K11n19

K11n18

K11n20

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n19 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X7,17,8,16 X2,9,3,10 X11,18,12,19 X13,20,14,21 X15,22,16,1 X17,7,18,6 X19,12,20,13 X21,14,22,15
Gauss code	1, -5, 2, -1, 3, 9, -4, -2, 5, -3, -6, 10, -7, 11, -8, 4, -9, 6, -10, 7, -11, 8
Dowker-Thistlethwaite code	4 8 10 -16 2 -18 -20 -22 -6 -12 -14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n19/ThurstonBennequinNumber Hyperbolic Volume 4.76989 A-Polynomial See Data:K11n19/A-polynomial

[edit Notes for K11n19'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 K11n19's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^3+2 t^2-1+2 t^{-2} - t^{-3}$
Conway polynomial	$-z^6-4 z^4-z^2+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 5, -4 }
Jones polynomial	$q^2-q+1- q^{-1} + q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$-a^6+z^4 a^4+5 z^2 a^4+4 a^4-z^6 a^2-6 z^4 a^2-10 z^2 a^2-5 a^2+z^4+4 z^2+3$
Kauffman polynomial (db, data sources)	$a^3 z^9+a z^9+a^4 z^8+2 a^2 z^8+z^8-7 a^3 z^7-7 a z^7-7 a^4 z^6-14 a^2 z^6-7 z^6+15 a^3 z^5+15 a z^5+15 a^4 z^4+30 a^2 z^4+15 z^4-a^5 z^3-12 a^3 z^3-11 a z^3-a^6 z^2-13 a^4 z^2-23 a^2 z^2-11 z^2+a^5 z+3 a^3 z+2 a z+a^6+4 a^4+5 a^2+3$
The A2 invariant	$-q^{20}+q^{16}+q^{14}+q^{12}-q^6-q^4+ q^{-2} + q^{-4} + q^{-6}$
The G2 invariant	Data:K11n19/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n19 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["K11n19"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 5, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^2-q+1- q^{-1} + q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^3 z^9+a z^9+a^4 z^8+2 a^2 z^8+z^8-7 a^3 z^7-7 a z^7-7 a^4 z^6-14 a^2 z^6-7 z^6+15 a^3 z^5+15 a z^5+15 a^4 z^4+30 a^2 z^4+15 z^4-a^5 z^3-12 a^3 z^3-11 a z^3-a^6 z^2-13 a^4 z^2-23 a^2 z^2-11 z^2+a^5 z+3 a^3 z+2 a z+a^6+4 a^4+5 a^2+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n135,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {4_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["K11n19"];

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]=

{ $-t^3+2 t^2-1+2 t^{-2} - t^{-3}$, $q^2-q+1- q^{-1} + q^{-2}$ }

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]=

{K11n135,}

In[6]:=

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

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

Out[6]=

{4_1,}

Vassiliev invariants

V2 and V3:

(-1, 0)

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 $-4$ $0$ $8$ $\frac{178}{3}$ $\frac{110}{3}$ $0$ $-128$ $-32$ $-64$ $-\frac{32}{3}$ $0$ $-\frac{712}{3}$ $-\frac{440}{3}$ $-\frac{2431}{30}$ $\frac{302}{15}$ $-\frac{8702}{45}$ $\frac{2335}{18}$ $-\frac{1951}{30}$

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 $s=$-4 is the signature of K11n19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 χ 5               1 1 3                 0 1           1 1   0 -1       1 1       0 -3       1 1       0 -5   1 1 1         1 -7                 0 -9 1 1             0

Integral Khovanov Homology (db, data source)

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

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.

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