K11a315

K11a314

K11a316

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a315 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a315/ThurstonBennequinNumber Hyperbolic Volume 18.2725 A-Polynomial See Data:K11a315/A-polynomial

[edit Notes for K11a315's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant 0

[edit Notes for K11a315's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-6 t^3+18 t^2-33 t+41-33 t^{-1} +18 t^{-2} -6 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+2 z^6+2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 157, 0 }
Jones polynomial	[math]\displaystyle{ q^6-4 q^5+9 q^4-16 q^3+22 q^2-25 q+26-22 q^{-1} +17 q^{-2} -10 q^{-3} +4 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +5 z^6-3 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} +11 z^4-4 a^2 z^2-9 z^2 a^{-2} +2 z^2 a^{-4} +12 z^2-2 a^2-4 a^{-2} + a^{-4} +6 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 3 z^{10} a^{-2} +3 z^{10}+10 a z^9+17 z^9 a^{-1} +7 z^9 a^{-3} +13 a^2 z^8+12 z^8 a^{-2} +7 z^8 a^{-4} +18 z^8+9 a^3 z^7-11 a z^7-32 z^7 a^{-1} -8 z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-25 a^2 z^6-41 z^6 a^{-2} -14 z^6 a^{-4} +z^6 a^{-6} -55 z^6+a^5 z^5-13 a^3 z^5-2 a z^5+15 z^5 a^{-1} -6 z^5 a^{-3} -9 z^5 a^{-5} -4 a^4 z^4+23 a^2 z^4+39 z^4 a^{-2} +8 z^4 a^{-4} -2 z^4 a^{-6} +56 z^4-a^5 z^3+7 a^3 z^3+9 a z^3+3 z^3 a^{-1} +8 z^3 a^{-3} +6 z^3 a^{-5} -11 a^2 z^2-19 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -26 z^2-2 a^3 z-4 a z-4 z a^{-1} -3 z a^{-3} -z a^{-5} +2 a^2+4 a^{-2} + a^{-4} +6 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{14}+2 q^{12}-4 q^{10}+2 q^8+q^6-3 q^4+7 q^2-3+5 q^{-2} - q^{-4} -2 q^{-6} +3 q^{-8} -5 q^{-10} +2 q^{-12} - q^{-16} + q^{-18} }[/math]
The G2 invariant	Data:K11a315/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a315 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["K11a315"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-6 t^3+18 t^2-33 t+41-33 t^{-1} +18 t^{-2} -6 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+2 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]=

{ 157, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^6-4 q^5+9 q^4-16 q^3+22 q^2-25 q+26-22 q^{-1} +17 q^{-2} -10 q^{-3} +4 q^{-4} - q^{-5} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +5 z^6-3 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} +11 z^4-4 a^2 z^2-9 z^2 a^{-2} +2 z^2 a^{-4} +12 z^2-2 a^2-4 a^{-2} + a^{-4} +6 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 3 z^{10} a^{-2} +3 z^{10}+10 a z^9+17 z^9 a^{-1} +7 z^9 a^{-3} +13 a^2 z^8+12 z^8 a^{-2} +7 z^8 a^{-4} +18 z^8+9 a^3 z^7-11 a z^7-32 z^7 a^{-1} -8 z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-25 a^2 z^6-41 z^6 a^{-2} -14 z^6 a^{-4} +z^6 a^{-6} -55 z^6+a^5 z^5-13 a^3 z^5-2 a z^5+15 z^5 a^{-1} -6 z^5 a^{-3} -9 z^5 a^{-5} -4 a^4 z^4+23 a^2 z^4+39 z^4 a^{-2} +8 z^4 a^{-4} -2 z^4 a^{-6} +56 z^4-a^5 z^3+7 a^3 z^3+9 a z^3+3 z^3 a^{-1} +8 z^3 a^{-3} +6 z^3 a^{-5} -11 a^2 z^2-19 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -26 z^2-2 a^3 z-4 a z-4 z a^{-1} -3 z a^{-3} -z a^{-5} +2 a^2+4 a^{-2} + a^{-4} +6 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a24, K11a26,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a24, K11a26,}

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["K11a315"];

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^4-6 t^3+18 t^2-33 t+41-33 t^{-1} +18 t^{-2} -6 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^6-4 q^5+9 q^4-16 q^3+22 q^2-25 q+26-22 q^{-1} +17 q^{-2} -10 q^{-3} +4 q^{-4} - q^{-5} }[/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]=

{K11a24, K11a26,}

In[6]:=

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

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

Out[6]=

{K11a24, K11a26,}

Vassiliev invariants

V2 and V3:

(1, -1)

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{ -8 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{34}{3} }[/math] [math]\displaystyle{ -\frac{38}{3} }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ -\frac{176}{3} }[/math] [math]\displaystyle{ \frac{160}{3} }[/math] [math]\displaystyle{ -72 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{136}{3} }[/math] [math]\displaystyle{ -\frac{152}{3} }[/math] [math]\displaystyle{ -\frac{209}{30} }[/math] [math]\displaystyle{ \frac{1738}{15} }[/math] [math]\displaystyle{ -\frac{3898}{45} }[/math] [math]\displaystyle{ -\frac{1039}{18} }[/math] [math]\displaystyle{ -\frac{1169}{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]0 is the signature of K11a315. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   6 1   5 7                 10 3     -7 5               12 6       6 3             13 10         -3 1           13 12           1 -1         10 14             4 -3       7 12               -5 -5     3 10                 7 -7   1 7                   -6 -9   3                     3 -11 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/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.

K11a314 K11a316