K11n62

K11n61

K11n63

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n62 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,5,15,6 X2837 X9,17,10,16 X11,20,12,21 X13,18,14,19 X6,15,7,16 X17,1,18,22 X19,12,20,13 X21,10,22,11
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, -5, 11, -6, 10, -7, -3, 8, 5, -9, 7, -10, 6, -11, 9
Dowker-Thistlethwaite code	4 8 14 2 -16 -20 -18 6 -22 -12 -10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n62/ThurstonBennequinNumber Hyperbolic Volume 9.61446 A-Polynomial See Data:K11n62/A-polynomial

[edit Notes for K11n62's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for K11n62's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2 t^2-8 t+13-8 t^{-1} +2 t^{-2}$
Conway polynomial	$2 z^4+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 33, 0 }
Jones polynomial	$-q^5+2 q^4-3 q^3+5 q^2-5 q+6-5 q^{-1} +3 q^{-2} -2 q^{-3} + q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$a^4-2 z^2 a^2-2 a^2+z^4+z^2+1+z^4 a^{-2} +2 z^2 a^{-2} +2 a^{-2} -z^2 a^{-4} - a^{-4}$
Kauffman polynomial (db, data sources)	$a z^9+z^9 a^{-1} +a^2 z^8+2 z^8 a^{-2} +3 z^8-5 a z^7-3 z^7 a^{-1} +2 z^7 a^{-3} -5 a^2 z^6-7 z^6 a^{-2} +2 z^6 a^{-4} -14 z^6+2 a^3 z^5+11 a z^5+3 z^5 a^{-1} -5 z^5 a^{-3} +z^5 a^{-5} +a^4 z^4+12 a^2 z^4+8 z^4 a^{-2} -6 z^4 a^{-4} +25 z^4-5 a^3 z^3-8 a z^3+z^3 a^{-1} +z^3 a^{-3} -3 z^3 a^{-5} -3 a^4 z^2-11 a^2 z^2-z^2 a^{-2} +3 z^2 a^{-4} -12 z^2+2 a^3 z+3 a z+z a^{-1} +z a^{-3} +z a^{-5} +a^4+2 a^2-2 a^{-2} - a^{-4} +1$
The A2 invariant	Data:K11n62/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n62/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n62 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["K11n62"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2 t^2-8 t+13-8 t^{-1} +2 t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$2 z^4+1$

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]=

{ 33, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a z^9+z^9 a^{-1} +a^2 z^8+2 z^8 a^{-2} +3 z^8-5 a z^7-3 z^7 a^{-1} +2 z^7 a^{-3} -5 a^2 z^6-7 z^6 a^{-2} +2 z^6 a^{-4} -14 z^6+2 a^3 z^5+11 a z^5+3 z^5 a^{-1} -5 z^5 a^{-3} +z^5 a^{-5} +a^4 z^4+12 a^2 z^4+8 z^4 a^{-2} -6 z^4 a^{-4} +25 z^4-5 a^3 z^3-8 a z^3+z^3 a^{-1} +z^3 a^{-3} -3 z^3 a^{-5} -3 a^4 z^2-11 a^2 z^2-z^2 a^{-2} +3 z^2 a^{-4} -12 z^2+2 a^3 z+3 a z+z a^{-1} +z a^{-3} +z a^{-5} +a^4+2 a^2-2 a^{-2} - a^{-4} +1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_146, K11n18,}

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["K11n62"];

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]=

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

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_146, K11n18,}

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:

(0, 2)

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 $0$ $16$ $0$ $-16$ $-16$ $0$ $\frac{64}{3}$ $-\frac{128}{3}$ $48$ $0$ $128$ $0$ $0$ $216$ $\frac{208}{3}$ $\frac{160}{3}$ $-\frac{104}{3}$ $-24$

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 K11n62. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 11                   1 -1 9                 1   1 7               2 1   -1 5             3 1     2 3           2 2       0 1         4 3         1 -1       2 3           1 -3     1 3             -2 -5   1 2               1 -7   1                 -1 -9 1                   1

Integral Khovanov Homology (db, data source)

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

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.

K11n61 K11n63