K11a355

K11a354

K11a356

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a355 at Knotilus!

Knot presentations

Planar diagram presentation	X8291 X14,4,15,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X22,12,1,11 X2,14,3,13 X4,16,5,15 X6,18,7,17 X12,20,13,19 X10,22,11,21
Gauss code	1, -7, 2, -8, 3, -9, 4, -1, 5, -11, 6, -10, 7, -2, 8, -3, 9, -4, 10, -5, 11, -6
Dowker-Thistlethwaite code	8 14 16 18 20 22 2 4 6 12 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 4 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a355/ThurstonBennequinNumber Hyperbolic Volume 8.42324 A-Polynomial See Data:K11a355/A-polynomial

[edit Notes for K11a355's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant -8

[edit Notes for K11a355's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^4-4 t^3+6 t^2-7 t+7-7 t^{-1} +6 t^{-2} -4 t^{-3} +2 t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^8+12 z^6+22 z^4+13 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 45, 8 }
Jones polynomial	[math]\displaystyle{ -q^{15}+2 q^{14}-4 q^{13}+5 q^{12}-6 q^{11}+7 q^{10}-6 q^9+5 q^8-4 q^7+3 q^6-q^5+q^4 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +7 z^6 a^{-8} +6 z^6 a^{-10} -z^6 a^{-12} +16 z^4 a^{-8} +11 z^4 a^{-10} -5 z^4 a^{-12} +13 z^2 a^{-8} +7 z^2 a^{-10} -7 z^2 a^{-12} +2 a^{-8} +2 a^{-10} -3 a^{-12} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-9} +3 z^9 a^{-11} +2 z^9 a^{-13} +z^8 a^{-8} -5 z^8 a^{-10} -3 z^8 a^{-12} +3 z^8 a^{-14} -5 z^7 a^{-9} -14 z^7 a^{-11} -6 z^7 a^{-13} +3 z^7 a^{-15} -7 z^6 a^{-8} +8 z^6 a^{-10} +3 z^6 a^{-12} -9 z^6 a^{-14} +3 z^6 a^{-16} +6 z^5 a^{-9} +18 z^5 a^{-11} +4 z^5 a^{-13} -5 z^5 a^{-15} +3 z^5 a^{-17} +16 z^4 a^{-8} -8 z^4 a^{-10} -9 z^4 a^{-12} +10 z^4 a^{-14} -3 z^4 a^{-16} +2 z^4 a^{-18} -8 z^3 a^{-11} -2 z^3 a^{-13} +2 z^3 a^{-15} -3 z^3 a^{-17} +z^3 a^{-19} -13 z^2 a^{-8} +7 z^2 a^{-10} +11 z^2 a^{-12} -7 z^2 a^{-14} +z^2 a^{-16} -z^2 a^{-18} -z a^{-9} +3 z a^{-11} -z a^{-15} +2 z a^{-17} -z a^{-19} +2 a^{-8} -2 a^{-10} -3 a^{-12} }[/math]
The A2 invariant	[math]\displaystyle{ q^{-14} +2 q^{-18} + q^{-22} +2 q^{-28} +2 q^{-32} - q^{-34} - q^{-36} - q^{-38} - q^{-40} - q^{-44} }[/math]
The G2 invariant	Data:K11a355/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a355 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["K11a355"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^4-4 t^3+6 t^2-7 t+7-7 t^{-1} +6 t^{-2} -4 t^{-3} +2 t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^8+12 z^6+22 z^4+13 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]=

{ 45, 8 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^{15}+2 q^{14}-4 q^{13}+5 q^{12}-6 q^{11}+7 q^{10}-6 q^9+5 q^8-4 q^7+3 q^6-q^5+q^4 }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^8 a^{-8} +z^8 a^{-10} +7 z^6 a^{-8} +6 z^6 a^{-10} -z^6 a^{-12} +16 z^4 a^{-8} +11 z^4 a^{-10} -5 z^4 a^{-12} +13 z^2 a^{-8} +7 z^2 a^{-10} -7 z^2 a^{-12} +2 a^{-8} +2 a^{-10} -3 a^{-12} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-9} +3 z^9 a^{-11} +2 z^9 a^{-13} +z^8 a^{-8} -5 z^8 a^{-10} -3 z^8 a^{-12} +3 z^8 a^{-14} -5 z^7 a^{-9} -14 z^7 a^{-11} -6 z^7 a^{-13} +3 z^7 a^{-15} -7 z^6 a^{-8} +8 z^6 a^{-10} +3 z^6 a^{-12} -9 z^6 a^{-14} +3 z^6 a^{-16} +6 z^5 a^{-9} +18 z^5 a^{-11} +4 z^5 a^{-13} -5 z^5 a^{-15} +3 z^5 a^{-17} +16 z^4 a^{-8} -8 z^4 a^{-10} -9 z^4 a^{-12} +10 z^4 a^{-14} -3 z^4 a^{-16} +2 z^4 a^{-18} -8 z^3 a^{-11} -2 z^3 a^{-13} +2 z^3 a^{-15} -3 z^3 a^{-17} +z^3 a^{-19} -13 z^2 a^{-8} +7 z^2 a^{-10} +11 z^2 a^{-12} -7 z^2 a^{-14} +z^2 a^{-16} -z^2 a^{-18} -z a^{-9} +3 z a^{-11} -z a^{-15} +2 z a^{-17} -z a^{-19} +2 a^{-8} -2 a^{-10} -3 a^{-12} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["K11a355"];

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{ 2 t^4-4 t^3+6 t^2-7 t+7-7 t^{-1} +6 t^{-2} -4 t^{-3} +2 t^{-4} }[/math], [math]\displaystyle{ -q^{15}+2 q^{14}-4 q^{13}+5 q^{12}-6 q^{11}+7 q^{10}-6 q^9+5 q^8-4 q^7+3 q^6-q^5+q^4 }[/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]=

{}

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:

(13, 45)

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{ 52 }[/math] [math]\displaystyle{ 360 }[/math] [math]\displaystyle{ 1352 }[/math] [math]\displaystyle{ \frac{9782}{3} }[/math] [math]\displaystyle{ \frac{1474}{3} }[/math] [math]\displaystyle{ 18720 }[/math] [math]\displaystyle{ 32688 }[/math] [math]\displaystyle{ 5792 }[/math] [math]\displaystyle{ 4168 }[/math] [math]\displaystyle{ \frac{70304}{3} }[/math] [math]\displaystyle{ 64800 }[/math] [math]\displaystyle{ \frac{508664}{3} }[/math] [math]\displaystyle{ \frac{76648}{3} }[/math] [math]\displaystyle{ \frac{10027603}{30} }[/math] [math]\displaystyle{ \frac{174674}{15} }[/math] [math]\displaystyle{ \frac{5685806}{45} }[/math] [math]\displaystyle{ \frac{35117}{18} }[/math] [math]\displaystyle{ \frac{488563}{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]8 is the signature of K11a355. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 11 χ 31                       1 -1 29                     1   1 27                   3 1   -2 25                 2 1     1 23               4 3       -1 21             3 2         1 19           3 4           1 17         2 3             -1 15       2 3               1 13     1 2                 -1 11     2                   2 9 1 1                     0 7 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=7 }[/math] [math]\displaystyle{ i=9 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=9 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=10 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=11 }[/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.

K11a354 K11a356