K11a237

K11a236

K11a238

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a237 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a237/ThurstonBennequinNumber Hyperbolic Volume 13.6547 A-Polynomial See Data:K11a237/A-polynomial

[edit Notes for K11a237's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant -4

[edit Notes for K11a237's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$8t^{2}-23t+31-23t^{-1}+8t^{-2}$
Conway polynomial	$8z^{4}+9z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 93, 4 }
Jones polynomial	$-q^{13}+2q^{12}-5q^{11}+9q^{10}-12q^{9}+14q^{8}-15q^{7}+14q^{6}-10q^{5}+7q^{4}-3q^{3}+q^{2}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-4}+3z^{4}a^{-6}+3z^{4}a^{-8}+z^{4}a^{-10}+z^{2}a^{-4}+5z^{2}a^{-6}+4z^{2}a^{-8}-z^{2}a^{-12}+2a^{-6}-a^{-12}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-10}+z^{10}a^{-12}+3z^{9}a^{-9}+5z^{9}a^{-11}+2z^{9}a^{-13}+6z^{8}a^{-8}+6z^{8}a^{-10}+2z^{8}a^{-12}+2z^{8}a^{-14}+7z^{7}a^{-7}+5z^{7}a^{-9}-7z^{7}a^{-11}-4z^{7}a^{-13}+z^{7}a^{-15}+6z^{6}a^{-6}-5z^{6}a^{-8}-13z^{6}a^{-10}-10z^{6}a^{-12}-8z^{6}a^{-14}+3z^{5}a^{-5}-8z^{5}a^{-7}-21z^{5}a^{-9}-9z^{5}a^{-11}-4z^{5}a^{-13}-5z^{5}a^{-15}+z^{4}a^{-4}-8z^{4}a^{-6}-2z^{4}a^{-8}+3z^{4}a^{-12}+10z^{4}a^{-14}-2z^{3}a^{-5}+4z^{3}a^{-7}+16z^{3}a^{-9}+12z^{3}a^{-11}+10z^{3}a^{-13}+8z^{3}a^{-15}-z^{2}a^{-4}+7z^{2}a^{-6}+2z^{2}a^{-8}+2z^{2}a^{-10}+4z^{2}a^{-12}-4z^{2}a^{-14}-4za^{-9}-3za^{-11}-3za^{-13}-4za^{-15}-2a^{-6}-a^{-12}$
The A2 invariant	Data:K11a237/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a237/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a237 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["K11a237"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$8t^{2}-23t+31-23t^{-1}+8t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$8z^{4}+9z^{2}+1$

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]=

{ 93, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-10}+z^{10}a^{-12}+3z^{9}a^{-9}+5z^{9}a^{-11}+2z^{9}a^{-13}+6z^{8}a^{-8}+6z^{8}a^{-10}+2z^{8}a^{-12}+2z^{8}a^{-14}+7z^{7}a^{-7}+5z^{7}a^{-9}-7z^{7}a^{-11}-4z^{7}a^{-13}+z^{7}a^{-15}+6z^{6}a^{-6}-5z^{6}a^{-8}-13z^{6}a^{-10}-10z^{6}a^{-12}-8z^{6}a^{-14}+3z^{5}a^{-5}-8z^{5}a^{-7}-21z^{5}a^{-9}-9z^{5}a^{-11}-4z^{5}a^{-13}-5z^{5}a^{-15}+z^{4}a^{-4}-8z^{4}a^{-6}-2z^{4}a^{-8}+3z^{4}a^{-12}+10z^{4}a^{-14}-2z^{3}a^{-5}+4z^{3}a^{-7}+16z^{3}a^{-9}+12z^{3}a^{-11}+10z^{3}a^{-13}+8z^{3}a^{-15}-z^{2}a^{-4}+7z^{2}a^{-6}+2z^{2}a^{-8}+2z^{2}a^{-10}+4z^{2}a^{-12}-4z^{2}a^{-14}-4za^{-9}-3za^{-11}-3za^{-13}-4za^{-15}-2a^{-6}-a^{-12}$

"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["K11a237"];

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]=

{ $8t^{2}-23t+31-23t^{-1}+8t^{-2}$, $-q^{13}+2q^{12}-5q^{11}+9q^{10}-12q^{9}+14q^{8}-15q^{7}+14q^{6}-10q^{5}+7q^{4}-3q^{3}+q^{2}$ }

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, 27)

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 $36$ $216$ $648$ $1658$ $254$ $7776$ $14064$ $2464$ $1880$ $7776$ $23328$ $59688$ $9144$ ${\frac {1214893}{10}}$ ${\frac {45482}{15}}$ ${\frac {711746}{15}}$ ${\frac {5363}{6}}$ ${\frac {61933}{10}}$

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=$4 is the signature of K11a237. 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 27                       1 -1 25                     1   1 23                   4 1   -3 21                 5 1     4 19               7 4       -3 17             7 5         2 15           8 7           -1 13         6 7             -1 11       4 8               4 9     3 6                 -3 7     4                   4 5 1 3                     -2 3 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a236 K11a238