K11a215

K11a214

K11a216

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a215 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a215/ThurstonBennequinNumber Hyperbolic Volume 16.7322 A-Polynomial See Data:K11a215/A-polynomial

[edit Notes for K11a215'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 K11a215's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-6 t^3+17 t^2-27 t+31-27 t^{-1} +17 t^{-2} -6 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+2 z^6+z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 133, 4 }
Jones polynomial	[math]\displaystyle{ q^{10}-4 q^9+8 q^8-14 q^7+19 q^6-21 q^5+21 q^4-18 q^3+14 q^2-8 q+4- q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-4} -z^6 a^{-2} +5 z^6 a^{-4} -2 z^6 a^{-6} -3 z^4 a^{-2} +10 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} -2 z^2 a^{-2} +10 z^2 a^{-4} -7 z^2 a^{-6} +2 z^2 a^{-8} +3 a^{-4} -2 a^{-6} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^{10} a^{-4} +2 z^{10} a^{-6} +5 z^9 a^{-3} +13 z^9 a^{-5} +8 z^9 a^{-7} +4 z^8 a^{-2} +10 z^8 a^{-4} +19 z^8 a^{-6} +13 z^8 a^{-8} +z^7 a^{-1} -12 z^7 a^{-3} -27 z^7 a^{-5} -2 z^7 a^{-7} +12 z^7 a^{-9} -14 z^6 a^{-2} -49 z^6 a^{-4} -64 z^6 a^{-6} -21 z^6 a^{-8} +8 z^6 a^{-10} -3 z^5 a^{-1} +z^5 a^{-3} -3 z^5 a^{-5} -26 z^5 a^{-7} -15 z^5 a^{-9} +4 z^5 a^{-11} +16 z^4 a^{-2} +54 z^4 a^{-4} +56 z^4 a^{-6} +11 z^4 a^{-8} -6 z^4 a^{-10} +z^4 a^{-12} +3 z^3 a^{-1} +10 z^3 a^{-3} +22 z^3 a^{-5} +23 z^3 a^{-7} +6 z^3 a^{-9} -2 z^3 a^{-11} -6 z^2 a^{-2} -21 z^2 a^{-4} -18 z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} -z a^{-1} -4 z a^{-3} -8 z a^{-5} -5 z a^{-7} +3 a^{-4} +2 a^{-6} }[/math]
The A2 invariant	[math]\displaystyle{ -q^2+2-2 q^{-2} +2 q^{-4} +2 q^{-6} -2 q^{-8} +5 q^{-10} -4 q^{-12} +3 q^{-14} - q^{-18} +3 q^{-20} -4 q^{-22} + q^{-24} - q^{-26} - q^{-28} + q^{-30} }[/math]
The G2 invariant	[math]\displaystyle{ q^{12}-3 q^{10}+9 q^8-19 q^6+28 q^4-33 q^2+17+26 q^{-2} -91 q^{-4} +165 q^{-6} -205 q^{-8} +166 q^{-10} -35 q^{-12} -176 q^{-14} +395 q^{-16} -518 q^{-18} +476 q^{-20} -232 q^{-22} -146 q^{-24} +522 q^{-26} -735 q^{-28} +685 q^{-30} -371 q^{-32} -82 q^{-34} +483 q^{-36} -670 q^{-38} +562 q^{-40} -208 q^{-42} -215 q^{-44} +535 q^{-46} -584 q^{-48} +345 q^{-50} +76 q^{-52} -513 q^{-54} +774 q^{-56} -745 q^{-58} +424 q^{-60} +89 q^{-62} -602 q^{-64} +937 q^{-66} -952 q^{-68} +640 q^{-70} -115 q^{-72} -433 q^{-74} +789 q^{-76} -837 q^{-78} +567 q^{-80} -103 q^{-82} -338 q^{-84} +580 q^{-86} -525 q^{-88} +214 q^{-90} +182 q^{-92} -488 q^{-94} +556 q^{-96} -372 q^{-98} +23 q^{-100} +335 q^{-102} -564 q^{-104} +587 q^{-106} -407 q^{-108} +116 q^{-110} +170 q^{-112} -372 q^{-114} +426 q^{-116} -361 q^{-118} +225 q^{-120} -58 q^{-122} -76 q^{-124} +164 q^{-126} -192 q^{-128} +168 q^{-130} -118 q^{-132} +59 q^{-134} -5 q^{-136} -35 q^{-138} +55 q^{-140} -60 q^{-142} +48 q^{-144} -29 q^{-146} +14 q^{-148} + q^{-150} -8 q^{-152} +10 q^{-154} -10 q^{-156} +6 q^{-158} -3 q^{-160} + q^{-162} }[/math]

