K11n95

K11n94

K11n96

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n95 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X14,6,15,5 X7,13,8,12 X2,10,3,9 X11,21,12,20 X18,14,19,13 X15,22,16,1 X6,18,7,17 X19,9,20,8 X21,16,22,17
Gauss code	1, -5, 2, -1, 3, -9, -4, 10, 5, -2, -6, 4, 7, -3, -8, 11, 9, -7, -10, 6, -11, 8
Dowker-Thistlethwaite code	4 10 14 -12 2 -20 18 -22 6 -8 -16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n95/ThurstonBennequinNumber Hyperbolic Volume 11.1971 A-Polynomial See Data:K11n95/A-polynomial

[edit Notes for K11n95's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n95's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+5t^{2}-7t+7-7t^{-1}+5t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 33, 4 }
Jones polynomial	$-2q^{9}+4q^{8}-5q^{7}+6q^{6}-6q^{5}+5q^{4}-3q^{3}+2q^{2}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-6}+2z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}a^{-8}+6z^{2}a^{-4}-5z^{2}a^{-6}+3z^{2}a^{-8}+3a^{-4}-3a^{-6}+2a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-6}+z^{8}a^{-8}+z^{7}a^{-5}+3z^{7}a^{-7}+2z^{7}a^{-9}-2z^{6}a^{-6}-z^{6}a^{-8}+z^{6}a^{-10}-z^{5}a^{-5}-5z^{5}a^{-7}-4z^{5}a^{-9}+3z^{4}a^{-4}+6z^{4}a^{-6}+5z^{4}a^{-8}+2z^{4}a^{-10}+4z^{3}a^{-7}+7z^{3}a^{-9}+3z^{3}a^{-11}-7z^{2}a^{-4}-9z^{2}a^{-6}-6z^{2}a^{-8}-4z^{2}a^{-10}-za^{-5}-za^{-7}-3za^{-9}-3za^{-11}+3a^{-4}+3a^{-6}+2a^{-8}+a^{-10}$
The A2 invariant	Data:K11n95/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n95/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n95 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["K11n95"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+5t^{2}-7t+7-7t^{-1}+5t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 33, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-6}+2z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}a^{-8}+6z^{2}a^{-4}-5z^{2}a^{-6}+3z^{2}a^{-8}+3a^{-4}-3a^{-6}+2a^{-8}-a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-6}+z^{8}a^{-8}+z^{7}a^{-5}+3z^{7}a^{-7}+2z^{7}a^{-9}-2z^{6}a^{-6}-z^{6}a^{-8}+z^{6}a^{-10}-z^{5}a^{-5}-5z^{5}a^{-7}-4z^{5}a^{-9}+3z^{4}a^{-4}+6z^{4}a^{-6}+5z^{4}a^{-8}+2z^{4}a^{-10}+4z^{3}a^{-7}+7z^{3}a^{-9}+3z^{3}a^{-11}-7z^{2}a^{-4}-9z^{2}a^{-6}-6z^{2}a^{-8}-4z^{2}a^{-10}-za^{-5}-za^{-7}-3za^{-9}-3za^{-11}+3a^{-4}+3a^{-6}+2a^{-8}+a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_11,}

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["K11n95"];

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]=

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

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]=

{9_11,}

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:

(4, 9)

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 $16$ $72$ $128$ ${\frac {1208}{3}}$ ${\frac {208}{3}}$ $1152$ $2416$ $448$ $360$ ${\frac {2048}{3}}$ $2592$ ${\frac {19328}{3}}$ ${\frac {3328}{3}}$ ${\frac {219182}{15}}$ ${\frac {384}{5}}$ ${\frac {285008}{45}}$ ${\frac {1090}{9}}$ ${\frac {12782}{15}}$

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=$4 is the signature of K11n95. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n94 K11n96