K11a216

K11a215

K11a217

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a216 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a216/ThurstonBennequinNumber Hyperbolic Volume 17.2795 A-Polynomial See Data:K11a216/A-polynomial

[edit Notes for K11a216's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 1 }[/math] Rasmussen s-Invariant -2

[edit Notes for K11a216's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+6 t^3-17 t^2+31 t-37+31 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-2 z^6-z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 147, 2 }
Jones polynomial	[math]\displaystyle{ -q^8+4 q^7-9 q^6+15 q^5-21 q^4+24 q^3-23 q^2+21 q-15+9 q^{-1} -4 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^8 a^{-2} -5 z^6 a^{-2} +2 z^6 a^{-4} +z^6-10 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +3 z^4-8 z^2 a^{-2} +8 z^2 a^{-4} -2 z^2 a^{-6} +3 z^2- a^{-2} +2 a^{-4} - a^{-6} +1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10} a^{-4} +6 z^9 a^{-1} +13 z^9 a^{-3} +7 z^9 a^{-5} +15 z^8 a^{-2} +18 z^8 a^{-4} +10 z^8 a^{-6} +7 z^8+4 a z^7-4 z^7 a^{-1} -16 z^7 a^{-3} +8 z^7 a^{-7} +a^2 z^6-44 z^6 a^{-2} -47 z^6 a^{-4} -15 z^6 a^{-6} +4 z^6 a^{-8} -15 z^6-9 a z^5-13 z^5 a^{-1} -9 z^5 a^{-3} -18 z^5 a^{-5} -12 z^5 a^{-7} +z^5 a^{-9} -2 a^2 z^4+38 z^4 a^{-2} +41 z^4 a^{-4} +10 z^4 a^{-6} -5 z^4 a^{-8} +10 z^4+6 a z^3+12 z^3 a^{-1} +16 z^3 a^{-3} +18 z^3 a^{-5} +7 z^3 a^{-7} -z^3 a^{-9} +a^2 z^2-15 z^2 a^{-2} -15 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} -4 z^2-a z-3 z a^{-1} -4 z a^{-3} -4 z a^{-5} -2 z a^{-7} + a^{-2} +2 a^{-4} + a^{-6} +1 }[/math]
The A2 invariant	[math]\displaystyle{ q^8-2 q^6+3 q^4-2 q^2-1+4 q^{-2} -4 q^{-4} +6 q^{-6} -2 q^{-8} + q^{-10} + q^{-12} -4 q^{-14} +4 q^{-16} -2 q^{-18} + q^{-22} - q^{-24} }[/math]
The G2 invariant	[math]\displaystyle{ q^{46}-3 q^{44}+8 q^{42}-16 q^{40}+22 q^{38}-25 q^{36}+14 q^{34}+18 q^{32}-66 q^{30}+129 q^{28}-176 q^{26}+173 q^{24}-98 q^{22}-67 q^{20}+290 q^{18}-494 q^{16}+587 q^{14}-481 q^{12}+158 q^{10}+304 q^8-746 q^6+998 q^4-914 q^2+492+127 q^{-2} -723 q^{-4} +1027 q^{-6} -919 q^{-8} +449 q^{-10} +188 q^{-12} -688 q^{-14} +841 q^{-16} -570 q^{-18} -12 q^{-20} +643 q^{-22} -1038 q^{-24} +998 q^{-26} -517 q^{-28} -239 q^{-30} +989 q^{-32} -1425 q^{-34} +1393 q^{-36} -879 q^{-38} +60 q^{-40} +769 q^{-42} -1323 q^{-44} +1389 q^{-46} -970 q^{-48} +253 q^{-50} +481 q^{-52} -920 q^{-54} +915 q^{-56} -499 q^{-58} -114 q^{-60} +636 q^{-62} -836 q^{-64} +614 q^{-66} -87 q^{-68} -512 q^{-70} +930 q^{-72} -980 q^{-74} +679 q^{-76} -154 q^{-78} -393 q^{-80} +751 q^{-82} -847 q^{-84} +677 q^{-86} -341 q^{-88} -17 q^{-90} +288 q^{-92} -413 q^{-94} +398 q^{-96} -287 q^{-98} +143 q^{-100} -9 q^{-102} -82 q^{-104} +116 q^{-106} -114 q^{-108} +83 q^{-110} -45 q^{-112} +16 q^{-114} +7 q^{-116} -16 q^{-118} +16 q^{-120} -13 q^{-122} +7 q^{-124} -3 q^{-126} + q^{-128} }[/math]

Further Quantum Invariants

K11a216 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["K11a216"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+6 t^3-17 t^2+31 t-37+31 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+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]=

{ 147, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^8+4 q^7-9 q^6+15 q^5-21 q^4+24 q^3-23 q^2+21 q-15+9 q^{-1} -4 q^{-2} + q^{-3} }[/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} -5 z^6 a^{-2} +2 z^6 a^{-4} +z^6-10 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +3 z^4-8 z^2 a^{-2} +8 z^2 a^{-4} -2 z^2 a^{-6} +3 z^2- a^{-2} +2 a^{-4} - a^{-6} +1 }[/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} +6 z^9 a^{-1} +13 z^9 a^{-3} +7 z^9 a^{-5} +15 z^8 a^{-2} +18 z^8 a^{-4} +10 z^8 a^{-6} +7 z^8+4 a z^7-4 z^7 a^{-1} -16 z^7 a^{-3} +8 z^7 a^{-7} +a^2 z^6-44 z^6 a^{-2} -47 z^6 a^{-4} -15 z^6 a^{-6} +4 z^6 a^{-8} -15 z^6-9 a z^5-13 z^5 a^{-1} -9 z^5 a^{-3} -18 z^5 a^{-5} -12 z^5 a^{-7} +z^5 a^{-9} -2 a^2 z^4+38 z^4 a^{-2} +41 z^4 a^{-4} +10 z^4 a^{-6} -5 z^4 a^{-8} +10 z^4+6 a z^3+12 z^3 a^{-1} +16 z^3 a^{-3} +18 z^3 a^{-5} +7 z^3 a^{-7} -z^3 a^{-9} +a^2 z^2-15 z^2 a^{-2} -15 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} -4 z^2-a z-3 z a^{-1} -4 z a^{-3} -4 z a^{-5} -2 z a^{-7} + a^{-2} +2 a^{-4} + a^{-6} +1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a196, K11a286,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a196,}

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["K11a216"];

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+31 t-37+31 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^8+4 q^7-9 q^6+15 q^5-21 q^4+24 q^3-23 q^2+21 q-15+9 q^{-1} -4 q^{-2} + q^{-3} }[/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]=

{K11a196, K11a286,}

In[6]:=

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

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

Out[6]=

{K11a196,}

Vassiliev invariants

V2 and V3:

(1, 2)

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{ 16 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{158}{3} }[/math] [math]\displaystyle{ \frac{34}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{448}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{632}{3} }[/math] [math]\displaystyle{ \frac{136}{3} }[/math] [math]\displaystyle{ \frac{16591}{30} }[/math] [math]\displaystyle{ -\frac{234}{5} }[/math] [math]\displaystyle{ \frac{10142}{45} }[/math] [math]\displaystyle{ \frac{689}{18} }[/math] [math]\displaystyle{ \frac{271}{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]2 is the signature of K11a216. 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 17                       1 -1 15                     3   3 13                   6 1   -5 11                 9 3     6 9               12 6       -6 7             12 9         3 5           11 12           1 3         10 12             -2 1       6 12               6 -1     3 9                 -6 -3   1 6                   5 -5   3                     -3 -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=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

K11a215 K11a217