K11n129

K11n128

K11n130

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n129 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X5,19,6,18 X7,14,8,15 X9,16,10,17 X2,11,3,12 X13,20,14,21 X15,8,16,9 X17,1,18,22 X19,12,20,13 X21,7,22,6
Gauss code	1, -6, 2, -1, -3, 11, -4, 8, -5, -2, 6, 10, -7, 4, -8, 5, -9, 3, -10, 7, -11, 9
Dowker-Thistlethwaite code	4 10 -18 -14 -16 2 -20 -8 -22 -12 -6

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:K11n129/ThurstonBennequinNumber Hyperbolic Volume 11.7204 A-Polynomial See Data:K11n129/A-polynomial

[edit Notes for K11n129'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 K11n129's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-4t^{2}+10t-13+10t^{-1}-4t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+2z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 43, -2 }
Jones polynomial	$q-3+5q^{-1}-6q^{-2}+8q^{-3}-7q^{-4}+6q^{-5}-4q^{-6}+2q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{6}-3z^{2}a^{6}-3a^{6}+z^{6}a^{4}+5z^{4}a^{4}+10z^{2}a^{4}+6a^{4}-2z^{4}a^{2}-5z^{2}a^{2}-3a^{2}+z^{2}+1$
Kauffman polynomial (db, data sources)	$z^{5}a^{9}-3z^{3}a^{9}+za^{9}+2z^{6}a^{8}-5z^{4}a^{8}+z^{2}a^{8}+3z^{7}a^{7}-9z^{5}a^{7}+8z^{3}a^{7}-3za^{7}+3z^{8}a^{6}-11z^{6}a^{6}+17z^{4}a^{6}-11z^{2}a^{6}+3a^{6}+z^{9}a^{5}-7z^{5}a^{5}+15z^{3}a^{5}-6za^{5}+4z^{8}a^{4}-16z^{6}a^{4}+30z^{4}a^{4}-21z^{2}a^{4}+6a^{4}+z^{9}a^{3}-3z^{7}a^{3}+6z^{5}a^{3}-z^{3}a^{3}-2za^{3}+z^{8}a^{2}-3z^{6}a^{2}+9z^{4}a^{2}-11z^{2}a^{2}+3a^{2}+3z^{5}a-5z^{3}a+z^{4}-2z^{2}+1$
The A2 invariant	$-q^{24}-2q^{20}-q^{18}+2q^{16}+3q^{12}+q^{10}+q^{8}+q^{6}-2q^{4}+q^{2}-1+q^{-4}$
The G2 invariant	Data:K11n129/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n129 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["K11n129"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-4t^{2}+10t-13+10t^{-1}-4t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 43, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}a^{6}-3z^{2}a^{6}-3a^{6}+z^{6}a^{4}+5z^{4}a^{4}+10z^{2}a^{4}+6a^{4}-2z^{4}a^{2}-5z^{2}a^{2}-3a^{2}+z^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{5}a^{9}-3z^{3}a^{9}+za^{9}+2z^{6}a^{8}-5z^{4}a^{8}+z^{2}a^{8}+3z^{7}a^{7}-9z^{5}a^{7}+8z^{3}a^{7}-3za^{7}+3z^{8}a^{6}-11z^{6}a^{6}+17z^{4}a^{6}-11z^{2}a^{6}+3a^{6}+z^{9}a^{5}-7z^{5}a^{5}+15z^{3}a^{5}-6za^{5}+4z^{8}a^{4}-16z^{6}a^{4}+30z^{4}a^{4}-21z^{2}a^{4}+6a^{4}+z^{9}a^{3}-3z^{7}a^{3}+6z^{5}a^{3}-z^{3}a^{3}-2za^{3}+z^{8}a^{2}-3z^{6}a^{2}+9z^{4}a^{2}-11z^{2}a^{2}+3a^{2}+3z^{5}a-5z^{3}a+z^{4}-2z^{2}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_151, K11n54,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {9_21,}

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["K11n129"];

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]=

{ $t^{3}-4t^{2}+10t-13+10t^{-1}-4t^{-2}+t^{-3}$, $q-3+5q^{-1}-6q^{-2}+8q^{-3}-7q^{-4}+6q^{-5}-4q^{-6}+2q^{-7}-q^{-8}$ }

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]=

{10_151, K11n54,}

In[6]:=

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

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

Out[6]=

{9_21,}

Vassiliev invariants

V2 and V3:

(3, -6)

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 $12$ $-48$ $72$ $190$ $18$ $-576$ $-1024$ $-96$ $-208$ $288$ $1152$ $2280$ $216$ ${\frac {52271}{10}}$ $-{\frac {3946}{15}}$ ${\frac {33742}{15}}$ ${\frac {913}{6}}$ ${\frac {2671}{10}}$

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 K11n129. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n128 K11n130