K11n29

K11n28

K11n30

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n29 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,13,6,12 X2837 X14,9,15,10 X18,12,19,11 X13,7,14,6 X22,15,1,16 X20,17,21,18 X10,20,11,19 X16,21,17,22
Gauss code	1, -4, 2, -1, -3, 7, 4, -2, 5, -10, 6, 3, -7, -5, 8, -11, 9, -6, 10, -9, 11, -8
Dowker-Thistlethwaite code	4 8 -12 2 14 18 -6 22 20 10 16

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:K11n29/ThurstonBennequinNumber Hyperbolic Volume 11.7974 A-Polynomial See Data:K11n29/A-polynomial

[edit Notes for K11n29's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n29's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{2}+13t-19+13t^{-1}-3t^{-2}$
Conway polynomial	$-3z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 51, 2 }
Jones polynomial	$-q^{8}+2q^{7}-4q^{6}+7q^{5}-8q^{4}+9q^{3}-8q^{2}+6q-4+2q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-2z^{4}a^{-2}-z^{4}a^{-4}-4z^{2}a^{-2}+z^{2}a^{-4}+2z^{2}a^{-6}+2z^{2}-3a^{-2}+2a^{-4}+a^{-6}-a^{-8}+2$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+4z^{8}a^{-4}+2z^{8}a^{-6}+z^{7}a^{-1}-z^{7}a^{-3}+2z^{7}a^{-7}-6z^{6}a^{-2}-11z^{6}a^{-4}-3z^{6}a^{-6}+2z^{6}a^{-8}+z^{5}a^{-1}+4z^{5}a^{-3}-2z^{5}a^{-7}+z^{5}a^{-9}+14z^{4}a^{-2}+18z^{4}a^{-4}+2z^{4}a^{-6}-5z^{4}a^{-8}+3z^{4}-5z^{3}a^{-1}-5z^{3}a^{-3}-3z^{3}a^{-7}-3z^{3}a^{-9}-14z^{2}a^{-2}-11z^{2}a^{-4}+3z^{2}a^{-8}-6z^{2}+2za^{-1}+3za^{-3}+za^{-5}+2za^{-7}+2za^{-9}+3a^{-2}+2a^{-4}-a^{-6}-a^{-8}+2$
The A2 invariant	Data:K11n29/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n29/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n29 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["K11n29"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{2}+13t-19+13t^{-1}-3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-3z^{4}+z^{2}+1$

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]=

{ 51, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+2q^{7}-4q^{6}+7q^{5}-8q^{4}+9q^{3}-8q^{2}+6q-4+2q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-3}+z^{9}a^{-5}+2z^{8}a^{-2}+4z^{8}a^{-4}+2z^{8}a^{-6}+z^{7}a^{-1}-z^{7}a^{-3}+2z^{7}a^{-7}-6z^{6}a^{-2}-11z^{6}a^{-4}-3z^{6}a^{-6}+2z^{6}a^{-8}+z^{5}a^{-1}+4z^{5}a^{-3}-2z^{5}a^{-7}+z^{5}a^{-9}+14z^{4}a^{-2}+18z^{4}a^{-4}+2z^{4}a^{-6}-5z^{4}a^{-8}+3z^{4}-5z^{3}a^{-1}-5z^{3}a^{-3}-3z^{3}a^{-7}-3z^{3}a^{-9}-14z^{2}a^{-2}-11z^{2}a^{-4}+3z^{2}a^{-8}-6z^{2}+2za^{-1}+3za^{-3}+za^{-5}+2za^{-7}+2za^{-9}+3a^{-2}+2a^{-4}-a^{-6}-a^{-8}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_36, K11a230,}

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["K11n29"];

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]=

{ $-3t^{2}+13t-19+13t^{-1}-3t^{-2}$, $-q^{8}+2q^{7}-4q^{6}+7q^{5}-8q^{4}+9q^{3}-8q^{2}+6q-4+2q^{-1}$ }

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]=

{10_36, K11a230,}

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, 4)

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 $4$ $32$ $8$ ${\frac {398}{3}}$ ${\frac {82}{3}}$ $128$ ${\frac {1472}{3}}$ ${\frac {224}{3}}$ $96$ ${\frac {32}{3}}$ $512$ ${\frac {1592}{3}}$ ${\frac {328}{3}}$ ${\frac {66751}{30}}$ $-{\frac {474}{5}}$ ${\frac {48062}{45}}$ ${\frac {737}{18}}$ ${\frac {3871}{30}}$

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=$2 is the signature of K11n29. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n28 K11n30