K11a141

K11a140

K11a142

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a141 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a141/ThurstonBennequinNumber Hyperbolic Volume 14.9456 A-Polynomial See Data:K11a141/A-polynomial

[edit Notes for K11a141'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 K11a141's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-11t^{2}+24t-29+24t^{-1}-11t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 103, -2 }
Jones polynomial	$-q^{4}+4q^{3}-7q^{2}+11q-14+16q^{-1}-16q^{-2}+14q^{-3}-10q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}+a^{6}-2z^{4}a^{4}-4z^{2}a^{4}-2a^{4}+z^{6}a^{2}+2z^{4}a^{2}+2z^{2}a^{2}+2a^{2}+z^{6}+2z^{4}-1-z^{4}a^{-2}-z^{2}a^{-2}+a^{-2}$
Kauffman polynomial (db, data sources)	$2a^{2}z^{10}+2z^{10}+5a^{3}z^{9}+10az^{9}+5z^{9}a^{-1}+6a^{4}z^{8}+5a^{2}z^{8}+4z^{8}a^{-2}+3z^{8}+6a^{5}z^{7}-6a^{3}z^{7}-30az^{7}-17z^{7}a^{-1}+z^{7}a^{-3}+5a^{6}z^{6}-5a^{4}z^{6}-23a^{2}z^{6}-15z^{6}a^{-2}-28z^{6}+3a^{7}z^{5}-5a^{5}z^{5}-a^{3}z^{5}+24az^{5}+14z^{5}a^{-1}-3z^{5}a^{-3}+a^{8}z^{4}-5a^{6}z^{4}-5a^{4}z^{4}+16a^{2}z^{4}+15z^{4}a^{-2}+30z^{4}-3a^{7}z^{3}-7az^{3}-2z^{3}a^{-1}+2z^{3}a^{-3}-a^{8}z^{2}+3a^{6}z^{2}+7a^{4}z^{2}-3z^{2}a^{-2}-6z^{2}+a^{7}z+a^{5}z+a^{3}z+az-a^{6}-2a^{4}-2a^{2}-a^{-2}-1$
The A2 invariant	Data:K11a141/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a141/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a141 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["K11a141"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-11t^{2}+24t-29+24t^{-1}-11t^{-2}+2t^{-3}$

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 103, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+4q^{3}-7q^{2}+11q-14+16q^{-1}-16q^{-2}+14q^{-3}-10q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{2}z^{10}+2z^{10}+5a^{3}z^{9}+10az^{9}+5z^{9}a^{-1}+6a^{4}z^{8}+5a^{2}z^{8}+4z^{8}a^{-2}+3z^{8}+6a^{5}z^{7}-6a^{3}z^{7}-30az^{7}-17z^{7}a^{-1}+z^{7}a^{-3}+5a^{6}z^{6}-5a^{4}z^{6}-23a^{2}z^{6}-15z^{6}a^{-2}-28z^{6}+3a^{7}z^{5}-5a^{5}z^{5}-a^{3}z^{5}+24az^{5}+14z^{5}a^{-1}-3z^{5}a^{-3}+a^{8}z^{4}-5a^{6}z^{4}-5a^{4}z^{4}+16a^{2}z^{4}+15z^{4}a^{-2}+30z^{4}-3a^{7}z^{3}-7az^{3}-2z^{3}a^{-1}+2z^{3}a^{-3}-a^{8}z^{2}+3a^{6}z^{2}+7a^{4}z^{2}-3z^{2}a^{-2}-6z^{2}+a^{7}z+a^{5}z+a^{3}z+az-a^{6}-2a^{4}-2a^{2}-a^{-2}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a12,}

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["K11a141"];

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]=

{ $2t^{3}-11t^{2}+24t-29+24t^{-1}-11t^{-2}+2t^{-3}$, $-q^{4}+4q^{3}-7q^{2}+11q-14+16q^{-1}-16q^{-2}+14q^{-3}-10q^{-4}+6q^{-5}-3q^{-6}+q^{-7}$ }

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]=

{K11a12,}

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, 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 $-8$ $16$ $32$ ${\frac {20}{3}}$ ${\frac {28}{3}}$ $-128$ $-{\frac {512}{3}}$ $-{\frac {224}{3}}$ $-16$ $-{\frac {256}{3}}$ $128$ $-{\frac {160}{3}}$ $-{\frac {224}{3}}$ ${\frac {5849}{15}}$ $-{\frac {252}{5}}$ ${\frac {11636}{45}}$ ${\frac {7}{9}}$ ${\frac {569}{15}}$

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 $s=$-2 is the signature of K11a141. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a140 K11a142