Further Quantum Invariants

K11a215 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["K11a215"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-6 t^3+17 t^2-27 t+31-27 t^{-1} +17 t^{-2} -6 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+2 z^6+z^4+3 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]=

{ 133, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{10}-4 q^9+8 q^8-14 q^7+19 q^6-21 q^5+21 q^4-18 q^3+14 q^2-8 q+4- q^{-1} }[/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} -z^6 a^{-2} +5 z^6 a^{-4} -2 z^6 a^{-6} -3 z^4 a^{-2} +10 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} -2 z^2 a^{-2} +10 z^2 a^{-4} -7 z^2 a^{-6} +2 z^2 a^{-8} +3 a^{-4} -2 a^{-6} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 z^{10} a^{-4} +2 z^{10} a^{-6} +5 z^9 a^{-3} +13 z^9 a^{-5} +8 z^9 a^{-7} +4 z^8 a^{-2} +10 z^8 a^{-4} +19 z^8 a^{-6} +13 z^8 a^{-8} +z^7 a^{-1} -12 z^7 a^{-3} -27 z^7 a^{-5} -2 z^7 a^{-7} +12 z^7 a^{-9} -14 z^6 a^{-2} -49 z^6 a^{-4} -64 z^6 a^{-6} -21 z^6 a^{-8} +8 z^6 a^{-10} -3 z^5 a^{-1} +z^5 a^{-3} -3 z^5 a^{-5} -26 z^5 a^{-7} -15 z^5 a^{-9} +4 z^5 a^{-11} +16 z^4 a^{-2} +54 z^4 a^{-4} +56 z^4 a^{-6} +11 z^4 a^{-8} -6 z^4 a^{-10} +z^4 a^{-12} +3 z^3 a^{-1} +10 z^3 a^{-3} +22 z^3 a^{-5} +23 z^3 a^{-7} +6 z^3 a^{-9} -2 z^3 a^{-11} -6 z^2 a^{-2} -21 z^2 a^{-4} -18 z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} -z a^{-1} -4 z a^{-3} -8 z a^{-5} -5 z a^{-7} +3 a^{-4} +2 a^{-6} }[/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["K11a215"];

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+17 t^2-27 t+31-27 t^{-1} +17 t^{-2} -6 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^{10}-4 q^9+8 q^8-14 q^7+19 q^6-21 q^5+21 q^4-18 q^3+14 q^2-8 q+4- q^{-1} }[/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:

(3, 5)

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{ 12 }[/math] [math]\displaystyle{ 40 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 174 }[/math] [math]\displaystyle{ 26 }[/math] [math]\displaystyle{ 480 }[/math] [math]\displaystyle{ \frac{2608}{3} }[/math] [math]\displaystyle{ \frac{544}{3} }[/math] [math]\displaystyle{ 104 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 800 }[/math] [math]\displaystyle{ 2088 }[/math] [math]\displaystyle{ 312 }[/math] [math]\displaystyle{ \frac{42751}{10} }[/math] [math]\displaystyle{ \frac{2734}{15} }[/math] [math]\displaystyle{ \frac{8434}{5} }[/math] [math]\displaystyle{ \frac{11}{2} }[/math] [math]\displaystyle{ \frac{2431}{10} }[/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 K11a215. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 21                       1 1 19                     3   -3 17                   5 1   4 15                 9 3     -6 13               10 5       5 11             11 9         -2 9           10 10           0 7         8 11             3 5       6 10               -4 3     3 9                 6 1   1 5                   -4 -1   3                     3 -3 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=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/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.

K11a214 K11a216