K11a221

K11a220

K11a222

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a221 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible 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:K11a221/ThurstonBennequinNumber Hyperbolic Volume 12.2428 A-Polynomial See Data:K11a221/A-polynomial

[edit Notes for K11a221'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 K11a221's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+5 t^3-10 t^2+15 t-17+15 t^{-1} -10 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-3 z^6+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 79, -2 }
Jones polynomial	[math]\displaystyle{ q^3-3 q^2+5 q-8+11 q^{-1} -12 q^{-2} +13 q^{-3} -10 q^{-4} +8 q^{-5} -5 q^{-6} +2 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^2 z^8+2 a^4 z^6-6 a^2 z^6+z^6-a^6 z^4+10 a^4 z^4-13 a^2 z^4+4 z^4-4 a^6 z^2+16 a^4 z^2-12 a^2 z^2+4 z^2-4 a^6+8 a^4-4 a^2+1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^4 z^{10}+a^2 z^{10}+2 a^5 z^9+5 a^3 z^9+3 a z^9+3 a^6 z^8+2 a^4 z^8+3 a^2 z^8+4 z^8+3 a^7 z^7-a^5 z^7-14 a^3 z^7-7 a z^7+3 z^7 a^{-1} +2 a^8 z^6-6 a^6 z^6-11 a^4 z^6-17 a^2 z^6+z^6 a^{-2} -13 z^6+a^9 z^5-6 a^7 z^5-4 a^5 z^5+14 a^3 z^5+a z^5-10 z^5 a^{-1} -4 a^8 z^4+9 a^6 z^4+24 a^4 z^4+25 a^2 z^4-3 z^4 a^{-2} +11 z^4-3 a^9 z^3+4 a^7 z^3+11 a^5 z^3+2 a^3 z^3+5 a z^3+7 z^3 a^{-1} +a^8 z^2-10 a^6 z^2-21 a^4 z^2-15 a^2 z^2+z^2 a^{-2} -4 z^2+2 a^9 z-2 a^7 z-7 a^5 z-5 a^3 z-3 a z-z a^{-1} +4 a^6+8 a^4+4 a^2+1 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{24}-q^{22}-q^{20}-2 q^{18}+2 q^{16}+3 q^{12}+3 q^{10}+3 q^6-3 q^4+q^2-1- q^{-2} + q^{-4} - q^{-6} + q^{-8} }[/math]
The G2 invariant	[math]\displaystyle{ q^{128}-q^{126}+3 q^{124}-4 q^{122}+4 q^{120}-3 q^{118}-q^{116}+8 q^{114}-15 q^{112}+21 q^{110}-22 q^{108}+14 q^{106}-2 q^{104}-18 q^{102}+39 q^{100}-53 q^{98}+53 q^{96}-41 q^{94}+9 q^{92}+23 q^{90}-57 q^{88}+78 q^{86}-84 q^{84}+66 q^{82}-32 q^{80}-20 q^{78}+63 q^{76}-91 q^{74}+91 q^{72}-62 q^{70}+11 q^{68}+39 q^{66}-70 q^{64}+68 q^{62}-27 q^{60}-25 q^{58}+76 q^{56}-83 q^{54}+50 q^{52}+20 q^{50}-89 q^{48}+143 q^{46}-137 q^{44}+89 q^{42}-q^{40}-84 q^{38}+151 q^{36}-164 q^{34}+128 q^{32}-53 q^{30}-32 q^{28}+98 q^{26}-124 q^{24}+104 q^{22}-46 q^{20}-23 q^{18}+69 q^{16}-85 q^{14}+50 q^{12}+10 q^{10}-72 q^8+103 q^6-93 q^4+38 q^2+35-100 q^{-2} +128 q^{-4} -112 q^{-6} +63 q^{-8} +2 q^{-10} -61 q^{-12} +93 q^{-14} -91 q^{-16} +68 q^{-18} -28 q^{-20} -8 q^{-22} +31 q^{-24} -40 q^{-26} +36 q^{-28} -24 q^{-30} +12 q^{-32} + q^{-34} -7 q^{-36} +7 q^{-38} -7 q^{-40} +4 q^{-42} -2 q^{-44} + q^{-46} }[/math]

Further Quantum Invariants

K11a221 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["K11a221"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+5 t^3-10 t^2+15 t-17+15 t^{-1} -10 t^{-2} +5 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 79, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^3-3 q^2+5 q-8+11 q^{-1} -12 q^{-2} +13 q^{-3} -10 q^{-4} +8 q^{-5} -5 q^{-6} +2 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^6-6 a^2 z^6+z^6-a^6 z^4+10 a^4 z^4-13 a^2 z^4+4 z^4-4 a^6 z^2+16 a^4 z^2-12 a^2 z^2+4 z^2-4 a^6+8 a^4-4 a^2+1 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^4 z^{10}+a^2 z^{10}+2 a^5 z^9+5 a^3 z^9+3 a z^9+3 a^6 z^8+2 a^4 z^8+3 a^2 z^8+4 z^8+3 a^7 z^7-a^5 z^7-14 a^3 z^7-7 a z^7+3 z^7 a^{-1} +2 a^8 z^6-6 a^6 z^6-11 a^4 z^6-17 a^2 z^6+z^6 a^{-2} -13 z^6+a^9 z^5-6 a^7 z^5-4 a^5 z^5+14 a^3 z^5+a z^5-10 z^5 a^{-1} -4 a^8 z^4+9 a^6 z^4+24 a^4 z^4+25 a^2 z^4-3 z^4 a^{-2} +11 z^4-3 a^9 z^3+4 a^7 z^3+11 a^5 z^3+2 a^3 z^3+5 a z^3+7 z^3 a^{-1} +a^8 z^2-10 a^6 z^2-21 a^4 z^2-15 a^2 z^2+z^2 a^{-2} -4 z^2+2 a^9 z-2 a^7 z-7 a^5 z-5 a^3 z-3 a z-z a^{-1} +4 a^6+8 a^4+4 a^2+1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a259,}

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["K11a221"];

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+5 t^3-10 t^2+15 t-17+15 t^{-1} -10 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^3-3 q^2+5 q-8+11 q^{-1} -12 q^{-2} +13 q^{-3} -10 q^{-4} +8 q^{-5} -5 q^{-6} +2 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]=

{K11a259,}

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:

(4, -8)

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{ 16 }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{968}{3} }[/math] [math]\displaystyle{ \frac{184}{3} }[/math] [math]\displaystyle{ -1024 }[/math] [math]\displaystyle{ -\frac{5536}{3} }[/math] [math]\displaystyle{ -\frac{736}{3} }[/math] [math]\displaystyle{ -384 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 2048 }[/math] [math]\displaystyle{ \frac{15488}{3} }[/math] [math]\displaystyle{ \frac{2944}{3} }[/math] [math]\displaystyle{ \frac{157382}{15} }[/math] [math]\displaystyle{ -\frac{8648}{15} }[/math] [math]\displaystyle{ \frac{229688}{45} }[/math] [math]\displaystyle{ \frac{1834}{9} }[/math] [math]\displaystyle{ \frac{11222}{15} }[/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 K11a221. 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                     2   -2 3                   3 1   2 1                 5 2     -3 -1               6 3       3 -3             7 6         -1 -5           6 5           1 -7         4 7             3 -9       4 6               -2 -11     1 4                 3 -13   1 4                   -3 -15   1                     1 -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

K11a220 K11a222