K11a353

K11a352

K11a354

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a353 at Knotilus!

Knot K11a353.

A graph, K11a353.

A part of a link and a part of a graph.

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{3,4\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a353/ThurstonBennequinNumber Hyperbolic Volume 16.151 A-Polynomial See Data:K11a353/A-polynomial

[edit Notes for K11a353's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -6

[edit Notes for K11a353's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$5t^{3}-15t^{2}+26t-31+26t^{-1}-15t^{-2}+5t^{-3}$
Conway polynomial	$5z^{6}+15z^{4}+11z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 123, 6 }
Jones polynomial	$-q^{14}+3q^{13}-8q^{12}+13q^{11}-17q^{10}+20q^{9}-20q^{8}+17q^{7}-12q^{6}+8q^{5}-3q^{4}+q^{3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-6}+3z^{6}a^{-8}+z^{6}a^{-10}+3z^{4}a^{-6}+12z^{4}a^{-8}+z^{4}a^{-10}-z^{4}a^{-12}+2z^{2}a^{-6}+14z^{2}a^{-8}-3z^{2}a^{-10}-2z^{2}a^{-12}+4a^{-8}-2a^{-10}-a^{-12}$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-10}+2z^{10}a^{-12}+5z^{9}a^{-9}+10z^{9}a^{-11}+5z^{9}a^{-13}+6z^{8}a^{-8}+6z^{8}a^{-10}+7z^{8}a^{-12}+7z^{8}a^{-14}+3z^{7}a^{-7}-8z^{7}a^{-9}-18z^{7}a^{-11}-z^{7}a^{-13}+6z^{7}a^{-15}+z^{6}a^{-6}-18z^{6}a^{-8}-21z^{6}a^{-10}-15z^{6}a^{-12}-10z^{6}a^{-14}+3z^{6}a^{-16}-7z^{5}a^{-7}-z^{5}a^{-9}+8z^{5}a^{-11}-9z^{5}a^{-13}-10z^{5}a^{-15}+z^{5}a^{-17}-3z^{4}a^{-6}+22z^{4}a^{-8}+19z^{4}a^{-10}+5z^{4}a^{-12}+7z^{4}a^{-14}-4z^{4}a^{-16}+3z^{3}a^{-7}+6z^{3}a^{-9}+3z^{3}a^{-11}+10z^{3}a^{-13}+8z^{3}a^{-15}-2z^{3}a^{-17}+2z^{2}a^{-6}-16z^{2}a^{-8}-9z^{2}a^{-10}+4z^{2}a^{-12}-4z^{2}a^{-14}+z^{2}a^{-16}-3za^{-9}-za^{-11}-3za^{-13}-4za^{-15}+za^{-17}+4a^{-8}+2a^{-10}-a^{-12}$
The A2 invariant	Data:K11a353/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a353/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a353 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["K11a353"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$5t^{3}-15t^{2}+26t-31+26t^{-1}-15t^{-2}+5t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$5z^{6}+15z^{4}+11z^{2}+1$

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]=

{ 123, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{14}+3q^{13}-8q^{12}+13q^{11}-17q^{10}+20q^{9}-20q^{8}+17q^{7}-12q^{6}+8q^{5}-3q^{4}+q^{3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-6}+3z^{6}a^{-8}+z^{6}a^{-10}+3z^{4}a^{-6}+12z^{4}a^{-8}+z^{4}a^{-10}-z^{4}a^{-12}+2z^{2}a^{-6}+14z^{2}a^{-8}-3z^{2}a^{-10}-2z^{2}a^{-12}+4a^{-8}-2a^{-10}-a^{-12}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-10}+2z^{10}a^{-12}+5z^{9}a^{-9}+10z^{9}a^{-11}+5z^{9}a^{-13}+6z^{8}a^{-8}+6z^{8}a^{-10}+7z^{8}a^{-12}+7z^{8}a^{-14}+3z^{7}a^{-7}-8z^{7}a^{-9}-18z^{7}a^{-11}-z^{7}a^{-13}+6z^{7}a^{-15}+z^{6}a^{-6}-18z^{6}a^{-8}-21z^{6}a^{-10}-15z^{6}a^{-12}-10z^{6}a^{-14}+3z^{6}a^{-16}-7z^{5}a^{-7}-z^{5}a^{-9}+8z^{5}a^{-11}-9z^{5}a^{-13}-10z^{5}a^{-15}+z^{5}a^{-17}-3z^{4}a^{-6}+22z^{4}a^{-8}+19z^{4}a^{-10}+5z^{4}a^{-12}+7z^{4}a^{-14}-4z^{4}a^{-16}+3z^{3}a^{-7}+6z^{3}a^{-9}+3z^{3}a^{-11}+10z^{3}a^{-13}+8z^{3}a^{-15}-2z^{3}a^{-17}+2z^{2}a^{-6}-16z^{2}a^{-8}-9z^{2}a^{-10}+4z^{2}a^{-12}-4z^{2}a^{-14}+z^{2}a^{-16}-3za^{-9}-za^{-11}-3za^{-13}-4za^{-15}+za^{-17}+4a^{-8}+2a^{-10}-a^{-12}$

"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["K11a353"];

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]=

{ $5t^{3}-15t^{2}+26t-31+26t^{-1}-15t^{-2}+5t^{-3}$, $-q^{14}+3q^{13}-8q^{12}+13q^{11}-17q^{10}+20q^{9}-20q^{8}+17q^{7}-12q^{6}+8q^{5}-3q^{4}+q^{3}$ }

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:

(11, 35)

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 $44$ $280$ $968$ ${\frac {7018}{3}}$ ${\frac {1070}{3}}$ $12320$ ${\frac {64816}{3}}$ ${\frac {11392}{3}}$ $2840$ ${\frac {42592}{3}}$ $39200$ ${\frac {308792}{3}}$ ${\frac {47080}{3}}$ ${\frac {6099941}{30}}$ ${\frac {28666}{5}}$ ${\frac {3532402}{45}}$ ${\frac {25051}{18}}$ ${\frac {306821}{30}}$

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=$6 is the signature of K11a353. 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 29                       1 -1 27                     2   2 25                   6 1   -5 23                 7 2     5 21               10 6       -4 19             10 7         3 17           10 10           0 15         7 10             -3 13       5 10               5 11     3 7                 -4 9     5                   5 7 1 3                     -2 5 1                       1

Integral Khovanov Homology (db, data source)

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

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.

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