K11n163

K11n162

K11n164

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n163 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X3,11,4,10 X5,12,6,13 X22,8,1,7 X9,16,10,17 X11,19,12,18 X8,14,9,13 X20,16,21,15 X17,4,18,5 X19,3,20,2 X14,22,15,21
Gauss code	1, 10, -2, 9, -3, -1, 4, -7, -5, 2, -6, 3, 7, -11, 8, 5, -9, 6, -10, -8, 11, -4
Dowker-Thistlethwaite code	6 -10 -12 22 -16 -18 8 20 -4 -2 14

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:K11n163/ThurstonBennequinNumber Hyperbolic Volume 16.1487 A-Polynomial See Data:K11n163/A-polynomial

[edit Notes for K11n163'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 K11n163's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-8t^{2}+22t-29+22t^{-1}-8t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 91, 2 }
Jones polynomial	$2q^{7}-6q^{6}+10q^{5}-14q^{4}+16q^{3}-15q^{2}+13q-9+5q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+2z^{4}a^{-2}-3z^{4}a^{-4}-z^{4}+2z^{2}a^{-2}-5z^{2}a^{-4}+2z^{2}a^{-6}+a^{-2}-2a^{-4}+a^{-6}+1$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-3}+2z^{9}a^{-5}+7z^{8}a^{-2}+10z^{8}a^{-4}+3z^{8}a^{-6}+9z^{7}a^{-1}+12z^{7}a^{-3}+4z^{7}a^{-5}+z^{7}a^{-7}-5z^{6}a^{-2}-11z^{6}a^{-4}-z^{6}a^{-6}+5z^{6}+az^{5}-15z^{5}a^{-1}-26z^{5}a^{-3}-5z^{5}a^{-5}+5z^{5}a^{-7}-9z^{4}a^{-2}-5z^{4}a^{-4}+z^{4}a^{-6}+3z^{4}a^{-8}-6z^{4}+4z^{3}a^{-1}+9z^{3}a^{-3}-2z^{3}a^{-5}-7z^{3}a^{-7}+5z^{2}a^{-2}+9z^{2}a^{-4}+z^{2}a^{-6}-3z^{2}a^{-8}+za^{-3}+3za^{-5}+2za^{-7}-a^{-2}-2a^{-4}-a^{-6}+1$
The A2 invariant	Data:K11n163/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n163/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n163 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["K11n163"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-8t^{2}+22t-29+22t^{-1}-8t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 91, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$2q^{7}-6q^{6}+10q^{5}-14q^{4}+16q^{3}-15q^{2}+13q-9+5q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+2z^{4}a^{-2}-3z^{4}a^{-4}-z^{4}+2z^{2}a^{-2}-5z^{2}a^{-4}+2z^{2}a^{-6}+a^{-2}-2a^{-4}+a^{-6}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-3}+2z^{9}a^{-5}+7z^{8}a^{-2}+10z^{8}a^{-4}+3z^{8}a^{-6}+9z^{7}a^{-1}+12z^{7}a^{-3}+4z^{7}a^{-5}+z^{7}a^{-7}-5z^{6}a^{-2}-11z^{6}a^{-4}-z^{6}a^{-6}+5z^{6}+az^{5}-15z^{5}a^{-1}-26z^{5}a^{-3}-5z^{5}a^{-5}+5z^{5}a^{-7}-9z^{4}a^{-2}-5z^{4}a^{-4}+z^{4}a^{-6}+3z^{4}a^{-8}-6z^{4}+4z^{3}a^{-1}+9z^{3}a^{-3}-2z^{3}a^{-5}-7z^{3}a^{-7}+5z^{2}a^{-2}+9z^{2}a^{-4}+z^{2}a^{-6}-3z^{2}a^{-8}+za^{-3}+3za^{-5}+2za^{-7}-a^{-2}-2a^{-4}-a^{-6}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_105,}

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["K11n163"];

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]=

{ $t^{3}-8t^{2}+22t-29+22t^{-1}-8t^{-2}+t^{-3}$, $2q^{7}-6q^{6}+10q^{5}-14q^{4}+16q^{3}-15q^{2}+13q-9+5q^{-1}-q^{-2}$ }

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]=

{10_105,}

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, -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 $-4$ $-16$ $8$ $-{\frac {14}{3}}$ ${\frac {62}{3}}$ $64$ ${\frac {416}{3}}$ ${\frac {224}{3}}$ $48$ $-{\frac {32}{3}}$ $128$ ${\frac {56}{3}}$ $-{\frac {248}{3}}$ ${\frac {14849}{30}}$ ${\frac {2302}{15}}$ ${\frac {5938}{45}}$ ${\frac {415}{18}}$ $-{\frac {991}{30}}$

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 K11n163. 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 χ 15                   2 2 13                 4   -4 11               6 2   4 9             8 4     -4 7           8 6       2 5         7 8         1 3       6 8           -2 1     4 8             4 -1   1 5               -4 -3   4                 4 -5 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

K11n162 K11n164