K11a217

K11a216

K11a218

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a217 at Knotilus!

Knot presentations

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

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:K11a217/ThurstonBennequinNumber Hyperbolic Volume 17.2462 A-Polynomial See Data:K11a217/A-polynomial

[edit Notes for K11a217's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a217's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+6 t^3-16 t^2+29 t-35+29 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-2 z^6+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 139, -2 }
Jones polynomial	[math]\displaystyle{ q^3-4 q^2+9 q-15+20 q^{-1} -22 q^{-2} +23 q^{-3} -19 q^{-4} +14 q^{-5} -8 q^{-6} +3 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^2 z^8+2 a^4 z^6-5 a^2 z^6+z^6-a^6 z^4+8 a^4 z^4-10 a^2 z^4+3 z^4-3 a^6 z^2+12 a^4 z^2-9 a^2 z^2+3 z^2-3 a^6+6 a^4-3 a^2+1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 a^4 z^{10}+2 a^2 z^{10}+6 a^5 z^9+12 a^3 z^9+6 a z^9+8 a^6 z^8+14 a^4 z^8+13 a^2 z^8+7 z^8+6 a^7 z^7-a^5 z^7-16 a^3 z^7-5 a z^7+4 z^7 a^{-1} +3 a^8 z^6-13 a^6 z^6-38 a^4 z^6-38 a^2 z^6+z^6 a^{-2} -15 z^6+a^9 z^5-9 a^7 z^5-13 a^5 z^5-5 a^3 z^5-11 a z^5-9 z^5 a^{-1} -4 a^8 z^4+14 a^6 z^4+38 a^4 z^4+31 a^2 z^4-2 z^4 a^{-2} +9 z^4-2 a^9 z^3+7 a^7 z^3+18 a^5 z^3+13 a^3 z^3+10 a z^3+6 z^3 a^{-1} +a^8 z^2-9 a^6 z^2-20 a^4 z^2-14 a^2 z^2+z^2 a^{-2} -3 z^2+a^9 z-3 a^7 z-7 a^5 z-5 a^3 z-3 a z-z a^{-1} +3 a^6+6 a^4+3 a^2+1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{24}-q^{20}-3 q^{18}+4 q^{16}-2 q^{14}+3 q^{12}+3 q^{10}-2 q^8+5 q^6-5 q^4+3 q^2-1-2 q^{-2} +3 q^{-4} -2 q^{-6} + q^{-8} }[/math]
The G2 invariant	[math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+10 q^{120}-10 q^{118}+4 q^{116}+10 q^{114}-29 q^{112}+52 q^{110}-73 q^{108}+76 q^{106}-58 q^{104}+5 q^{102}+83 q^{100}-182 q^{98}+271 q^{96}-306 q^{94}+242 q^{92}-75 q^{90}-188 q^{88}+468 q^{86}-660 q^{84}+663 q^{82}-438 q^{80}+11 q^{78}+468 q^{76}-819 q^{74}+889 q^{72}-622 q^{70}+115 q^{68}+415 q^{66}-748 q^{64}+719 q^{62}-337 q^{60}-214 q^{58}+689 q^{56}-838 q^{54}+581 q^{52}-653 q^{48}+1119 q^{46}-1184 q^{44}+816 q^{42}-131 q^{40}-614 q^{38}+1165 q^{36}-1312 q^{34}+1024 q^{32}-406 q^{30}-307 q^{28}+840 q^{26}-1013 q^{24}+772 q^{22}-239 q^{20}-347 q^{18}+723 q^{16}-730 q^{14}+362 q^{12}+203 q^{10}-707 q^8+918 q^6-745 q^4+257 q^2+326-774 q^{-2} +931 q^{-4} -752 q^{-6} +356 q^{-8} +92 q^{-10} -432 q^{-12} +563 q^{-14} -494 q^{-16} +304 q^{-18} -82 q^{-20} -88 q^{-22} +171 q^{-24} -178 q^{-26} +131 q^{-28} -67 q^{-30} +18 q^{-32} +14 q^{-34} -25 q^{-36} +22 q^{-38} -16 q^{-40} +8 q^{-42} -3 q^{-44} + q^{-46} }[/math]

Further Quantum Invariants

K11a217 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["K11a217"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+6 t^3-16 t^2+29 t-35+29 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 139, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^3-4 q^2+9 q-15+20 q^{-1} -22 q^{-2} +23 q^{-3} -19 q^{-4} +14 q^{-5} -8 q^{-6} +3 q^{-7} - q^{-8} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -a^2 z^8+2 a^4 z^6-5 a^2 z^6+z^6-a^6 z^4+8 a^4 z^4-10 a^2 z^4+3 z^4-3 a^6 z^2+12 a^4 z^2-9 a^2 z^2+3 z^2-3 a^6+6 a^4-3 a^2+1 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 a^4 z^{10}+2 a^2 z^{10}+6 a^5 z^9+12 a^3 z^9+6 a z^9+8 a^6 z^8+14 a^4 z^8+13 a^2 z^8+7 z^8+6 a^7 z^7-a^5 z^7-16 a^3 z^7-5 a z^7+4 z^7 a^{-1} +3 a^8 z^6-13 a^6 z^6-38 a^4 z^6-38 a^2 z^6+z^6 a^{-2} -15 z^6+a^9 z^5-9 a^7 z^5-13 a^5 z^5-5 a^3 z^5-11 a z^5-9 z^5 a^{-1} -4 a^8 z^4+14 a^6 z^4+38 a^4 z^4+31 a^2 z^4-2 z^4 a^{-2} +9 z^4-2 a^9 z^3+7 a^7 z^3+18 a^5 z^3+13 a^3 z^3+10 a z^3+6 z^3 a^{-1} +a^8 z^2-9 a^6 z^2-20 a^4 z^2-14 a^2 z^2+z^2 a^{-2} -3 z^2+a^9 z-3 a^7 z-7 a^5 z-5 a^3 z-3 a z-z a^{-1} +3 a^6+6 a^4+3 a^2+1 }[/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["K11a217"];

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^3-16 t^2+29 t-35+29 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^3-4 q^2+9 q-15+20 q^{-1} -22 q^{-2} +23 q^{-3} -19 q^{-4} +14 q^{-5} -8 q^{-6} +3 q^{-7} - q^{-8} }[/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, -6)

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{ -48 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 206 }[/math] [math]\displaystyle{ 34 }[/math] [math]\displaystyle{ -576 }[/math] [math]\displaystyle{ -1056 }[/math] [math]\displaystyle{ -128 }[/math] [math]\displaystyle{ -208 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 1152 }[/math] [math]\displaystyle{ 2472 }[/math] [math]\displaystyle{ 408 }[/math] [math]\displaystyle{ \frac{54591}{10} }[/math] [math]\displaystyle{ -\frac{1022}{5} }[/math] [math]\displaystyle{ \frac{36782}{15} }[/math] [math]\displaystyle{ \frac{737}{6} }[/math] [math]\displaystyle{ \frac{3231}{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]-2 is the signature of K11a217. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   6 1   5 1                 9 3     -6 -1               11 6       5 -3             12 10         -2 -5           11 10           1 -7         8 12             4 -9       6 11               -5 -11     2 8                 6 -13   1 6                   -5 -15   2                     2 -17 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=-1 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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.

K11a216 K11a218