K11a241

K11a240

K11a242

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a241 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 X8,20,9,19 X6,22,7,21
Gauss code	1, -7, 2, -1, 3, -11, 4, -10, 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 8 6

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a241/ThurstonBennequinNumber Hyperbolic Volume 13.7014 A-Polynomial See Data:K11a241/A-polynomial

[edit Notes for K11a241's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -6

[edit Notes for K11a241's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$4t^{3}-12t^{2}+20t-23+20t^{-1}-12t^{-2}+4t^{-3}$
Conway polynomial	$4z^{6}+12z^{4}+8z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 95, 6 }
Jones polynomial	$-q^{14}+3q^{13}-6q^{12}+10q^{11}-14q^{10}+15q^{9}-15q^{8}+13q^{7}-9q^{6}+6q^{5}-2q^{4}+q^{3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-6}+2z^{6}a^{-8}+z^{6}a^{-10}+4z^{4}a^{-6}+7z^{4}a^{-8}+2z^{4}a^{-10}-z^{4}a^{-12}+5z^{2}a^{-6}+6z^{2}a^{-8}-z^{2}a^{-10}-2z^{2}a^{-12}+2a^{-6}+a^{-8}-2a^{-10}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-10}+z^{10}a^{-12}+2z^{9}a^{-9}+5z^{9}a^{-11}+3z^{9}a^{-13}+3z^{8}a^{-8}+3z^{8}a^{-10}+5z^{8}a^{-12}+5z^{8}a^{-14}+2z^{7}a^{-7}+z^{7}a^{-9}-6z^{7}a^{-11}+5z^{7}a^{-15}+z^{6}a^{-6}-7z^{6}a^{-8}-6z^{6}a^{-10}-10z^{6}a^{-12}-9z^{6}a^{-14}+3z^{6}a^{-16}-5z^{5}a^{-7}-11z^{5}a^{-9}-2z^{5}a^{-11}-8z^{5}a^{-13}-11z^{5}a^{-15}+z^{5}a^{-17}-4z^{4}a^{-6}+5z^{4}a^{-8}+2z^{4}a^{-10}+6z^{4}a^{-12}+7z^{4}a^{-14}-6z^{4}a^{-16}+2z^{3}a^{-7}+10z^{3}a^{-9}+9z^{3}a^{-11}+12z^{3}a^{-13}+9z^{3}a^{-15}-2z^{3}a^{-17}+5z^{2}a^{-6}-3z^{2}a^{-8}-4z^{2}a^{-10}+z^{2}a^{-12}-z^{2}a^{-14}+2z^{2}a^{-16}+za^{-7}-4za^{-9}-6za^{-11}-4za^{-13}-3za^{-15}-2a^{-6}+a^{-8}+2a^{-10}$
The A2 invariant	Data:K11a241/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a241/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a241 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["K11a241"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$4t^{3}-12t^{2}+20t-23+20t^{-1}-12t^{-2}+4t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$4z^{6}+12z^{4}+8z^{2}+1$

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]=

{ 95, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a186,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a186,}

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["K11a241"];

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]=

{ $4t^{3}-12t^{2}+20t-23+20t^{-1}-12t^{-2}+4t^{-3}$, $-q^{14}+3q^{13}-6q^{12}+10q^{11}-14q^{10}+15q^{9}-15q^{8}+13q^{7}-9q^{6}+6q^{5}-2q^{4}+q^{3}$ }

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]=

{K11a186,}

In[6]:=

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

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

Out[6]=

{K11a186,}

Vassiliev invariants

V2 and V3:

(8, 22)

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 $32$ $176$ $512$ ${\frac {3664}{3}}$ ${\frac {464}{3}}$ $5632$ ${\frac {28544}{3}}$ ${\frac {4640}{3}}$ $1072$ ${\frac {16384}{3}}$ $15488$ ${\frac {117248}{3}}$ ${\frac {14848}{3}}$ ${\frac {1146124}{15}}$ ${\frac {18528}{5}}$ ${\frac {1151536}{45}}$ ${\frac {6020}{9}}$ ${\frac {46204}{15}}$

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=$6 is the signature of K11a241. 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 29                       1 -1 27                     2   2 25                   4 1   -3 23                 6 2     4 21               8 4       -4 19             7 6         1 17           8 8           0 15         5 7             -2 13       4 8               4 11     2 5                 -3 9     4                   4 7 1 2                     -1 5 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a240 K11a242