K11a346

K11a345

K11a347

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a346 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X16,3,17,4 X12,6,13,5 X18,8,19,7 X20,10,21,9 X4,12,5,11 X22,13,1,14 X10,16,11,15 X2,17,3,18 X8,20,9,19 X14,21,15,22
Gauss code	1, -9, 2, -6, 3, -1, 4, -10, 5, -8, 6, -3, 7, -11, 8, -2, 9, -4, 10, -5, 11, -7
Dowker-Thistlethwaite code	6 16 12 18 20 4 22 10 2 8 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{2,3\} }[/math] 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a346/ThurstonBennequinNumber Hyperbolic Volume 14.6898 A-Polynomial See Data:K11a346/A-polynomial

[edit Notes for K11a346's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant -4

[edit Notes for K11a346's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-5 t^3+12 t^2-18 t+21-18 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+3 z^6+2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 93, 4 }
Jones polynomial	[math]\displaystyle{ -q^9+3 q^8-6 q^7+10 q^6-13 q^5+14 q^4-14 q^3+13 q^2-9 q+6-3 q^{-1} + q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-4} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -9 z^4 a^{-2} +14 z^4 a^{-4} -4 z^4 a^{-6} +z^4-12 z^2 a^{-2} +15 z^2 a^{-4} -5 z^2 a^{-6} +3 z^2-4 a^{-2} +5 a^{-4} -2 a^{-6} +2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10} a^{-4} +3 z^9 a^{-1} +10 z^9 a^{-3} +7 z^9 a^{-5} -3 z^8 a^{-2} +8 z^8 a^{-4} +12 z^8 a^{-6} +z^8-15 z^7 a^{-1} -40 z^7 a^{-3} -12 z^7 a^{-5} +13 z^7 a^{-7} -17 z^6 a^{-2} -53 z^6 a^{-4} -31 z^6 a^{-6} +10 z^6 a^{-8} -5 z^6+24 z^5 a^{-1} +42 z^5 a^{-3} -17 z^5 a^{-5} -29 z^5 a^{-7} +6 z^5 a^{-9} +41 z^4 a^{-2} +68 z^4 a^{-4} +19 z^4 a^{-6} -14 z^4 a^{-8} +3 z^4 a^{-10} +9 z^4-12 z^3 a^{-1} -6 z^3 a^{-3} +27 z^3 a^{-5} +17 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} -26 z^2 a^{-2} -30 z^2 a^{-4} -7 z^2 a^{-6} +4 z^2 a^{-8} -7 z^2-3 z a^{-3} -7 z a^{-5} -4 z a^{-7} +4 a^{-2} +5 a^{-4} +2 a^{-6} +2 }[/math]
The A2 invariant	[math]\displaystyle{ q^6+1-2 q^{-2} +2 q^{-4} +2 q^{-10} -3 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} + q^{-20} -2 q^{-22} + q^{-24} - q^{-26} }[/math]
The G2 invariant	Data:K11a346/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a346 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["K11a346"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-5 t^3+12 t^2-18 t+21-18 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+3 z^6+2 z^4+z^2+1 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 93, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^9+3 q^8-6 q^7+10 q^6-13 q^5+14 q^4-14 q^3+13 q^2-9 q+6-3 q^{-1} + q^{-2} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^8 a^{-4} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -9 z^4 a^{-2} +14 z^4 a^{-4} -4 z^4 a^{-6} +z^4-12 z^2 a^{-2} +15 z^2 a^{-4} -5 z^2 a^{-6} +3 z^2-4 a^{-2} +5 a^{-4} -2 a^{-6} +2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10} a^{-4} +3 z^9 a^{-1} +10 z^9 a^{-3} +7 z^9 a^{-5} -3 z^8 a^{-2} +8 z^8 a^{-4} +12 z^8 a^{-6} +z^8-15 z^7 a^{-1} -40 z^7 a^{-3} -12 z^7 a^{-5} +13 z^7 a^{-7} -17 z^6 a^{-2} -53 z^6 a^{-4} -31 z^6 a^{-6} +10 z^6 a^{-8} -5 z^6+24 z^5 a^{-1} +42 z^5 a^{-3} -17 z^5 a^{-5} -29 z^5 a^{-7} +6 z^5 a^{-9} +41 z^4 a^{-2} +68 z^4 a^{-4} +19 z^4 a^{-6} -14 z^4 a^{-8} +3 z^4 a^{-10} +9 z^4-12 z^3 a^{-1} -6 z^3 a^{-3} +27 z^3 a^{-5} +17 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} -26 z^2 a^{-2} -30 z^2 a^{-4} -7 z^2 a^{-6} +4 z^2 a^{-8} -7 z^2-3 z a^{-3} -7 z a^{-5} -4 z a^{-7} +4 a^{-2} +5 a^{-4} +2 a^{-6} +2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a106, K11a194,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

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["K11a346"];

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]=

{ [math]\displaystyle{ t^4-5 t^3+12 t^2-18 t+21-18 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q^9+3 q^8-6 q^7+10 q^6-13 q^5+14 q^4-14 q^3+13 q^2-9 q+6-3 q^{-1} + q^{-2} }[/math] }

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]=

{K11a106, K11a194,}

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, 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 [math]\displaystyle{ 4 }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{158}{3} }[/math] [math]\displaystyle{ -\frac{38}{3} }[/math] [math]\displaystyle{ 96 }[/math] [math]\displaystyle{ 176 }[/math] [math]\displaystyle{ -128 }[/math] [math]\displaystyle{ 120 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ \frac{632}{3} }[/math] [math]\displaystyle{ -\frac{152}{3} }[/math] [math]\displaystyle{ \frac{25711}{30} }[/math] [math]\displaystyle{ -\frac{3634}{5} }[/math] [math]\displaystyle{ \frac{39302}{45} }[/math] [math]\displaystyle{ \frac{1265}{18} }[/math] [math]\displaystyle{ \frac{3631}{30} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]4 is the signature of K11a346. 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 19                       1 -1 17                     2   2 15                   4 1   -3 13                 6 2     4 11               7 4       -3 9             7 6         1 7           7 7           0 5         6 7             -1 3       4 8               4 1     2 5                 -3 -1   1 4                   3 -3   2                     -2 -5 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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.

K11a345 K11a347