K11a165

K11a164

K11a166

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a165 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X18,6,19,5 X16,8,17,7 X2,10,3,9 X20,11,21,12 X22,13,1,14 X8,16,9,15 X6,18,7,17 X14,19,15,20 X12,21,13,22
Gauss code	1, -5, 2, -1, 3, -9, 4, -8, 5, -2, 6, -11, 7, -10, 8, -4, 9, -3, 10, -6, 11, -7
Dowker-Thistlethwaite code	4 10 18 16 2 20 22 8 6 14 12

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a165/ThurstonBennequinNumber Hyperbolic Volume 12.5729 A-Polynomial See Data:K11a165/A-polynomial

[edit Notes for K11a165's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a165's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2 t^3+9 t^2-18 t+23-18 t^{-1} +9 t^{-2} -2 t^{-3}$
Conway polynomial	$-2 z^6-3 z^4+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^2+t+1\right\}$
Determinant and Signature	{ 81, 0 }
Jones polynomial	$-q^7+2 q^6-4 q^5+8 q^4-10 q^3+12 q^2-13 q+12-9 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^6 a^{-2} -z^6+a^2 z^4-3 z^4 a^{-2} +2 z^4 a^{-4} -3 z^4+2 a^2 z^2-4 z^2 a^{-2} +6 z^2 a^{-4} -z^2 a^{-6} -3 z^2+a^2-3 a^{-2} +5 a^{-4} -2 a^{-6}$
Kauffman polynomial (db, data sources)	$z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +5 z^9 a^{-3} +2 z^9 a^{-5} +4 z^8 a^{-2} +z^8 a^{-4} +2 z^8 a^{-6} +5 z^8+6 a z^7-z^7 a^{-1} -14 z^7 a^{-3} -6 z^7 a^{-5} +z^7 a^{-7} +5 a^2 z^6-15 z^6 a^{-2} -13 z^6 a^{-4} -9 z^6 a^{-6} -6 z^6+3 a^3 z^5-9 a z^5-8 z^5 a^{-1} +11 z^5 a^{-3} +2 z^5 a^{-5} -5 z^5 a^{-7} +a^4 z^4-6 a^2 z^4+15 z^4 a^{-2} +21 z^4 a^{-4} +13 z^4 a^{-6} -3 a^3 z^3+7 a z^3+9 z^3 a^{-1} -4 z^3 a^{-3} +4 z^3 a^{-5} +7 z^3 a^{-7} -a^4 z^2+3 a^2 z^2-10 z^2 a^{-2} -18 z^2 a^{-4} -8 z^2 a^{-6} +4 z^2-2 a z-2 z a^{-1} -2 z a^{-5} -2 z a^{-7} -a^2+3 a^{-2} +5 a^{-4} +2 a^{-6}$
The A2 invariant	Data:K11a165/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a165/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a165 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["K11a165"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2 t^3+9 t^2-18 t+23-18 t^{-1} +9 t^{-2} -2 t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-2 z^6-3 z^4+1$

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]=

$\left\{2,t^2+t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 81, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^7+2 q^6-4 q^5+8 q^4-10 q^3+12 q^2-13 q+12-9 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +5 z^9 a^{-3} +2 z^9 a^{-5} +4 z^8 a^{-2} +z^8 a^{-4} +2 z^8 a^{-6} +5 z^8+6 a z^7-z^7 a^{-1} -14 z^7 a^{-3} -6 z^7 a^{-5} +z^7 a^{-7} +5 a^2 z^6-15 z^6 a^{-2} -13 z^6 a^{-4} -9 z^6 a^{-6} -6 z^6+3 a^3 z^5-9 a z^5-8 z^5 a^{-1} +11 z^5 a^{-3} +2 z^5 a^{-5} -5 z^5 a^{-7} +a^4 z^4-6 a^2 z^4+15 z^4 a^{-2} +21 z^4 a^{-4} +13 z^4 a^{-6} -3 a^3 z^3+7 a z^3+9 z^3 a^{-1} -4 z^3 a^{-3} +4 z^3 a^{-5} +7 z^3 a^{-7} -a^4 z^2+3 a^2 z^2-10 z^2 a^{-2} -18 z^2 a^{-4} -8 z^2 a^{-6} +4 z^2-2 a z-2 z a^{-1} -2 z a^{-5} -2 z a^{-7} -a^2+3 a^{-2} +5 a^{-4} +2 a^{-6}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_87, 10_98, K11a58, K11n72,}

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["K11a165"];

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]=

{ $-2 t^3+9 t^2-18 t+23-18 t^{-1} +9 t^{-2} -2 t^{-3}$, $-q^7+2 q^6-4 q^5+8 q^4-10 q^3+12 q^2-13 q+12-9 q^{-1} +6 q^{-2} -3 q^{-3} + q^{-4}$ }

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_87, 10_98, K11a58, K11n72,}

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, 3)

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 $0$ $24$ $0$ $80$ $24$ $0$ $144$ $-64$ $120$ $0$ $288$ $0$ $0$ $728$ $-\frac{856}{3}$ $488$ $136$ $56$

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 K11a165. 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 6 7 χ 15                       1 -1 13                     1   1 11                   3 1   -2 9                 5 1     4 7               5 3       -2 5             7 5         2 3           6 5           -1 1         6 7             -1 -1       4 7               3 -3     2 5                 -3 -5   1 4                   3 -7   2                     -2 -9 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

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.

K11a164 K11a166