K11n135

K11n134

K11n136

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n135 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X12,3,13,4 X14,6,15,5 X7,17,8,16 X22,10,1,9 X11,19,12,18 X2,13,3,14 X15,20,16,21 X17,11,18,10 X19,7,20,6 X8,22,9,21
Gauss code	1, -7, 2, -1, 3, 10, -4, -11, 5, 9, -6, -2, 7, -3, -8, 4, -9, 6, -10, 8, 11, -5
Dowker-Thistlethwaite code	4 12 14 -16 22 -18 2 -20 -10 -6 8

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:K11n135/ThurstonBennequinNumber Hyperbolic Volume 8.03988 A-Polynomial See Data:K11n135/A-polynomial

[edit Notes for K11n135'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 K11n135's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+2t^{2}-1+2t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-4z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 5, 4 }
Jones polynomial	$q^{9}-q^{8}-q^{5}+q^{4}-q^{3}+2q^{2}-q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}+z^{4}a^{-2}-5z^{4}a^{-4}+4z^{2}a^{-2}-6z^{2}a^{-4}+z^{2}a^{-8}+3a^{-2}-2a^{-4}-a^{-6}+a^{-8}$
Kauffman polynomial (db, data sources)	$z^{7}a^{-3}+z^{7}a^{-9}+z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}a^{-10}-4z^{5}a^{-3}+z^{5}a^{-5}-z^{5}a^{-7}-6z^{5}a^{-9}-5z^{4}a^{-2}-9z^{4}a^{-4}-z^{4}a^{-8}-5z^{4}a^{-10}+2z^{3}a^{-3}-4z^{3}a^{-5}+3z^{3}a^{-7}+9z^{3}a^{-9}+7z^{2}a^{-2}+9z^{2}a^{-4}-2z^{2}a^{-6}+z^{2}a^{-8}+5z^{2}a^{-10}+2za^{-3}+2za^{-5}-3za^{-7}-3za^{-9}-3a^{-2}-2a^{-4}+a^{-6}+a^{-8}$
The A2 invariant	$1+q^{-2}+q^{-4}+q^{-6}+q^{-8}+q^{-10}-q^{-12}-2q^{-16}-q^{-18}-q^{-20}+q^{-24}+q^{-28}$
The G2 invariant	Data:K11n135/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n135 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["K11n135"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 5, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n19,}

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["K11n135"];

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]=

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

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]=

{K11n19,}

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, -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 $-4$ $-32$ $8$ $-{\frac {158}{3}}$ ${\frac {110}{3}}$ $128$ ${\frac {736}{3}}$ ${\frac {352}{3}}$ $192$ $-{\frac {32}{3}}$ $512$ ${\frac {632}{3}}$ $-{\frac {440}{3}}$ ${\frac {57329}{30}}$ $-{\frac {806}{5}}$ ${\frac {55378}{45}}$ ${\frac {4495}{18}}$ ${\frac {1649}{30}}$

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=$4 is the signature of K11n135. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n134 K11n136