K11a239

K11a238

K11a240

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a239 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a239/ThurstonBennequinNumber Hyperbolic Volume 19.5737 A-Polynomial See Data:K11a239/A-polynomial

[edit Notes for K11a239's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a239's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+7t^{3}-22t^{2}+42t-51+42t^{-1}-22t^{-2}+7t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-z^{6}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{3,t^{2}+1\right\}$
Determinant and Signature	{ 195, 2 }
Jones polynomial	$-q^{8}+5q^{7}-12q^{6}+20q^{5}-28q^{4}+32q^{3}-31q^{2}+28q-20+12q^{-1}-5q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-4z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-6z^{4}a^{-2}+5z^{4}a^{-4}-z^{4}a^{-6}+2z^{4}-2z^{2}a^{-2}+3z^{2}a^{-4}-z^{2}a^{-6}+z^{2}+2a^{-2}-a^{-4}$
Kauffman polynomial (db, data sources)	$4z^{10}a^{-2}+4z^{10}a^{-4}+11z^{9}a^{-1}+24z^{9}a^{-3}+13z^{9}a^{-5}+23z^{8}a^{-2}+29z^{8}a^{-4}+17z^{8}a^{-6}+11z^{8}+5az^{7}-12z^{7}a^{-1}-33z^{7}a^{-3}-4z^{7}a^{-5}+12z^{7}a^{-7}+a^{2}z^{6}-66z^{6}a^{-2}-72z^{6}a^{-4}-24z^{6}a^{-6}+5z^{6}a^{-8}-22z^{6}-8az^{5}-8z^{5}a^{-1}-6z^{5}a^{-3}-21z^{5}a^{-5}-14z^{5}a^{-7}+z^{5}a^{-9}-a^{2}z^{4}+45z^{4}a^{-2}+45z^{4}a^{-4}+13z^{4}a^{-6}-3z^{4}a^{-8}+15z^{4}+4az^{3}+8z^{3}a^{-1}+10z^{3}a^{-3}+12z^{3}a^{-5}+6z^{3}a^{-7}-8z^{2}a^{-2}-8z^{2}a^{-4}-3z^{2}a^{-6}-3z^{2}+za^{-3}+za^{-5}-2a^{-2}-a^{-4}$
The A2 invariant	$q^{8}-3q^{6}+4q^{4}-3q^{2}-1+6q^{-2}-5q^{-4}+8q^{-6}-3q^{-8}+q^{-10}+q^{-12}-6q^{-14}+5q^{-16}-3q^{-18}+2q^{-22}-q^{-24}$
The G2 invariant	Data:K11a239/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a239 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["K11a239"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+7t^{3}-22t^{2}+42t-51+42t^{-1}-22t^{-2}+7t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-z^{6}+z^{2}+1$

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]=

$\left\{3,t^{2}+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 195, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+5q^{7}-12q^{6}+20q^{5}-28q^{4}+32q^{3}-31q^{2}+28q-20+12q^{-1}-5q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$4z^{10}a^{-2}+4z^{10}a^{-4}+11z^{9}a^{-1}+24z^{9}a^{-3}+13z^{9}a^{-5}+23z^{8}a^{-2}+29z^{8}a^{-4}+17z^{8}a^{-6}+11z^{8}+5az^{7}-12z^{7}a^{-1}-33z^{7}a^{-3}-4z^{7}a^{-5}+12z^{7}a^{-7}+a^{2}z^{6}-66z^{6}a^{-2}-72z^{6}a^{-4}-24z^{6}a^{-6}+5z^{6}a^{-8}-22z^{6}-8az^{5}-8z^{5}a^{-1}-6z^{5}a^{-3}-21z^{5}a^{-5}-14z^{5}a^{-7}+z^{5}a^{-9}-a^{2}z^{4}+45z^{4}a^{-2}+45z^{4}a^{-4}+13z^{4}a^{-6}-3z^{4}a^{-8}+15z^{4}+4az^{3}+8z^{3}a^{-1}+10z^{3}a^{-3}+12z^{3}a^{-5}+6z^{3}a^{-7}-8z^{2}a^{-2}-8z^{2}a^{-4}-3z^{2}a^{-6}-3z^{2}+za^{-3}+za^{-5}-2a^{-2}-a^{-4}$

"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["K11a239"];

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]=

{ $-t^{4}+7t^{3}-22t^{2}+42t-51+42t^{-1}-22t^{-2}+7t^{-3}-t^{-4}$, $-q^{8}+5q^{7}-12q^{6}+20q^{5}-28q^{4}+32q^{3}-31q^{2}+28q-20+12q^{-1}-5q^{-2}+q^{-3}$ }

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]=

{}

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:

(1, 1)

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 $4$ $8$ $8$ ${\frac {62}{3}}$ ${\frac {10}{3}}$ $32$ ${\frac {176}{3}}$ $-{\frac {64}{3}}$ $40$ ${\frac {32}{3}}$ $32$ ${\frac {248}{3}}$ ${\frac {40}{3}}$ ${\frac {5071}{30}}$ $-{\frac {1382}{15}}$ ${\frac {5942}{45}}$ ${\frac {593}{18}}$ ${\frac {271}{30}}$

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 K11a239. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     4   4 13                   8 1   -7 11                 12 4     8 9               16 8       -8 7             16 12         4 5           15 16           1 3         13 16             -3 1       8 16               8 -1     4 12                 -8 -3   1 8                   7 -5   4                     -4 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{15}$ $r=2$ ${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{16}$ ${\mathbb {Z} }^{16}$ $r=3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{16}$ ${\mathbb {Z} }^{16}$ $r=4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a238 K11a240