K11a32

K11a31

K11a33

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a32 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,6,15,5 X2837 X16,10,17,9 X18,11,19,12 X6,14,7,13 X22,16,1,15 X20,17,21,18 X12,19,13,20 X10,22,11,21
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -6, 10, -9, 11, -8
Dowker-Thistlethwaite code	4 8 14 2 16 18 6 22 20 12 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a32/ThurstonBennequinNumber Hyperbolic Volume 16.0547 A-Polynomial See Data:K11a32/A-polynomial

[edit Notes for K11a32'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 K11a32's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{3}-14t^{2}+32t-41+32t^{-1}-14t^{-2}+3t^{-3}$
Conway polynomial	$3z^{6}+4z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 139, 2 }
Jones polynomial	$q^{9}-4q^{8}+8q^{7}-14q^{6}+19q^{5}-22q^{4}+23q^{3}-19q^{2}+15q-9+4q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+2z^{6}a^{-4}+z^{4}a^{-2}+7z^{4}a^{-4}-3z^{4}a^{-6}-z^{4}-z^{2}a^{-2}+11z^{2}a^{-4}-7z^{2}a^{-6}+z^{2}a^{-8}-z^{2}-a^{-2}+6a^{-4}-5a^{-6}+a^{-8}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-4}+z^{10}a^{-6}+5z^{9}a^{-3}+9z^{9}a^{-5}+4z^{9}a^{-7}+9z^{8}a^{-2}+20z^{8}a^{-4}+17z^{8}a^{-6}+6z^{8}a^{-8}+8z^{7}a^{-1}+9z^{7}a^{-3}+2z^{7}a^{-5}+5z^{7}a^{-7}+4z^{7}a^{-9}-11z^{6}a^{-2}-45z^{6}a^{-4}-43z^{6}a^{-6}-12z^{6}a^{-8}+z^{6}a^{-10}+4z^{6}+az^{5}-12z^{5}a^{-1}-32z^{5}a^{-3}-40z^{5}a^{-5}-31z^{5}a^{-7}-10z^{5}a^{-9}+5z^{4}a^{-2}+37z^{4}a^{-4}+35z^{4}a^{-6}+6z^{4}a^{-8}-2z^{4}a^{-10}-5z^{4}-az^{3}+7z^{3}a^{-1}+26z^{3}a^{-3}+41z^{3}a^{-5}+31z^{3}a^{-7}+8z^{3}a^{-9}-3z^{2}a^{-2}-19z^{2}a^{-4}-16z^{2}a^{-6}-z^{2}a^{-8}+z^{2}a^{-10}+2z^{2}-2za^{-1}-7za^{-3}-13za^{-5}-10za^{-7}-2za^{-9}+a^{-2}+6a^{-4}+5a^{-6}+a^{-8}$
The A2 invariant	Data:K11a32/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a32/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a32 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["K11a32"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{3}-14t^{2}+32t-41+32t^{-1}-14t^{-2}+3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 139, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{9}-4q^{8}+8q^{7}-14q^{6}+19q^{5}-22q^{4}+23q^{3}-19q^{2}+15q-9+4q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+2z^{6}a^{-4}+z^{4}a^{-2}+7z^{4}a^{-4}-3z^{4}a^{-6}-z^{4}-z^{2}a^{-2}+11z^{2}a^{-4}-7z^{2}a^{-6}+z^{2}a^{-8}-z^{2}-a^{-2}+6a^{-4}-5a^{-6}+a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-4}+z^{10}a^{-6}+5z^{9}a^{-3}+9z^{9}a^{-5}+4z^{9}a^{-7}+9z^{8}a^{-2}+20z^{8}a^{-4}+17z^{8}a^{-6}+6z^{8}a^{-8}+8z^{7}a^{-1}+9z^{7}a^{-3}+2z^{7}a^{-5}+5z^{7}a^{-7}+4z^{7}a^{-9}-11z^{6}a^{-2}-45z^{6}a^{-4}-43z^{6}a^{-6}-12z^{6}a^{-8}+z^{6}a^{-10}+4z^{6}+az^{5}-12z^{5}a^{-1}-32z^{5}a^{-3}-40z^{5}a^{-5}-31z^{5}a^{-7}-10z^{5}a^{-9}+5z^{4}a^{-2}+37z^{4}a^{-4}+35z^{4}a^{-6}+6z^{4}a^{-8}-2z^{4}a^{-10}-5z^{4}-az^{3}+7z^{3}a^{-1}+26z^{3}a^{-3}+41z^{3}a^{-5}+31z^{3}a^{-7}+8z^{3}a^{-9}-3z^{2}a^{-2}-19z^{2}a^{-4}-16z^{2}a^{-6}-z^{2}a^{-8}+z^{2}a^{-10}+2z^{2}-2za^{-1}-7za^{-3}-13za^{-5}-10za^{-7}-2za^{-9}+a^{-2}+6a^{-4}+5a^{-6}+a^{-8}$

"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["K11a32"];

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]=

{ $3t^{3}-14t^{2}+32t-41+32t^{-1}-14t^{-2}+3t^{-3}$, $q^{9}-4q^{8}+8q^{7}-14q^{6}+19q^{5}-22q^{4}+23q^{3}-19q^{2}+15q-9+4q^{-1}-q^{-2}$ }

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:

(3, 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 $12$ $32$ $72$ $110$ $2$ $384$ ${\frac {1472}{3}}$ ${\frac {224}{3}}$ $32$ $288$ $512$ $1320$ $24$ ${\frac {21471}{10}}$ ${\frac {3494}{15}}$ ${\frac {2234}{5}}$ ${\frac {43}{2}}$ ${\frac {191}{10}}$

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 K11a32. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     3   -3 15                   5 1   4 13                 9 3     -6 11               10 5       5 9             12 9         -3 7           11 10           1 5         8 12             4 3       7 11               -4 1     3 9                 6 -1   1 6                   -5 -3   3                     3 -5 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11a31 K11a33