K11a345

K11a344

K11a346

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a345 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a345/ThurstonBennequinNumber Hyperbolic Volume 14.216 A-Polynomial See Data:K11a345/A-polynomial

[edit Notes for K11a345's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a345's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-5t^{2}+23t-35+23t^{-1}-5t^{-2}$
Conway polynomial	$-5z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 91, 2 }
Jones polynomial	$-q^{10}+3q^{9}-6q^{8}+9q^{7}-12q^{6}+14q^{5}-14q^{4}+13q^{3}-9q^{2}+6q-3+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{-2}-2z^{4}a^{-4}-2z^{4}a^{-6}+z^{2}a^{-2}-2z^{2}a^{-6}+3z^{2}a^{-8}+z^{2}+a^{-2}+a^{-4}-2a^{-6}+2a^{-8}-a^{-10}$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-6}+2z^{10}a^{-8}+5z^{9}a^{-5}+9z^{9}a^{-7}+4z^{9}a^{-9}+6z^{8}a^{-4}+2z^{8}a^{-6}-z^{8}a^{-8}+3z^{8}a^{-10}+6z^{7}a^{-3}-11z^{7}a^{-5}-33z^{7}a^{-7}-15z^{7}a^{-9}+z^{7}a^{-11}+5z^{6}a^{-2}-10z^{6}a^{-4}-20z^{6}a^{-6}-17z^{6}a^{-8}-12z^{6}a^{-10}+3z^{5}a^{-1}-8z^{5}a^{-3}+10z^{5}a^{-5}+40z^{5}a^{-7}+15z^{5}a^{-9}-4z^{5}a^{-11}-6z^{4}a^{-2}+6z^{4}a^{-4}+26z^{4}a^{-6}+26z^{4}a^{-8}+13z^{4}a^{-10}+z^{4}-3z^{3}a^{-1}+4z^{3}a^{-3}-5z^{3}a^{-5}-22z^{3}a^{-7}-6z^{3}a^{-9}+4z^{3}a^{-11}+3z^{2}a^{-2}-z^{2}a^{-4}-13z^{2}a^{-6}-13z^{2}a^{-8}-5z^{2}a^{-10}-z^{2}+za^{-5}+4za^{-7}+2za^{-9}-za^{-11}-a^{-2}+a^{-4}+2a^{-6}+2a^{-8}+a^{-10}$
The A2 invariant	Data:K11a345/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a345/QuantumInvariant/G2/1,0

Further Quantum Invariants[show]

K11a345 Quantum Invariants.

Computer Talk[show]

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["K11a345"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-5t^{2}+23t-35+23t^{-1}-5t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-5z^{4}+3z^{2}+1$

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]=

{ 91, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-6}+2z^{10}a^{-8}+5z^{9}a^{-5}+9z^{9}a^{-7}+4z^{9}a^{-9}+6z^{8}a^{-4}+2z^{8}a^{-6}-z^{8}a^{-8}+3z^{8}a^{-10}+6z^{7}a^{-3}-11z^{7}a^{-5}-33z^{7}a^{-7}-15z^{7}a^{-9}+z^{7}a^{-11}+5z^{6}a^{-2}-10z^{6}a^{-4}-20z^{6}a^{-6}-17z^{6}a^{-8}-12z^{6}a^{-10}+3z^{5}a^{-1}-8z^{5}a^{-3}+10z^{5}a^{-5}+40z^{5}a^{-7}+15z^{5}a^{-9}-4z^{5}a^{-11}-6z^{4}a^{-2}+6z^{4}a^{-4}+26z^{4}a^{-6}+26z^{4}a^{-8}+13z^{4}a^{-10}+z^{4}-3z^{3}a^{-1}+4z^{3}a^{-3}-5z^{3}a^{-5}-22z^{3}a^{-7}-6z^{3}a^{-9}+4z^{3}a^{-11}+3z^{2}a^{-2}-z^{2}a^{-4}-13z^{2}a^{-6}-13z^{2}a^{-8}-5z^{2}a^{-10}-z^{2}+za^{-5}+4za^{-7}+2za^{-9}-za^{-11}-a^{-2}+a^{-4}+2a^{-6}+2a^{-8}+a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Computer Talk[show]

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["K11a345"];

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]=

{ $-5t^{2}+23t-35+23t^{-1}-5t^{-2}$, $-q^{10}+3q^{9}-6q^{8}+9q^{7}-12q^{6}+14q^{5}-14q^{4}+13q^{3}-9q^{2}+6q-3+q^{-1}$ }

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:

(3, 7)

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 $12$ $56$ $72$ $318$ $74$ $672$ ${\frac {5456}{3}}$ ${\frac {800}{3}}$ $472$ $288$ $1568$ $3816$ $888$ ${\frac {101391}{10}}$ $-{\frac {25826}{15}}$ ${\frac {32754}{5}}$ ${\frac {635}{2}}$ ${\frac {10351}{10}}$

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=$2 is the signature of K11a345. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 21                       1 -1 19                     2   2 17                   4 1   -3 15                 5 2     3 13               7 4       -3 11             7 5         2 9           7 7           0 7         6 7             -1 5       3 7               4 3     3 6                 -3 1   1 4                   3 -1   2                     -2 -3 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a344 K11a346