K11a240

K11a239

K11a241

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a240 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 4 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a240/ThurstonBennequinNumber Hyperbolic Volume 10.6981 A-Polynomial See Data:K11a240/A-polynomial

[edit Notes for K11a240's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant -8

[edit Notes for K11a240's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{4}-5t^{3}+8t^{2}-10t+11-10t^{-1}+8t^{-2}-5t^{-3}+2t^{-4}$
Conway polynomial	$2z^{8}+11z^{6}+18z^{4}+9z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 61, 8 }
Jones polynomial	$-q^{15}+3q^{14}-5q^{13}+7q^{12}-9q^{11}+9q^{10}-9q^{9}+7q^{8}-5q^{7}+4q^{6}-q^{5}+q^{4}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-8}+z^{8}a^{-10}+7z^{6}a^{-8}+5z^{6}a^{-10}-z^{6}a^{-12}+17z^{4}a^{-8}+5z^{4}a^{-10}-4z^{4}a^{-12}+17z^{2}a^{-8}-5z^{2}a^{-10}-3z^{2}a^{-12}+6a^{-8}-6a^{-10}+a^{-12}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-10}+z^{10}a^{-12}+z^{9}a^{-9}+4z^{9}a^{-11}+3z^{9}a^{-13}+z^{8}a^{-8}-3z^{8}a^{-10}+z^{8}a^{-12}+5z^{8}a^{-14}-4z^{7}a^{-9}-16z^{7}a^{-11}-6z^{7}a^{-13}+6z^{7}a^{-15}-7z^{6}a^{-8}-3z^{6}a^{-10}-13z^{6}a^{-12}-11z^{6}a^{-14}+6z^{6}a^{-16}+z^{5}a^{-9}+13z^{5}a^{-11}-3z^{5}a^{-13}-10z^{5}a^{-15}+5z^{5}a^{-17}+17z^{4}a^{-8}+15z^{4}a^{-10}+12z^{4}a^{-12}+4z^{4}a^{-14}-7z^{4}a^{-16}+3z^{4}a^{-18}+9z^{3}a^{-9}+7z^{3}a^{-11}+5z^{3}a^{-13}+2z^{3}a^{-15}-4z^{3}a^{-17}+z^{3}a^{-19}-17z^{2}a^{-8}-15z^{2}a^{-10}-z^{2}a^{-12}-z^{2}a^{-14}+z^{2}a^{-16}-z^{2}a^{-18}-7za^{-9}-7za^{-11}-za^{-13}+za^{-17}+6a^{-8}+6a^{-10}+a^{-12}$
The A2 invariant	$q^{-14}+3q^{-18}+q^{-20}+3q^{-22}+q^{-24}-q^{-26}-4q^{-30}-2q^{-34}+q^{-38}+q^{-42}-q^{-44}$
The G2 invariant	Data:K11a240/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a240 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["K11a240"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{4}-5t^{3}+8t^{2}-10t+11-10t^{-1}+8t^{-2}-5t^{-3}+2t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{8}+11z^{6}+18z^{4}+9z^{2}+1$

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]=

{ 61, 8 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{15}+3q^{14}-5q^{13}+7q^{12}-9q^{11}+9q^{10}-9q^{9}+7q^{8}-5q^{7}+4q^{6}-q^{5}+q^{4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-8}+z^{8}a^{-10}+7z^{6}a^{-8}+5z^{6}a^{-10}-z^{6}a^{-12}+17z^{4}a^{-8}+5z^{4}a^{-10}-4z^{4}a^{-12}+17z^{2}a^{-8}-5z^{2}a^{-10}-3z^{2}a^{-12}+6a^{-8}-6a^{-10}+a^{-12}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-10}+z^{10}a^{-12}+z^{9}a^{-9}+4z^{9}a^{-11}+3z^{9}a^{-13}+z^{8}a^{-8}-3z^{8}a^{-10}+z^{8}a^{-12}+5z^{8}a^{-14}-4z^{7}a^{-9}-16z^{7}a^{-11}-6z^{7}a^{-13}+6z^{7}a^{-15}-7z^{6}a^{-8}-3z^{6}a^{-10}-13z^{6}a^{-12}-11z^{6}a^{-14}+6z^{6}a^{-16}+z^{5}a^{-9}+13z^{5}a^{-11}-3z^{5}a^{-13}-10z^{5}a^{-15}+5z^{5}a^{-17}+17z^{4}a^{-8}+15z^{4}a^{-10}+12z^{4}a^{-12}+4z^{4}a^{-14}-7z^{4}a^{-16}+3z^{4}a^{-18}+9z^{3}a^{-9}+7z^{3}a^{-11}+5z^{3}a^{-13}+2z^{3}a^{-15}-4z^{3}a^{-17}+z^{3}a^{-19}-17z^{2}a^{-8}-15z^{2}a^{-10}-z^{2}a^{-12}-z^{2}a^{-14}+z^{2}a^{-16}-z^{2}a^{-18}-7za^{-9}-7za^{-11}-za^{-13}+za^{-17}+6a^{-8}+6a^{-10}+a^{-12}$

"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["K11a240"];

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]=

{ $2t^{4}-5t^{3}+8t^{2}-10t+11-10t^{-1}+8t^{-2}-5t^{-3}+2t^{-4}$, $-q^{15}+3q^{14}-5q^{13}+7q^{12}-9q^{11}+9q^{10}-9q^{9}+7q^{8}-5q^{7}+4q^{6}-q^{5}+q^{4}$ }

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:

(9, 25)

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 $36$ $200$ $648$ $1434$ $174$ $7200$ ${\frac {34640}{3}}$ ${\frac {5600}{3}}$ $1192$ $7776$ $20000$ $51624$ $6264$ ${\frac {962893}{10}}$ ${\frac {30094}{5}}$ ${\frac {150022}{5}}$ ${\frac {1361}{2}}$ ${\frac {35053}{10}}$

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=$8 is the signature of K11a240. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 11 χ 31                       1 -1 29                     2   2 27                   3 1   -2 25                 4 2     2 23               5 3       -2 21             4 4         0 19           5 5           0 17         2 4             -2 15       3 5               2 13     1 2                 -1 11     3                   3 9 1 1                     0 7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=7$ $i=9$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=7$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=8$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=9$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=10$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=11$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a239 K11a241