K11a306

K11a305

K11a307

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a306 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a306/ThurstonBennequinNumber Hyperbolic Volume 13.5238 A-Polynomial See Data:K11a306/A-polynomial

[edit Notes for K11a306's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a306's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+13t^{2}-21t+25-21t^{-1}+13t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+3z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 105, -4 }
Jones polynomial	$-q+3-5q^{-1}+10q^{-2}-13q^{-3}+16q^{-4}-17q^{-5}+15q^{-6}-12q^{-7}+8q^{-8}-4q^{-9}+q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{8}+2z^{2}a^{8}+a^{8}-2z^{6}a^{6}-8z^{4}a^{6}-10z^{2}a^{6}-4a^{6}+z^{8}a^{4}+6z^{6}a^{4}+14z^{4}a^{4}+14z^{2}a^{4}+4a^{4}-z^{6}a^{2}-4z^{4}a^{2}-4z^{2}a^{2}$
Kauffman polynomial (db, data sources)	$z^{4}a^{12}+4z^{5}a^{11}-2z^{3}a^{11}+8z^{6}a^{10}-8z^{4}a^{10}+2z^{2}a^{10}+10z^{7}a^{9}-13z^{5}a^{9}+6z^{3}a^{9}-za^{9}+8z^{8}a^{8}-7z^{6}a^{8}-3z^{4}a^{8}+z^{2}a^{8}+a^{8}+4z^{9}a^{7}+5z^{7}a^{7}-23z^{5}a^{7}+15z^{3}a^{7}-3za^{7}+z^{10}a^{6}+11z^{8}a^{6}-32z^{6}a^{6}+27z^{4}a^{6}-14z^{2}a^{6}+4a^{6}+7z^{9}a^{5}-15z^{7}a^{5}+4z^{5}a^{5}+3z^{3}a^{5}-za^{5}+z^{10}a^{4}+6z^{8}a^{4}-30z^{6}a^{4}+39z^{4}a^{4}-21z^{2}a^{4}+4a^{4}+3z^{9}a^{3}-9z^{7}a^{3}+6z^{5}a^{3}+za^{3}+3z^{8}a^{2}-13z^{6}a^{2}+18z^{4}a^{2}-8z^{2}a^{2}+z^{7}a-4z^{5}a+4z^{3}a$
The A2 invariant	$q^{30}-q^{28}+q^{24}-3q^{22}+2q^{20}-2q^{18}+q^{14}-3q^{12}+4q^{10}-q^{8}+3q^{6}+2q^{4}-q^{2}+1-q^{-2}$
The G2 invariant	Data:K11a306/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a306 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["K11a306"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+13t^{2}-21t+25-21t^{-1}+13t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 105, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q+3-5q^{-1}+10q^{-2}-13q^{-3}+16q^{-4}-17q^{-5}+15q^{-6}-12q^{-7}+8q^{-8}-4q^{-9}+q^{-10}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{4}a^{12}+4z^{5}a^{11}-2z^{3}a^{11}+8z^{6}a^{10}-8z^{4}a^{10}+2z^{2}a^{10}+10z^{7}a^{9}-13z^{5}a^{9}+6z^{3}a^{9}-za^{9}+8z^{8}a^{8}-7z^{6}a^{8}-3z^{4}a^{8}+z^{2}a^{8}+a^{8}+4z^{9}a^{7}+5z^{7}a^{7}-23z^{5}a^{7}+15z^{3}a^{7}-3za^{7}+z^{10}a^{6}+11z^{8}a^{6}-32z^{6}a^{6}+27z^{4}a^{6}-14z^{2}a^{6}+4a^{6}+7z^{9}a^{5}-15z^{7}a^{5}+4z^{5}a^{5}+3z^{3}a^{5}-za^{5}+z^{10}a^{4}+6z^{8}a^{4}-30z^{6}a^{4}+39z^{4}a^{4}-21z^{2}a^{4}+4a^{4}+3z^{9}a^{3}-9z^{7}a^{3}+6z^{5}a^{3}+za^{3}+3z^{8}a^{2}-13z^{6}a^{2}+18z^{4}a^{2}-8z^{2}a^{2}+z^{7}a-4z^{5}a+4z^{3}a$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a175,}

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["K11a306"];

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]=

{ $t^{4}-5t^{3}+13t^{2}-21t+25-21t^{-1}+13t^{-2}-5t^{-3}+t^{-4}$, $-q+3-5q^{-1}+10q^{-2}-13q^{-3}+16q^{-4}-17q^{-5}+15q^{-6}-12q^{-7}+8q^{-8}-4q^{-9}+q^{-10}$ }

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]=

{K11a175,}

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:

(2, -2)

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 $8$ $-16$ $32$ ${\frac {76}{3}}$ $-{\frac {28}{3}}$ $-128$ $-{\frac {64}{3}}$ $-{\frac {160}{3}}$ $112$ ${\frac {256}{3}}$ $128$ ${\frac {608}{3}}$ $-{\frac {224}{3}}$ $-{\frac {1289}{15}}$ ${\frac {10276}{15}}$ $-{\frac {37556}{45}}$ $-{\frac {1159}{9}}$ $-{\frac {2009}{15}}$

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 $s=$-4 is the signature of K11a306. 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 3                       1 -1 1                     2   2 -1                   3 1   -2 -3                 7 2     5 -5               7 4       -3 -7             9 6         3 -9           8 7           -1 -11         7 9             -2 -13       5 8               3 -15     3 7                 -4 -17   1 5                   4 -19   3                     -3 -21 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a305 K11a307