K11a232

K11a231

K11a233

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a232 at Knotilus!

Knot presentations

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

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:K11a232/ThurstonBennequinNumber Hyperbolic Volume 16.4523 A-Polynomial See Data:K11a232/A-polynomial

[edit Notes for K11a232'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 K11a232's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+5 t^3-14 t^2+26 t-31+26 t^{-1} -14 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-3 z^6-4 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 123, -2 }
Jones polynomial	[math]\displaystyle{ -q^4+4 q^3-8 q^2+13 q-17+20 q^{-1} -19 q^{-2} +17 q^{-3} -13 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^2 z^8+a^4 z^6-6 a^2 z^6+2 z^6+4 a^4 z^4-15 a^2 z^4-z^4 a^{-2} +8 z^4+6 a^4 z^2-16 a^2 z^2-2 z^2 a^{-2} +11 z^2+2 a^4-5 a^2- a^{-2} +5 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 a^2 z^{10}+2 z^{10}+7 a^3 z^9+12 a z^9+5 z^9 a^{-1} +11 a^4 z^8+15 a^2 z^8+4 z^8 a^{-2} +8 z^8+10 a^5 z^7-3 a^3 z^7-27 a z^7-13 z^7 a^{-1} +z^7 a^{-3} +6 a^6 z^6-21 a^4 z^6-56 a^2 z^6-14 z^6 a^{-2} -43 z^6+3 a^7 z^5-17 a^5 z^5-24 a^3 z^5+2 a z^5+3 z^5 a^{-1} -3 z^5 a^{-3} +a^8 z^4-5 a^6 z^4+18 a^4 z^4+58 a^2 z^4+15 z^4 a^{-2} +49 z^4-2 a^7 z^3+16 a^5 z^3+29 a^3 z^3+17 a z^3+9 z^3 a^{-1} +3 z^3 a^{-3} -a^8 z^2+2 a^6 z^2-8 a^4 z^2-29 a^2 z^2-5 z^2 a^{-2} -23 z^2-4 a^5 z-9 a^3 z-8 a z-4 z a^{-1} -z a^{-3} +2 a^4+5 a^2+ a^{-2} +5 }[/math]
The A2 invariant	[math]\displaystyle{ q^{20}-q^{18}+3 q^{16}-2 q^{14}-2 q^{12}+2 q^{10}-4 q^8+4 q^6-2 q^4+q^2+2-2 q^{-2} +4 q^{-4} - q^{-6} + q^{-10} - q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+6 q^{106}-5 q^{104}+8 q^{100}-17 q^{98}+27 q^{96}-36 q^{94}+35 q^{92}-24 q^{90}+2 q^{88}+34 q^{86}-69 q^{84}+102 q^{82}-125 q^{80}+113 q^{78}-69 q^{76}-18 q^{74}+134 q^{72}-233 q^{70}+303 q^{68}-286 q^{66}+170 q^{64}+31 q^{62}-264 q^{60}+441 q^{58}-476 q^{56}+329 q^{54}-59 q^{52}-238 q^{50}+451 q^{48}-459 q^{46}+263 q^{44}+53 q^{42}-355 q^{40}+479 q^{38}-373 q^{36}+46 q^{34}+338 q^{32}-602 q^{30}+642 q^{28}-417 q^{26}+7 q^{24}+423 q^{22}-723 q^{20}+769 q^{18}-561 q^{16}+159 q^{14}+279 q^{12}-582 q^{10}+674 q^8-505 q^6+172 q^4+194 q^2-455+489 q^{-2} -291 q^{-4} -46 q^{-6} +377 q^{-8} -528 q^{-10} +446 q^{-12} -151 q^{-14} -218 q^{-16} +509 q^{-18} -603 q^{-20} +468 q^{-22} -178 q^{-24} -151 q^{-26} +385 q^{-28} -448 q^{-30} +361 q^{-32} -175 q^{-34} -14 q^{-36} +139 q^{-38} -187 q^{-40} +156 q^{-42} -91 q^{-44} +29 q^{-46} +15 q^{-48} -32 q^{-50} +28 q^{-52} -19 q^{-54} +9 q^{-56} -3 q^{-58} + q^{-60} }[/math]

Further Quantum Invariants

K11a232 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["K11a232"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+5 t^3-14 t^2+26 t-31+26 t^{-1} -14 t^{-2} +5 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 123, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^4+4 q^3-8 q^2+13 q-17+20 q^{-1} -19 q^{-2} +17 q^{-3} -13 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7} }[/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+a^4 z^6-6 a^2 z^6+2 z^6+4 a^4 z^4-15 a^2 z^4-z^4 a^{-2} +8 z^4+6 a^4 z^2-16 a^2 z^2-2 z^2 a^{-2} +11 z^2+2 a^4-5 a^2- a^{-2} +5 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 a^2 z^{10}+2 z^{10}+7 a^3 z^9+12 a z^9+5 z^9 a^{-1} +11 a^4 z^8+15 a^2 z^8+4 z^8 a^{-2} +8 z^8+10 a^5 z^7-3 a^3 z^7-27 a z^7-13 z^7 a^{-1} +z^7 a^{-3} +6 a^6 z^6-21 a^4 z^6-56 a^2 z^6-14 z^6 a^{-2} -43 z^6+3 a^7 z^5-17 a^5 z^5-24 a^3 z^5+2 a z^5+3 z^5 a^{-1} -3 z^5 a^{-3} +a^8 z^4-5 a^6 z^4+18 a^4 z^4+58 a^2 z^4+15 z^4 a^{-2} +49 z^4-2 a^7 z^3+16 a^5 z^3+29 a^3 z^3+17 a z^3+9 z^3 a^{-1} +3 z^3 a^{-3} -a^8 z^2+2 a^6 z^2-8 a^4 z^2-29 a^2 z^2-5 z^2 a^{-2} -23 z^2-4 a^5 z-9 a^3 z-8 a z-4 z a^{-1} -z a^{-3} +2 a^4+5 a^2+ a^{-2} +5 }[/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["K11a232"];

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^3-14 t^2+26 t-31+26 t^{-1} -14 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^4+4 q^3-8 q^2+13 q-17+20 q^{-1} -19 q^{-2} +17 q^{-3} -13 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7} }[/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:

(-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{82}{3} }[/math] [math]\displaystyle{ \frac{110}{3} }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ -\frac{608}{3} }[/math] [math]\displaystyle{ -\frac{320}{3} }[/math] [math]\displaystyle{ -48 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ -\frac{328}{3} }[/math] [math]\displaystyle{ -\frac{440}{3} }[/math] [math]\displaystyle{ \frac{9569}{30} }[/math] [math]\displaystyle{ \frac{6622}{15} }[/math] [math]\displaystyle{ -\frac{17342}{45} }[/math] [math]\displaystyle{ \frac{1759}{18} }[/math] [math]\displaystyle{ -\frac{4831}{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 K11a232. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

K11a231 K11a233