K11a313

K11a312

K11a314

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a313 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a313/ThurstonBennequinNumber Hyperbolic Volume 13.3553 A-Polynomial See Data:K11a313/A-polynomial

[edit Notes for K11a313's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for K11a313's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$5t^{2}-19t+29-19t^{-1}+5t^{-2}$
Conway polynomial	$5z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 77, 0 }
Jones polynomial	$-q^{3}+3q^{2}-5q+8-10q^{-1}+12q^{-2}-11q^{-3}+10q^{-4}-8q^{-5}+5q^{-6}-3q^{-7}+q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$a^{8}-3z^{2}a^{6}-3a^{6}+2z^{4}a^{4}+3z^{2}a^{4}+2a^{4}+2z^{4}a^{2}+2z^{2}a^{2}+a^{2}+z^{4}-z^{2}a^{-2}$
Kauffman polynomial (db, data sources)	$2a^{6}z^{10}+2a^{4}z^{10}+3a^{7}z^{9}+9a^{5}z^{9}+6a^{3}z^{9}+a^{8}z^{8}-5a^{6}z^{8}+2a^{4}z^{8}+8a^{2}z^{8}-16a^{7}z^{7}-42a^{5}z^{7}-18a^{3}z^{7}+8az^{7}-5a^{8}z^{6}-10a^{6}z^{6}-33a^{4}z^{6}-21a^{2}z^{6}+7z^{6}+27a^{7}z^{5}+59a^{5}z^{5}+12a^{3}z^{5}-15az^{5}+5z^{5}a^{-1}+8a^{8}z^{4}+31a^{6}z^{4}+47a^{4}z^{4}+12a^{2}z^{4}+3z^{4}a^{-2}-9z^{4}-16a^{7}z^{3}-29a^{5}z^{3}-5a^{3}z^{3}+4az^{3}-3z^{3}a^{-1}+z^{3}a^{-3}-5a^{8}z^{2}-19a^{6}z^{2}-19a^{4}z^{2}-2a^{2}z^{2}-z^{2}a^{-2}+2z^{2}+2a^{7}z+5a^{5}z+3a^{3}z+a^{8}+3a^{6}+2a^{4}-a^{2}$
The A2 invariant	Data:K11a313/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a313/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a313 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["K11a313"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$5t^{2}-19t+29-19t^{-1}+5t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 77, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{3}+3q^{2}-5q+8-10q^{-1}+12q^{-2}-11q^{-3}+10q^{-4}-8q^{-5}+5q^{-6}-3q^{-7}+q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{6}z^{10}+2a^{4}z^{10}+3a^{7}z^{9}+9a^{5}z^{9}+6a^{3}z^{9}+a^{8}z^{8}-5a^{6}z^{8}+2a^{4}z^{8}+8a^{2}z^{8}-16a^{7}z^{7}-42a^{5}z^{7}-18a^{3}z^{7}+8az^{7}-5a^{8}z^{6}-10a^{6}z^{6}-33a^{4}z^{6}-21a^{2}z^{6}+7z^{6}+27a^{7}z^{5}+59a^{5}z^{5}+12a^{3}z^{5}-15az^{5}+5z^{5}a^{-1}+8a^{8}z^{4}+31a^{6}z^{4}+47a^{4}z^{4}+12a^{2}z^{4}+3z^{4}a^{-2}-9z^{4}-16a^{7}z^{3}-29a^{5}z^{3}-5a^{3}z^{3}+4az^{3}-3z^{3}a^{-1}+z^{3}a^{-3}-5a^{8}z^{2}-19a^{6}z^{2}-19a^{4}z^{2}-2a^{2}z^{2}-z^{2}a^{-2}+2z^{2}+2a^{7}z+5a^{5}z+3a^{3}z+a^{8}+3a^{6}+2a^{4}-a^{2}$

"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["K11a313"];

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]=

{ $5t^{2}-19t+29-19t^{-1}+5t^{-2}$, $-q^{3}+3q^{2}-5q+8-10q^{-1}+12q^{-2}-11q^{-3}+10q^{-4}-8q^{-5}+5q^{-6}-3q^{-7}+q^{-8}$ }

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:

(1, 0)

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 $4$ $0$ $8$ $-{\frac {178}{3}}$ $-{\frac {110}{3}}$ $0$ $320$ $-32$ $256$ ${\frac {32}{3}}$ $0$ $-{\frac {712}{3}}$ $-{\frac {440}{3}}$ $-{\frac {34049}{30}}$ ${\frac {18898}{15}}$ $-{\frac {83458}{45}}$ $-{\frac {4639}{18}}$ $-{\frac {11489}{30}}$

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=$0 is the signature of K11a313. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a312 K11a314