K11n36

K11n35

K11n37

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n36 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,13,6,12 X2837 X9,17,10,16 X11,18,12,19 X13,7,14,6 X15,20,16,21 X17,1,18,22 X19,14,20,15 X21,10,22,11
Gauss code	1, -4, 2, -1, -3, 7, 4, -2, -5, 11, -6, 3, -7, 10, -8, 5, -9, 6, -10, 8, -11, 9
Dowker-Thistlethwaite code	4 8 -12 2 -16 -18 -6 -20 -22 -14 -10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n36/ThurstonBennequinNumber Hyperbolic Volume 14.4828 A-Polynomial See Data:K11n36/A-polynomial

[edit Notes for K11n36's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n36's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+4t^{3}-8t^{2}+13t-15+13t^{-1}-8t^{-2}+4t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-4z^{6}-4z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 67, 2 }
Jones polynomial	$-2q^{6}+5q^{5}-8q^{4}+11q^{3}-11q^{2}+11q-9+6q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-13z^{4}a^{-2}+5z^{4}a^{-4}+4z^{4}-12z^{2}a^{-2}+9z^{2}a^{-4}-z^{2}a^{-6}+5z^{2}-4a^{-2}+5a^{-4}-2a^{-6}+2$
Kauffman polynomial (db, data sources)	$2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+5z^{8}a^{-4}+4z^{8}+3az^{7}+z^{7}a^{-1}+2z^{7}a^{-3}+4z^{7}a^{-5}+a^{2}z^{6}-27z^{6}a^{-2}-14z^{6}a^{-4}+z^{6}a^{-6}-11z^{6}-9az^{5}-15z^{5}a^{-1}-14z^{5}a^{-3}-8z^{5}a^{-5}-3a^{2}z^{4}+30z^{4}a^{-2}+23z^{4}a^{-4}+4z^{4}a^{-6}+8z^{4}+7az^{3}+14z^{3}a^{-1}+17z^{3}a^{-3}+13z^{3}a^{-5}+3z^{3}a^{-7}+2a^{2}z^{2}-18z^{2}a^{-2}-16z^{2}a^{-4}-5z^{2}a^{-6}-5z^{2}-2az-5za^{-1}-6za^{-3}-6za^{-5}-3za^{-7}+4a^{-2}+5a^{-4}+2a^{-6}+2$
The A2 invariant	$q^{8}-q^{6}+2q^{4}-q^{2}+q^{-2}-3q^{-4}+3q^{-6}-2q^{-8}+3q^{-10}+q^{-12}+2q^{-16}-2q^{-18}-q^{-22}$
The G2 invariant	Data:K11n36/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n36 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["K11n36"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+4t^{3}-8t^{2}+13t-15+13t^{-1}-8t^{-2}+4t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-4z^{6}-4z^{4}+z^{2}+1$

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]=

{ 67, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-2q^{6}+5q^{5}-8q^{4}+11q^{3}-11q^{2}+11q-9+6q^{-1}-3q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-13z^{4}a^{-2}+5z^{4}a^{-4}+4z^{4}-12z^{2}a^{-2}+9z^{2}a^{-4}-z^{2}a^{-6}+5z^{2}-4a^{-2}+5a^{-4}-2a^{-6}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+5z^{8}a^{-4}+4z^{8}+3az^{7}+z^{7}a^{-1}+2z^{7}a^{-3}+4z^{7}a^{-5}+a^{2}z^{6}-27z^{6}a^{-2}-14z^{6}a^{-4}+z^{6}a^{-6}-11z^{6}-9az^{5}-15z^{5}a^{-1}-14z^{5}a^{-3}-8z^{5}a^{-5}-3a^{2}z^{4}+30z^{4}a^{-2}+23z^{4}a^{-4}+4z^{4}a^{-6}+8z^{4}+7az^{3}+14z^{3}a^{-1}+17z^{3}a^{-3}+13z^{3}a^{-5}+3z^{3}a^{-7}+2a^{2}z^{2}-18z^{2}a^{-2}-16z^{2}a^{-4}-5z^{2}a^{-6}-5z^{2}-2az-5za^{-1}-6za^{-3}-6za^{-5}-3za^{-7}+4a^{-2}+5a^{-4}+2a^{-6}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n44,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n7, K11n44,}

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["K11n36"];

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]=

{ $-t^{4}+4t^{3}-8t^{2}+13t-15+13t^{-1}-8t^{-2}+4t^{-3}-t^{-4}$, $-2q^{6}+5q^{5}-8q^{4}+11q^{3}-11q^{2}+11q-9+6q^{-1}-3q^{-2}+q^{-3}$ }

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]=

{K11n44,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n7, K11n44,}

Vassiliev invariants

V2 and V3:

(1, 3)

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 $4$ $24$ $8$ ${\frac {302}{3}}$ ${\frac {106}{3}}$ $96$ $304$ $64$ $56$ ${\frac {32}{3}}$ $288$ ${\frac {1208}{3}}$ ${\frac {424}{3}}$ ${\frac {37951}{30}}$ ${\frac {366}{5}}$ ${\frac {22022}{45}}$ ${\frac {1121}{18}}$ ${\frac {1471}{30}}$

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 K11n36. 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 χ 13                   2 -2 11                 3   3 9               5 2   -3 7             6 3     3 5           5 5       0 3         6 6         0 1       4 6           2 -1     2 5             -3 -3   1 4               3 -5   2                 -2 -7 1                   1

Integral Khovanov Homology (db, data source)

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

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.

K11n35 K11n37