K11a29

K11a28

K11a30

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a29 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X12,6,13,5 X16,7,17,8 X2,9,3,10 X18,11,19,12 X22,14,1,13 X20,16,21,15 X10,17,11,18 X6,19,7,20 X14,22,15,21
Gauss code	1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -3, 7, -11, 8, -4, 9, -6, 10, -8, 11, -7
Dowker-Thistlethwaite code	4 8 12 16 2 18 22 20 10 6 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a29/ThurstonBennequinNumber Hyperbolic Volume 15.7181 A-Polynomial See Data:K11a29/A-polynomial

[edit Notes for K11a29's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for K11a29's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-12t^{2}+27t-33+27t^{-1}-12t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 115, -2 }
Jones polynomial	$-q^{4}+4q^{3}-7q^{2}+12q-16+18q^{-1}-18q^{-2}+16q^{-3}-12q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}+a^{6}-2z^{4}a^{4}-3z^{2}a^{4}-a^{4}+z^{6}a^{2}+z^{4}a^{2}-z^{2}a^{2}+z^{6}+2z^{4}+z^{2}-z^{4}a^{-2}-z^{2}a^{-2}+a^{-2}$
Kauffman polynomial (db, data sources)	$2a^{2}z^{10}+2z^{10}+6a^{3}z^{9}+11az^{9}+5z^{9}a^{-1}+9a^{4}z^{8}+10a^{2}z^{8}+4z^{8}a^{-2}+5z^{8}+9a^{5}z^{7}-3a^{3}z^{7}-29az^{7}-16z^{7}a^{-1}+z^{7}a^{-3}+6a^{6}z^{6}-13a^{4}z^{6}-37a^{2}z^{6}-15z^{6}a^{-2}-33z^{6}+3a^{7}z^{5}-14a^{5}z^{5}-15a^{3}z^{5}+18az^{5}+13z^{5}a^{-1}-3z^{5}a^{-3}+a^{8}z^{4}-6a^{6}z^{4}+4a^{4}z^{4}+31a^{2}z^{4}+16z^{4}a^{-2}+36z^{4}-2a^{7}z^{3}+12a^{5}z^{3}+14a^{3}z^{3}-4az^{3}-2z^{3}a^{-1}+2z^{3}a^{-3}-a^{8}z^{2}+4a^{6}z^{2}+3a^{4}z^{2}-10a^{2}z^{2}-3z^{2}a^{-2}-11z^{2}-3a^{5}z-3a^{3}z-a^{6}-a^{4}-a^{-2}$
The A2 invariant	Data:K11a29/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a29/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a29 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["K11a29"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-12t^{2}+27t-33+27t^{-1}-12t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{6}-3z^{2}+1$

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]=

{ 115, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+4q^{3}-7q^{2}+12q-16+18q^{-1}-18q^{-2}+16q^{-3}-12q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{2}z^{10}+2z^{10}+6a^{3}z^{9}+11az^{9}+5z^{9}a^{-1}+9a^{4}z^{8}+10a^{2}z^{8}+4z^{8}a^{-2}+5z^{8}+9a^{5}z^{7}-3a^{3}z^{7}-29az^{7}-16z^{7}a^{-1}+z^{7}a^{-3}+6a^{6}z^{6}-13a^{4}z^{6}-37a^{2}z^{6}-15z^{6}a^{-2}-33z^{6}+3a^{7}z^{5}-14a^{5}z^{5}-15a^{3}z^{5}+18az^{5}+13z^{5}a^{-1}-3z^{5}a^{-3}+a^{8}z^{4}-6a^{6}z^{4}+4a^{4}z^{4}+31a^{2}z^{4}+16z^{4}a^{-2}+36z^{4}-2a^{7}z^{3}+12a^{5}z^{3}+14a^{3}z^{3}-4az^{3}-2z^{3}a^{-1}+2z^{3}a^{-3}-a^{8}z^{2}+4a^{6}z^{2}+3a^{4}z^{2}-10a^{2}z^{2}-3z^{2}a^{-2}-11z^{2}-3a^{5}z-3a^{3}z-a^{6}-a^{4}-a^{-2}$

"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["K11a29"];

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]=

{ $2t^{3}-12t^{2}+27t-33+27t^{-1}-12t^{-2}+2t^{-3}$, $-q^{4}+4q^{3}-7q^{2}+12q-16+18q^{-1}-18q^{-2}+16q^{-3}-12q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$ }

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:

(-3, 3)

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 $-12$ $24$ $72$ $66$ $38$ $-288$ $-432$ $-160$ $-40$ $-288$ $288$ $-792$ $-456$ ${\frac {1009}{10}}$ ${\frac {4546}{15}}$ $-{\frac {7142}{15}}$ ${\frac {559}{6}}$ $-{\frac {1551}{10}}$

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 $s=$-2 is the signature of K11a29. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 9                       1 -1 7                     3   3 5                   4 1   -3 3                 8 3     5 1               8 4       -4 -1             10 8         2 -3           9 9           0 -5         7 9             -2 -7       5 9               4 -9     2 7                 -5 -11   1 5                   4 -13   2                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a28 K11a30