K11a316

K11a315

K11a317

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a316 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a316/ThurstonBennequinNumber Hyperbolic Volume 15.8582 A-Polynomial See Data:K11a316/A-polynomial

[edit Notes for K11a316's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $[0,4]$ Rasmussen s-Invariant 0

[edit Notes for K11a316's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+14t^{2}-25t+31-25t^{-1}+14t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+4z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 121, 0 }
Jones polynomial	$q^{6}-4q^{5}+8q^{4}-13q^{3}+17q^{2}-19q+20-16q^{-1}+12q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-8z^{4}a^{-2}+z^{4}a^{-4}+15z^{4}-6a^{2}z^{2}-11z^{2}a^{-2}+2z^{2}a^{-4}+17z^{2}-3a^{2}-5a^{-2}+a^{-4}+8$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+6az^{9}+11z^{9}a^{-1}+5z^{9}a^{-3}+8a^{2}z^{8}+7z^{8}a^{-2}+6z^{8}a^{-4}+9z^{8}+6a^{3}z^{7}-10az^{7}-26z^{7}a^{-1}-6z^{7}a^{-3}+4z^{7}a^{-5}+3a^{4}z^{6}-21a^{2}z^{6}-26z^{6}a^{-2}-13z^{6}a^{-4}+z^{6}a^{-6}-36z^{6}+a^{5}z^{5}-13a^{3}z^{5}+4az^{5}+23z^{5}a^{-1}-5z^{5}a^{-3}-10z^{5}a^{-5}-5a^{4}z^{4}+27a^{2}z^{4}+27z^{4}a^{-2}+5z^{4}a^{-4}-2z^{4}a^{-6}+52z^{4}-2a^{5}z^{3}+10a^{3}z^{3}+10az^{3}-5z^{3}a^{-1}+3z^{3}a^{-3}+6z^{3}a^{-5}-14a^{2}z^{2}-18z^{2}a^{-2}-z^{2}a^{-4}+z^{2}a^{-6}-30z^{2}-3a^{3}z-5az-3za^{-1}-za^{-3}+3a^{2}+5a^{-2}+a^{-4}+8$
The A2 invariant	$-q^{14}+q^{12}-3q^{10}+q^{8}+q^{6}-2q^{4}+6q^{2}-1+4q^{-2}-2q^{-6}+2q^{-8}-4q^{-10}+q^{-12}-q^{-16}+q^{-18}$
The G2 invariant	Data:K11a316/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a316 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["K11a316"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+14t^{2}-25t+31-25t^{-1}+14t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 121, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-4q^{5}+8q^{4}-13q^{3}+17q^{2}-19q+20-16q^{-1}+12q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-8z^{4}a^{-2}+z^{4}a^{-4}+15z^{4}-6a^{2}z^{2}-11z^{2}a^{-2}+2z^{2}a^{-4}+17z^{2}-3a^{2}-5a^{-2}+a^{-4}+8$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+6az^{9}+11z^{9}a^{-1}+5z^{9}a^{-3}+8a^{2}z^{8}+7z^{8}a^{-2}+6z^{8}a^{-4}+9z^{8}+6a^{3}z^{7}-10az^{7}-26z^{7}a^{-1}-6z^{7}a^{-3}+4z^{7}a^{-5}+3a^{4}z^{6}-21a^{2}z^{6}-26z^{6}a^{-2}-13z^{6}a^{-4}+z^{6}a^{-6}-36z^{6}+a^{5}z^{5}-13a^{3}z^{5}+4az^{5}+23z^{5}a^{-1}-5z^{5}a^{-3}-10z^{5}a^{-5}-5a^{4}z^{4}+27a^{2}z^{4}+27z^{4}a^{-2}+5z^{4}a^{-4}-2z^{4}a^{-6}+52z^{4}-2a^{5}z^{3}+10a^{3}z^{3}+10az^{3}-5z^{3}a^{-1}+3z^{3}a^{-3}+6z^{3}a^{-5}-14a^{2}z^{2}-18z^{2}a^{-2}-z^{2}a^{-4}+z^{2}a^{-6}-30z^{2}-3a^{3}z-5az-3za^{-1}-za^{-3}+3a^{2}+5a^{-2}+a^{-4}+8$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a35,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a35, K11a36,}

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["K11a316"];

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]=

{ $t^{4}-5t^{3}+14t^{2}-25t+31-25t^{-1}+14t^{-2}-5t^{-3}+t^{-4}$, $q^{6}-4q^{5}+8q^{4}-13q^{3}+17q^{2}-19q+20-16q^{-1}+12q^{-2}-7q^{-3}+3q^{-4}-q^{-5}$ }

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]=

{K11a35,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11a35, K11a36,}

Vassiliev invariants

V2 and V3:

(2, -1)

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 $8$ $-8$ $32$ ${\frac {28}{3}}$ $-{\frac {52}{3}}$ $-64$ $-{\frac {464}{3}}$ ${\frac {256}{3}}$ $-168$ ${\frac {256}{3}}$ $32$ ${\frac {224}{3}}$ $-{\frac {416}{3}}$ ${\frac {751}{15}}$ ${\frac {4156}{15}}$ $-{\frac {15116}{45}}$ $-{\frac {415}{9}}$ $-{\frac {1649}{15}}$

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 K11a316. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   5 1   4 7                 8 3     -5 5               9 5       4 3             10 8         -2 1           10 9           1 -1         7 11             4 -3       5 9               -4 -5     2 7                 5 -7   1 5                   -4 -9   2                     2 -11 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11a315 K11a317