K11n130

K11n129

K11n131

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n130 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n130/ThurstonBennequinNumber Hyperbolic Volume 13.3479 A-Polynomial See Data:K11n130/A-polynomial

[edit Notes for K11n130's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for K11n130's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^3+5 t^2-12 t+17-12 t^{-1} +5 t^{-2} - t^{-3}$
Conway polynomial	$-z^6-z^4-z^2+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 53, 0 }
Jones polynomial	$-q^3+4 q^2-6 q+8-9 q^{-1} +9 q^{-2} -7 q^{-3} +5 q^{-4} -3 q^{-5} + q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$-a^2 z^6+a^4 z^4-4 a^2 z^4+2 z^4+2 a^4 z^2-6 a^2 z^2-z^2 a^{-2} +4 z^2+a^4-2 a^2+2$
Kauffman polynomial (db, data sources)	$2 a^3 z^9+2 a z^9+4 a^4 z^8+7 a^2 z^8+3 z^8+3 a^5 z^7-3 a^3 z^7-5 a z^7+z^7 a^{-1} +a^6 z^6-14 a^4 z^6-25 a^2 z^6-10 z^6-10 a^5 z^5-5 a^3 z^5+7 a z^5+2 z^5 a^{-1} -3 a^6 z^4+13 a^4 z^4+31 a^2 z^4+4 z^4 a^{-2} +19 z^4+7 a^5 z^3+6 a^3 z^3-4 a z^3-2 z^3 a^{-1} +z^3 a^{-3} +a^6 z^2-5 a^4 z^2-15 a^2 z^2-3 z^2 a^{-2} -12 z^2-a^5 z-a^3 z+a^4+2 a^2+2$
The A2 invariant	$q^{18}-q^{16}+q^{14}-2 q^{10}+2 q^8-q^6+q^4-1+2 q^{-2} - q^{-4} +2 q^{-6} + q^{-8} - q^{-10}$
The G2 invariant	Data:K11n130/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n130 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["K11n130"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^3+5 t^2-12 t+17-12 t^{-1} +5 t^{-2} - t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^6-z^4-z^2+1$

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]=

{ 53, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2 a^3 z^9+2 a z^9+4 a^4 z^8+7 a^2 z^8+3 z^8+3 a^5 z^7-3 a^3 z^7-5 a z^7+z^7 a^{-1} +a^6 z^6-14 a^4 z^6-25 a^2 z^6-10 z^6-10 a^5 z^5-5 a^3 z^5+7 a z^5+2 z^5 a^{-1} -3 a^6 z^4+13 a^4 z^4+31 a^2 z^4+4 z^4 a^{-2} +19 z^4+7 a^5 z^3+6 a^3 z^3-4 a z^3-2 z^3 a^{-1} +z^3 a^{-3} +a^6 z^2-5 a^4 z^2-15 a^2 z^2-3 z^2 a^{-2} -12 z^2-a^5 z-a^3 z+a^4+2 a^2+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_30,}

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["K11n130"];

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]=

{ $-t^3+5 t^2-12 t+17-12 t^{-1} +5 t^{-2} - t^{-3}$, $-q^3+4 q^2-6 q+8-9 q^{-1} +9 q^{-2} -7 q^{-3} +5 q^{-4} -3 q^{-5} + q^{-6}$ }

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]=

{9_30,}

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, 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 $-4$ $8$ $8$ $\frac{34}{3}$ $\frac{38}{3}$ $-32$ $-\frac{208}{3}$ $-\frac{160}{3}$ $8$ $-\frac{32}{3}$ $32$ $-\frac{136}{3}$ $-\frac{152}{3}$ $\frac{2129}{30}$ $\frac{1142}{15}$ $-\frac{4742}{45}$ $\frac{1039}{18}$ $-\frac{751}{30}$

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 $t^rq^j$ 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 K11n130. 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 χ 7                   1 -1 5                 3   3 3               3 1   -2 1             5 3     2 -1           5 4       -1 -3         4 4         0 -5       3 5           2 -7     2 4             -2 -9   1 3               2 -11   2                 -2 -13 1                   1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

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.

K11n129 K11n131