K11a132

K11a131

K11a133

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a132 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a132/ThurstonBennequinNumber Hyperbolic Volume 16.7788 A-Polynomial See Data:K11a132/A-polynomial

[edit Notes for K11a132'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 K11a132's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-13t^{2}+32t-41+32t^{-1}-13t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}-z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{2}+t+1\right\}$
Determinant and Signature	{ 135, 2 }
Jones polynomial	$-q^{8}+4q^{7}-9q^{6}+15q^{5}-19q^{4}+22q^{3}-22q^{2}+18q-13+8q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}-4z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}a^{-6}-2z^{2}+a^{2}-4a^{-2}+4a^{-4}-a^{-6}+1$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}a^{-4}+5z^{9}a^{-1}+12z^{9}a^{-3}+7z^{9}a^{-5}+9z^{8}a^{-2}+14z^{8}a^{-4}+10z^{8}a^{-6}+5z^{8}+3az^{7}-7z^{7}a^{-1}-23z^{7}a^{-3}-5z^{7}a^{-5}+8z^{7}a^{-7}+a^{2}z^{6}-30z^{6}a^{-2}-40z^{6}a^{-4}-17z^{6}a^{-6}+4z^{6}a^{-8}-10z^{6}-7az^{5}+2z^{5}a^{-1}+17z^{5}a^{-3}-5z^{5}a^{-5}-12z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+33z^{4}a^{-2}+42z^{4}a^{-4}+12z^{4}a^{-6}-5z^{4}a^{-8}+5z^{4}+4az^{3}-4z^{3}a^{-1}-8z^{3}a^{-3}+6z^{3}a^{-5}+5z^{3}a^{-7}-z^{3}a^{-9}+3a^{2}z^{2}-20z^{2}a^{-2}-21z^{2}a^{-4}-5z^{2}a^{-6}+z^{2}a^{-8}-2z^{2}+3za^{-1}+3za^{-3}-za^{-5}-za^{-7}-a^{2}+4a^{-2}+4a^{-4}+a^{-6}+1$
The A2 invariant	Data:K11a132/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a132/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a132 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["K11a132"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-13t^{2}+32t-41+32t^{-1}-13t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\left\{2,t^{2}+t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 135, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+4q^{7}-9q^{6}+15q^{5}-19q^{4}+22q^{3}-22q^{2}+18q-13+8q^{-1}-3q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}a^{-4}+5z^{9}a^{-1}+12z^{9}a^{-3}+7z^{9}a^{-5}+9z^{8}a^{-2}+14z^{8}a^{-4}+10z^{8}a^{-6}+5z^{8}+3az^{7}-7z^{7}a^{-1}-23z^{7}a^{-3}-5z^{7}a^{-5}+8z^{7}a^{-7}+a^{2}z^{6}-30z^{6}a^{-2}-40z^{6}a^{-4}-17z^{6}a^{-6}+4z^{6}a^{-8}-10z^{6}-7az^{5}+2z^{5}a^{-1}+17z^{5}a^{-3}-5z^{5}a^{-5}-12z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+33z^{4}a^{-2}+42z^{4}a^{-4}+12z^{4}a^{-6}-5z^{4}a^{-8}+5z^{4}+4az^{3}-4z^{3}a^{-1}-8z^{3}a^{-3}+6z^{3}a^{-5}+5z^{3}a^{-7}-z^{3}a^{-9}+3a^{2}z^{2}-20z^{2}a^{-2}-21z^{2}a^{-4}-5z^{2}a^{-6}+z^{2}a^{-8}-2z^{2}+3za^{-1}+3za^{-3}-za^{-5}-za^{-7}-a^{2}+4a^{-2}+4a^{-4}+a^{-6}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a6, K11a352,}

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["K11a132"];

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]=

{ $2t^{3}-13t^{2}+32t-41+32t^{-1}-13t^{-2}+2t^{-3}$, $-q^{8}+4q^{7}-9q^{6}+15q^{5}-19q^{4}+22q^{3}-22q^{2}+18q-13+8q^{-1}-3q^{-2}+q^{-3}$ }

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]=

{K11a6, K11a352,}

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:

(-2, 0)

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 $-8$ $0$ $32$ ${\frac {212}{3}}$ ${\frac {76}{3}}$ $0$ $32$ $-32$ $32$ $-{\frac {256}{3}}$ $0$ $-{\frac {1696}{3}}$ $-{\frac {608}{3}}$ $-{\frac {9511}{15}}$ $-{\frac {1796}{15}}$ $-{\frac {14524}{45}}$ ${\frac {391}{9}}$ $-{\frac {871}{15}}$

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=$2 is the signature of K11a132. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     3   3 13                   6 1   -5 11                 9 3     6 9               10 6       -4 7             12 9         3 5           10 10           0 3         8 12             -4 1       6 11               5 -1     2 7                 -5 -3   1 6                   5 -5   2                     -2 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a131 K11a133