K11a135

K11a134

K11a136

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a135 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X16,6,17,5 X14,7,15,8 X12,10,13,9 X2,11,3,12 X18,14,19,13 X20,15,21,16 X22,18,1,17 X8,20,9,19 X6,21,7,22
Gauss code	1, -6, 2, -1, 3, -11, 4, -10, 5, -2, 6, -5, 7, -4, 8, -3, 9, -7, 10, -8, 11, -9
Dowker-Thistlethwaite code	4 10 16 14 12 2 18 20 22 8 6

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:K11a135/ThurstonBennequinNumber Hyperbolic Volume 17.6564 A-Polynomial See Data:K11a135/A-polynomial

[edit Notes for K11a135's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $[2,3]$ Rasmussen s-Invariant 0

[edit Notes for K11a135's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{3}+13t^{2}-36t+51-36t^{-1}+13t^{-2}-2t^{-3}$
Conway polynomial	$-2z^{6}+z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 153, 0 }
Jones polynomial	$q^{6}-5q^{5}+11q^{4}-16q^{3}+22q^{2}-25q+24-21q^{-1}+15q^{-2}-8q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-2}-z^{6}+2a^{2}z^{4}-z^{4}a^{-2}+z^{4}a^{-4}-z^{4}-a^{4}z^{2}+2a^{2}z^{2}-3z^{2}+2a^{2}+2a^{-2}-3$
Kauffman polynomial (db, data sources)	$3z^{10}a^{-2}+3z^{10}+7az^{9}+16z^{9}a^{-1}+9z^{9}a^{-3}+8a^{2}z^{8}+13z^{8}a^{-2}+10z^{8}a^{-4}+11z^{8}+7a^{3}z^{7}-2az^{7}-29z^{7}a^{-1}-15z^{7}a^{-3}+5z^{7}a^{-5}+4a^{4}z^{6}-5a^{2}z^{6}-41z^{6}a^{-2}-23z^{6}a^{-4}+z^{6}a^{-6}-26z^{6}+a^{5}z^{5}-9a^{3}z^{5}-10az^{5}+10z^{5}a^{-1}+z^{5}a^{-3}-9z^{5}a^{-5}-6a^{4}z^{4}-8a^{2}z^{4}+24z^{4}a^{-2}+12z^{4}a^{-4}-z^{4}a^{-6}+9z^{4}-a^{5}z^{3}+4a^{3}z^{3}+11az^{3}+7z^{3}a^{-1}+3z^{3}a^{-3}+2z^{3}a^{-5}+3a^{4}z^{2}+9a^{2}z^{2}+3z^{2}a^{-2}+9z^{2}-a^{3}z-5az-5za^{-1}-za^{-3}-2a^{2}-2a^{-2}-3$
The A2 invariant	Data:K11a135/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a135/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a135 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["K11a135"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{3}+13t^{2}-36t+51-36t^{-1}+13t^{-2}-2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{3,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 153, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-5q^{5}+11q^{4}-16q^{3}+22q^{2}-25q+24-21q^{-1}+15q^{-2}-8q^{-3}+4q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3z^{10}a^{-2}+3z^{10}+7az^{9}+16z^{9}a^{-1}+9z^{9}a^{-3}+8a^{2}z^{8}+13z^{8}a^{-2}+10z^{8}a^{-4}+11z^{8}+7a^{3}z^{7}-2az^{7}-29z^{7}a^{-1}-15z^{7}a^{-3}+5z^{7}a^{-5}+4a^{4}z^{6}-5a^{2}z^{6}-41z^{6}a^{-2}-23z^{6}a^{-4}+z^{6}a^{-6}-26z^{6}+a^{5}z^{5}-9a^{3}z^{5}-10az^{5}+10z^{5}a^{-1}+z^{5}a^{-3}-9z^{5}a^{-5}-6a^{4}z^{4}-8a^{2}z^{4}+24z^{4}a^{-2}+12z^{4}a^{-4}-z^{4}a^{-6}+9z^{4}-a^{5}z^{3}+4a^{3}z^{3}+11az^{3}+7z^{3}a^{-1}+3z^{3}a^{-3}+2z^{3}a^{-5}+3a^{4}z^{2}+9a^{2}z^{2}+3z^{2}a^{-2}+9z^{2}-a^{3}z-5az-5za^{-1}-za^{-3}-2a^{2}-2a^{-2}-3$

"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["K11a135"];

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]=

{ $-2t^{3}+13t^{2}-36t+51-36t^{-1}+13t^{-2}-2t^{-3}$, $q^{6}-5q^{5}+11q^{4}-16q^{3}+22q^{2}-25q+24-21q^{-1}+15q^{-2}-8q^{-3}+4q^{-4}-q^{-5}$ }

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:

(-2, 0)

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 $-8$ $0$ $32$ ${\frac {116}{3}}$ ${\frac {28}{3}}$ $0$ $0$ $32$ $-32$ $-{\frac {256}{3}}$ $0$ $-{\frac {928}{3}}$ $-{\frac {224}{3}}$ $-{\frac {4471}{15}}$ ${\frac {364}{15}}$ $-{\frac {9964}{45}}$ ${\frac {439}{9}}$ $-{\frac {631}{15}}$

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=$0 is the signature of K11a135. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a134 K11a136