K11a233

K11a232

K11a234

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a233 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X14,4,15,3 X10,5,11,6 X20,8,21,7 X22,9,1,10 X18,11,19,12 X2,14,3,13 X8,15,9,16 X6,18,7,17 X12,19,13,20 X16,22,17,21
Gauss code	1, -7, 2, -1, 3, -9, 4, -8, 5, -3, 6, -10, 7, -2, 8, -11, 9, -6, 10, -4, 11, -5
Dowker-Thistlethwaite code	4 14 10 20 22 18 2 8 6 12 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:K11a233/ThurstonBennequinNumber Hyperbolic Volume 18.7296 A-Polynomial See Data:K11a233/A-polynomial

[edit Notes for K11a233's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant 0

[edit Notes for K11a233's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-6 t^3+19 t^2-37 t+47-37 t^{-1} +19 t^{-2} -6 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+2 z^6+3 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 173, 0 }
Jones polynomial	[math]\displaystyle{ q^6-5 q^5+11 q^4-18 q^3+25 q^2-28 q+28-24 q^{-1} +18 q^{-2} -10 q^{-3} +4 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +5 z^6-3 a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} +11 z^4-4 a^2 z^2-6 z^2 a^{-2} +z^2 a^{-4} +10 z^2-a^2- a^{-2} +3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 3 z^{10} a^{-2} +3 z^{10}+10 a z^9+19 z^9 a^{-1} +9 z^9 a^{-3} +13 a^2 z^8+21 z^8 a^{-2} +10 z^8 a^{-4} +24 z^8+9 a^3 z^7-4 a z^7-26 z^7 a^{-1} -8 z^7 a^{-3} +5 z^7 a^{-5} +4 a^4 z^6-21 a^2 z^6-62 z^6 a^{-2} -21 z^6 a^{-4} +z^6 a^{-6} -65 z^6+a^5 z^5-12 a^3 z^5-16 a z^5-8 z^5 a^{-1} -14 z^5 a^{-3} -9 z^5 a^{-5} -4 a^4 z^4+18 a^2 z^4+48 z^4 a^{-2} +13 z^4 a^{-4} -z^4 a^{-6} +56 z^4-a^5 z^3+8 a^3 z^3+17 a z^3+17 z^3 a^{-1} +13 z^3 a^{-3} +4 z^3 a^{-5} +a^4 z^2-8 a^2 z^2-14 z^2 a^{-2} -2 z^2 a^{-4} -21 z^2-2 a^3 z-5 a z-5 z a^{-1} -2 z a^{-3} +a^2+ a^{-2} +3 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{14}+2 q^{12}-4 q^{10}+3 q^8+2 q^6-4 q^4+6 q^2-5+4 q^{-2} - q^{-6} +5 q^{-8} -5 q^{-10} +2 q^{-12} -2 q^{-16} + q^{-18} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+17 q^{72}-19 q^{70}+12 q^{68}+10 q^{66}-43 q^{64}+89 q^{62}-134 q^{60}+156 q^{58}-137 q^{56}+47 q^{54}+124 q^{52}-341 q^{50}+547 q^{48}-650 q^{46}+539 q^{44}-191 q^{42}-364 q^{40}+970 q^{38}-1370 q^{36}+1362 q^{34}-847 q^{32}-58 q^{30}+1029 q^{28}-1692 q^{26}+1741 q^{24}-1115 q^{22}+53 q^{20}+995 q^{18}-1553 q^{16}+1370 q^{14}-502 q^{12}-631 q^{10}+1504 q^8-1703 q^6+1078 q^4+119 q^2-1412+2279 q^{-2} -2327 q^{-4} +1526 q^{-6} -137 q^{-8} -1329 q^{-10} +2335 q^{-12} -2541 q^{-14} +1875 q^{-16} -610 q^{-18} -769 q^{-20} +1750 q^{-22} -1955 q^{-24} +1364 q^{-26} -237 q^{-28} -899 q^{-30} +1524 q^{-32} -1392 q^{-34} +552 q^{-36} +590 q^{-38} -1509 q^{-40} +1818 q^{-42} -1366 q^{-44} +367 q^{-46} +751 q^{-48} -1552 q^{-50} +1747 q^{-52} -1340 q^{-54} +554 q^{-56} +263 q^{-58} -838 q^{-60} +1029 q^{-62} -862 q^{-64} +498 q^{-66} -99 q^{-68} -190 q^{-70} +312 q^{-72} -306 q^{-74} +213 q^{-76} -101 q^{-78} +20 q^{-80} +30 q^{-82} -43 q^{-84} +36 q^{-86} -24 q^{-88} +11 q^{-90} -4 q^{-92} + q^{-94} }[/math]

Further Quantum Invariants

K11a233 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["K11a233"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-6 t^3+19 t^2-37 t+47-37 t^{-1} +19 t^{-2} -6 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 173, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^6-5 q^5+11 q^4-18 q^3+25 q^2-28 q+28-24 q^{-1} +18 q^{-2} -10 q^{-3} +4 q^{-4} - q^{-5} }[/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 z^6-2 z^6 a^{-2} +5 z^6-3 a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} +11 z^4-4 a^2 z^2-6 z^2 a^{-2} +z^2 a^{-4} +10 z^2-a^2- a^{-2} +3 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 3 z^{10} a^{-2} +3 z^{10}+10 a z^9+19 z^9 a^{-1} +9 z^9 a^{-3} +13 a^2 z^8+21 z^8 a^{-2} +10 z^8 a^{-4} +24 z^8+9 a^3 z^7-4 a z^7-26 z^7 a^{-1} -8 z^7 a^{-3} +5 z^7 a^{-5} +4 a^4 z^6-21 a^2 z^6-62 z^6 a^{-2} -21 z^6 a^{-4} +z^6 a^{-6} -65 z^6+a^5 z^5-12 a^3 z^5-16 a z^5-8 z^5 a^{-1} -14 z^5 a^{-3} -9 z^5 a^{-5} -4 a^4 z^4+18 a^2 z^4+48 z^4 a^{-2} +13 z^4 a^{-4} -z^4 a^{-6} +56 z^4-a^5 z^3+8 a^3 z^3+17 a z^3+17 z^3 a^{-1} +13 z^3 a^{-3} +4 z^3 a^{-5} +a^4 z^2-8 a^2 z^2-14 z^2 a^{-2} -2 z^2 a^{-4} -21 z^2-2 a^3 z-5 a z-5 z a^{-1} -2 z a^{-3} +a^2+ a^{-2} +3 }[/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["K11a233"];

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+19 t^2-37 t+47-37 t^{-1} +19 t^{-2} -6 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^6-5 q^5+11 q^4-18 q^3+25 q^2-28 q+28-24 q^{-1} +18 q^{-2} -10 q^{-3} +4 q^{-4} - q^{-5} }[/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, 0)

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{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{34}{3} }[/math] [math]\displaystyle{ -\frac{62}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{136}{3} }[/math] [math]\displaystyle{ -\frac{248}{3} }[/math] [math]\displaystyle{ -\frac{1649}{30} }[/math] [math]\displaystyle{ \frac{86}{5} }[/math] [math]\displaystyle{ -\frac{3778}{45} }[/math] [math]\displaystyle{ -\frac{271}{18} }[/math] [math]\displaystyle{ -\frac{689}{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]0 is the signature of K11a233. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     4   -4 9                   7 1   6 7                 11 4     -7 5               14 7       7 3             14 11         -3 1           14 14           0 -1         11 15             4 -3       7 13               -6 -5     3 11                 8 -7   1 7                   -6 -9   3                     3 -11 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=1 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/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.

K11a232 K11a234