K11a346

K11a345

K11a347

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a346 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X16,3,17,4 X12,6,13,5 X18,8,19,7 X20,10,21,9 X4,12,5,11 X22,13,1,14 X10,16,11,15 X2,17,3,18 X8,20,9,19 X14,21,15,22
Gauss code	1, -9, 2, -6, 3, -1, 4, -10, 5, -8, 6, -3, 7, -11, 8, -2, 9, -4, 10, -5, 11, -7
Dowker-Thistlethwaite code	6 16 12 18 20 4 22 10 2 8 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:K11a346/ThurstonBennequinNumber Hyperbolic Volume 14.6898 A-Polynomial See Data:K11a346/A-polynomial

[edit Notes for K11a346'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 K11a346's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+12t^{2}-18t+21-18t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 93, 4 }
Jones polynomial	$-q^{9}+3q^{8}-6q^{7}+10q^{6}-13q^{5}+14q^{4}-14q^{3}+13q^{2}-9q+6-3q^{-1}+q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-2z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-9z^{4}a^{-2}+14z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}-12z^{2}a^{-2}+15z^{2}a^{-4}-5z^{2}a^{-6}+3z^{2}-4a^{-2}+5a^{-4}-2a^{-6}+2$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}a^{-4}+3z^{9}a^{-1}+10z^{9}a^{-3}+7z^{9}a^{-5}-3z^{8}a^{-2}+8z^{8}a^{-4}+12z^{8}a^{-6}+z^{8}-15z^{7}a^{-1}-40z^{7}a^{-3}-12z^{7}a^{-5}+13z^{7}a^{-7}-17z^{6}a^{-2}-53z^{6}a^{-4}-31z^{6}a^{-6}+10z^{6}a^{-8}-5z^{6}+24z^{5}a^{-1}+42z^{5}a^{-3}-17z^{5}a^{-5}-29z^{5}a^{-7}+6z^{5}a^{-9}+41z^{4}a^{-2}+68z^{4}a^{-4}+19z^{4}a^{-6}-14z^{4}a^{-8}+3z^{4}a^{-10}+9z^{4}-12z^{3}a^{-1}-6z^{3}a^{-3}+27z^{3}a^{-5}+17z^{3}a^{-7}-3z^{3}a^{-9}+z^{3}a^{-11}-26z^{2}a^{-2}-30z^{2}a^{-4}-7z^{2}a^{-6}+4z^{2}a^{-8}-7z^{2}-3za^{-3}-7za^{-5}-4za^{-7}+4a^{-2}+5a^{-4}+2a^{-6}+2$
The A2 invariant	$q^{6}+1-2q^{-2}+2q^{-4}+2q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}+q^{-20}-2q^{-22}+q^{-24}-q^{-26}$
The G2 invariant	Data:K11a346/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a346 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["K11a346"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+12t^{2}-18t+21-18t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 93, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{9}+3q^{8}-6q^{7}+10q^{6}-13q^{5}+14q^{4}-14q^{3}+13q^{2}-9q+6-3q^{-1}+q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-4}-2z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-9z^{4}a^{-2}+14z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}-12z^{2}a^{-2}+15z^{2}a^{-4}-5z^{2}a^{-6}+3z^{2}-4a^{-2}+5a^{-4}-2a^{-6}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}a^{-4}+3z^{9}a^{-1}+10z^{9}a^{-3}+7z^{9}a^{-5}-3z^{8}a^{-2}+8z^{8}a^{-4}+12z^{8}a^{-6}+z^{8}-15z^{7}a^{-1}-40z^{7}a^{-3}-12z^{7}a^{-5}+13z^{7}a^{-7}-17z^{6}a^{-2}-53z^{6}a^{-4}-31z^{6}a^{-6}+10z^{6}a^{-8}-5z^{6}+24z^{5}a^{-1}+42z^{5}a^{-3}-17z^{5}a^{-5}-29z^{5}a^{-7}+6z^{5}a^{-9}+41z^{4}a^{-2}+68z^{4}a^{-4}+19z^{4}a^{-6}-14z^{4}a^{-8}+3z^{4}a^{-10}+9z^{4}-12z^{3}a^{-1}-6z^{3}a^{-3}+27z^{3}a^{-5}+17z^{3}a^{-7}-3z^{3}a^{-9}+z^{3}a^{-11}-26z^{2}a^{-2}-30z^{2}a^{-4}-7z^{2}a^{-6}+4z^{2}a^{-8}-7z^{2}-3za^{-3}-7za^{-5}-4za^{-7}+4a^{-2}+5a^{-4}+2a^{-6}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a106, K11a194,}

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["K11a346"];

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]=

{ $t^{4}-5t^{3}+12t^{2}-18t+21-18t^{-1}+12t^{-2}-5t^{-3}+t^{-4}$, $-q^{9}+3q^{8}-6q^{7}+10q^{6}-13q^{5}+14q^{4}-14q^{3}+13q^{2}-9q+6-3q^{-1}+q^{-2}$ }

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]=

{K11a106, K11a194,}

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, 3)

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 $4$ $24$ $8$ ${\frac {158}{3}}$ $-{\frac {38}{3}}$ $96$ $176$ $-128$ $120$ ${\frac {32}{3}}$ $288$ ${\frac {632}{3}}$ $-{\frac {152}{3}}$ ${\frac {25711}{30}}$ $-{\frac {3634}{5}}$ ${\frac {39302}{45}}$ ${\frac {1265}{18}}$ ${\frac {3631}{30}}$

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=$4 is the signature of K11a346. 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 19                       1 -1 17                     2   2 15                   4 1   -3 13                 6 2     4 11               7 4       -3 9             7 6         1 7           7 7           0 5         6 7             -1 3       4 8               4 1     2 5                 -3 -1   1 4                   3 -3   2                     -2 -5 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a345 K11a347