^{2}-a^{-2}+1$

"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["K11a188"];

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]=

{ $2t^{3}-9t^{2}+15t-15+15t^{-1}-9t^{-2}+2t^{-3}$, $-q^{6}+3q^{5}-5q^{4}+8q^{3}-9q^{2}+10q-10+8q^{-1}-6q^{-2}+4q^{-3}-2q^{-4}+q^{-5}$ }

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:

(-3, 2)

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 $-12$ $16$ $72$ $66$ $14$ $-192$ $-{\frac {992}{3}}$ $-{\frac {416}{3}}$ $-16$ $-288$ $128$ $-792$ $-168$ $-{\frac {3631}{10}}$ $-{\frac {4694}{15}}$ ${\frac {2578}{15}}$ $-{\frac {17}{6}}$ ${\frac {529}{10}}$

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 K11a188. 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 13                       1 -1 11                     2   2 9                   3 1   -2 7                 5 2     3 5               4 3       -1 3             6 5         1 1           5 5           0 -1         3 5             -2 -3       3 5               2 -5     1 3                 -2 -7   1 3                   2 -9   1                     -1 -11 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a187 K11a189