K11a354

K11a353

K11a355

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a354 at Knotilus!

Knot K11a354.

A graph, K11a354.

Knot presentations

Planar diagram presentation	X8291 X12,4,13,3 X16,6,17,5 X20,8,21,7 X2,10,3,9 X18,12,19,11 X22,14,1,13 X6,16,7,15 X4,18,5,17 X10,20,11,19 X14,22,15,21
Gauss code	1, -5, 2, -9, 3, -8, 4, -1, 5, -10, 6, -2, 7, -11, 8, -3, 9, -6, 10, -4, 11, -7
Dowker-Thistlethwaite code	8 12 16 20 2 18 22 6 4 10 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3,4\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a354/ThurstonBennequinNumber Hyperbolic Volume 15.1817 A-Polynomial See Data:K11a354/A-polynomial

[edit Notes for K11a354'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 K11a354's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$9t^{2}-26t+35-26t^{-1}+9t^{-2}$
Conway polynomial	$9z^{4}+10z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{2}+t+1\right\}$
Determinant and Signature	{ 105, 4 }
Jones polynomial	$-q^{13}+3q^{12}-7q^{11}+10q^{10}-14q^{9}+17q^{8}-16q^{7}+15q^{6}-11q^{5}+7q^{4}-3q^{3}+q^{2}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-4}+3z^{4}a^{-6}+4z^{4}a^{-8}+z^{4}a^{-10}+z^{2}a^{-4}+4z^{2}a^{-6}+8z^{2}a^{-8}-2z^{2}a^{-10}-z^{2}a^{-12}+5a^{-8}-4a^{-10}$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-10}+2z^{10}a^{-12}+6z^{9}a^{-9}+10z^{9}a^{-11}+4z^{9}a^{-13}+9z^{8}a^{-8}+7z^{8}a^{-10}+z^{8}a^{-12}+3z^{8}a^{-14}+8z^{7}a^{-7}-9z^{7}a^{-9}-31z^{7}a^{-11}-13z^{7}a^{-13}+z^{7}a^{-15}+6z^{6}a^{-6}-20z^{6}a^{-8}-34z^{6}a^{-10}-19z^{6}a^{-12}-11z^{6}a^{-14}+3z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}+30z^{5}a^{-11}+11z^{5}a^{-13}-4z^{5}a^{-15}+z^{4}a^{-4}-7z^{4}a^{-6}+25z^{4}a^{-8}+42z^{4}a^{-10}+20z^{4}a^{-12}+11z^{4}a^{-14}-2z^{3}a^{-5}+6z^{3}a^{-7}+3z^{3}a^{-9}-15z^{3}a^{-11}-5z^{3}a^{-13}+5z^{3}a^{-15}-z^{2}a^{-4}+4z^{2}a^{-6}-18z^{2}a^{-8}-25z^{2}a^{-10}-5z^{2}a^{-12}-3z^{2}a^{-14}-2za^{-9}+4za^{-11}+4za^{-13}-2za^{-15}+5a^{-8}+4a^{-10}$
The A2 invariant	Data:K11a354/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a354/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a354 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["K11a354"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$9t^{2}-26t+35-26t^{-1}+9t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$9z^{4}+10z^{2}+1$

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]=

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

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 105, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{13}+3q^{12}-7q^{11}+10q^{10}-14q^{9}+17q^{8}-16q^{7}+15q^{6}-11q^{5}+7q^{4}-3q^{3}+q^{2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-10}+2z^{10}a^{-12}+6z^{9}a^{-9}+10z^{9}a^{-11}+4z^{9}a^{-13}+9z^{8}a^{-8}+7z^{8}a^{-10}+z^{8}a^{-12}+3z^{8}a^{-14}+8z^{7}a^{-7}-9z^{7}a^{-9}-31z^{7}a^{-11}-13z^{7}a^{-13}+z^{7}a^{-15}+6z^{6}a^{-6}-20z^{6}a^{-8}-34z^{6}a^{-10}-19z^{6}a^{-12}-11z^{6}a^{-14}+3z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}+30z^{5}a^{-11}+11z^{5}a^{-13}-4z^{5}a^{-15}+z^{4}a^{-4}-7z^{4}a^{-6}+25z^{4}a^{-8}+42z^{4}a^{-10}+20z^{4}a^{-12}+11z^{4}a^{-14}-2z^{3}a^{-5}+6z^{3}a^{-7}+3z^{3}a^{-9}-15z^{3}a^{-11}-5z^{3}a^{-13}+5z^{3}a^{-15}-z^{2}a^{-4}+4z^{2}a^{-6}-18z^{2}a^{-8}-25z^{2}a^{-10}-5z^{2}a^{-12}-3z^{2}a^{-14}-2za^{-9}+4za^{-11}+4za^{-13}-2za^{-15}+5a^{-8}+4a^{-10}$

"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["K11a354"];

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]=

{ $9t^{2}-26t+35-26t^{-1}+9t^{-2}$, $-q^{13}+3q^{12}-7q^{11}+10q^{10}-14q^{9}+17q^{8}-16q^{7}+15q^{6}-11q^{5}+7q^{4}-3q^{3}+q^{2}$ }

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:

(10, 30)

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 $40$ $240$ $800$ ${\frac {5804}{3}}$ ${\frac {964}{3}}$ $9600$ $17056$ $3008$ $2480$ ${\frac {32000}{3}}$ $28800$ ${\frac {232160}{3}}$ ${\frac {38560}{3}}$ ${\frac {457711}{3}}$ ${\frac {3860}{3}}$ ${\frac {573772}{9}}$ ${\frac {10165}{9}}$ ${\frac {26095}{3}}$

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 K11a354. 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                     2   2 23                   5 1   -4 21                 5 2     3 19               9 5       -4 17             8 5         3 15           8 9           1 13         7 8             -1 11       4 8               4 9     3 7                 -4 7     4                   4 5 1 3                     -2 3 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a353 K11a355