K11a293

K11a292

K11a294

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a293 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,4,11,3 X16,5,17,6 X22,8,1,7 X4,10,5,9 X18,11,19,12 X20,13,21,14 X8,15,9,16 X2,17,3,18 X12,19,13,20 X14,21,15,22
Gauss code	1, -9, 2, -5, 3, -1, 4, -8, 5, -2, 6, -10, 7, -11, 8, -3, 9, -6, 10, -7, 11, -4
Dowker-Thistlethwaite code	6 10 16 22 4 18 20 8 2 12 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a293/ThurstonBennequinNumber Hyperbolic Volume 13.4596 A-Polynomial See Data:K11a293/A-polynomial

[edit Notes for K11a293's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a293's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+12t^{2}-15t+15-15t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+2z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 81, -4 }
Jones polynomial	$q^{2}-3q+5-8q^{-1}+11q^{-2}-11q^{-3}+13q^{-4}-11q^{-5}+8q^{-6}-6q^{-7}+3q^{-8}-q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$a^{4}z^{8}-a^{6}z^{6}+6a^{4}z^{6}-2a^{2}z^{6}-4a^{6}z^{4}+14a^{4}z^{4}-9a^{2}z^{4}+z^{4}-5a^{6}z^{2}+17a^{4}z^{2}-11a^{2}z^{2}+3z^{2}-4a^{6}+8a^{4}-4a^{2}+1$
Kauffman polynomial (db, data sources)	$z^{3}a^{11}+3z^{4}a^{10}+6z^{5}a^{9}-5z^{3}a^{9}+2za^{9}+8z^{6}a^{8}-10z^{4}a^{8}+2z^{2}a^{8}+9z^{7}a^{7}-16z^{5}a^{7}+4z^{3}a^{7}-2za^{7}+9z^{8}a^{6}-24z^{6}a^{6}+18z^{4}a^{6}-11z^{2}a^{6}+4a^{6}+6z^{9}a^{5}-15z^{7}a^{5}+z^{5}a^{5}+10z^{3}a^{5}-6za^{5}+2z^{10}a^{4}+3z^{8}a^{4}-36z^{6}a^{4}+51z^{4}a^{4}-25z^{2}a^{4}+8a^{4}+9z^{9}a^{3}-40z^{7}a^{3}+50z^{5}a^{3}-15z^{3}a^{3}-2za^{3}+2z^{10}a^{2}-5z^{8}a^{2}-9z^{6}a^{2}+28z^{4}a^{2}-17z^{2}a^{2}+4a^{2}+3z^{9}a-16z^{7}a+27z^{5}a-15z^{3}a+z^{8}-5z^{6}+8z^{4}-5z^{2}+1$
The A2 invariant	$-q^{26}+q^{24}-2q^{22}-q^{20}-q^{18}-q^{16}+4q^{14}+4q^{10}+q^{8}+q^{4}-2q^{2}-q^{-2}+q^{-6}$
The G2 invariant	Data:K11a293/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a293 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["K11a293"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+12t^{2}-15t+15-15t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+3z^{6}+2z^{4}+4z^{2}+1$

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]=

$\{3,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 81, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{2}-3q+5-8q^{-1}+11q^{-2}-11q^{-3}+13q^{-4}-11q^{-5}+8q^{-6}-6q^{-7}+3q^{-8}-q^{-9}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{4}z^{8}-a^{6}z^{6}+6a^{4}z^{6}-2a^{2}z^{6}-4a^{6}z^{4}+14a^{4}z^{4}-9a^{2}z^{4}+z^{4}-5a^{6}z^{2}+17a^{4}z^{2}-11a^{2}z^{2}+3z^{2}-4a^{6}+8a^{4}-4a^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{3}a^{11}+3z^{4}a^{10}+6z^{5}a^{9}-5z^{3}a^{9}+2za^{9}+8z^{6}a^{8}-10z^{4}a^{8}+2z^{2}a^{8}+9z^{7}a^{7}-16z^{5}a^{7}+4z^{3}a^{7}-2za^{7}+9z^{8}a^{6}-24z^{6}a^{6}+18z^{4}a^{6}-11z^{2}a^{6}+4a^{6}+6z^{9}a^{5}-15z^{7}a^{5}+z^{5}a^{5}+10z^{3}a^{5}-6za^{5}+2z^{10}a^{4}+3z^{8}a^{4}-36z^{6}a^{4}+51z^{4}a^{4}-25z^{2}a^{4}+8a^{4}+9z^{9}a^{3}-40z^{7}a^{3}+50z^{5}a^{3}-15z^{3}a^{3}-2za^{3}+2z^{10}a^{2}-5z^{8}a^{2}-9z^{6}a^{2}+28z^{4}a^{2}-17z^{2}a^{2}+4a^{2}+3z^{9}a-16z^{7}a+27z^{5}a-15z^{3}a+z^{8}-5z^{6}+8z^{4}-5z^{2}+1$

"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["K11a293"];

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]=

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

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:

(4, -8)

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 $16$ $-64$ $128$ ${\frac {920}{3}}$ ${\frac {136}{3}}$ $-1024$ $-{\frac {5440}{3}}$ $-{\frac {544}{3}}$ $-416$ ${\frac {2048}{3}}$ $2048$ ${\frac {14720}{3}}$ ${\frac {2176}{3}}$ ${\frac {152942}{15}}$ $-{\frac {3976}{5}}$ ${\frac {226328}{45}}$ ${\frac {1810}{9}}$ ${\frac {10382}{15}}$

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 $s=$-4 is the signature of K11a293. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 5                       1 1 3                     2   -2 1                   3 1   2 -1                 5 2     -3 -3               6 3       3 -5             6 6         0 -7           7 5           2 -9         4 6             2 -11       4 7               -3 -13     2 4                 2 -15   1 4                   -3 -17   2                     2 -19 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

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