K11n23

K11n22

K11n24

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n23 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,13,6,12 X2837 X9,15,10,14 X11,19,12,18 X13,7,14,6 X15,20,16,21 X17,11,18,10 X19,22,20,1 X21,16,22,17
Gauss code	1, -4, 2, -1, -3, 7, 4, -2, -5, 9, -6, 3, -7, 5, -8, 11, -9, 6, -10, 8, -11, 10
Dowker-Thistlethwaite code	4 8 -12 2 -14 -18 -6 -20 -10 -22 -16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number [math]\displaystyle{ \{2,3\} }[/math] 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n23/ThurstonBennequinNumber Hyperbolic Volume 10.8083 A-Polynomial See Data:K11n23/A-polynomial

[edit Notes for K11n23's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n23's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-3 t^3+5 t^2-4 t+3-4 t^{-1} +5 t^{-2} -3 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+5 z^6+7 z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 29, 4 }
Jones polynomial	[math]\displaystyle{ -2 q^7+3 q^6-4 q^5+5 q^4-4 q^3+5 q^2-3 q+2- q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-4} -z^6 a^{-2} +7 z^6 a^{-4} -z^6 a^{-6} -5 z^4 a^{-2} +18 z^4 a^{-4} -6 z^4 a^{-6} -7 z^2 a^{-2} +22 z^2 a^{-4} -11 z^2 a^{-6} +z^2 a^{-8} -3 a^{-2} +10 a^{-4} -7 a^{-6} + a^{-8} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +5 z^8 a^{-4} +3 z^8 a^{-6} +z^7 a^{-1} -z^7 a^{-3} +2 z^7 a^{-7} -10 z^6 a^{-2} -25 z^6 a^{-4} -15 z^6 a^{-6} -5 z^5 a^{-1} -12 z^5 a^{-3} -16 z^5 a^{-5} -9 z^5 a^{-7} +15 z^4 a^{-2} +39 z^4 a^{-4} +25 z^4 a^{-6} +z^4 a^{-8} +7 z^3 a^{-1} +21 z^3 a^{-3} +27 z^3 a^{-5} +13 z^3 a^{-7} -10 z^2 a^{-2} -29 z^2 a^{-4} -20 z^2 a^{-6} -z^2 a^{-8} -3 z a^{-1} -9 z a^{-3} -13 z a^{-5} -6 z a^{-7} +z a^{-9} +3 a^{-2} +10 a^{-4} +7 a^{-6} + a^{-8} }[/math]
The A2 invariant	[math]\displaystyle{ -q^2- q^{-2} + q^{-6} +2 q^{-8} +4 q^{-10} + q^{-12} +3 q^{-14} - q^{-16} - q^{-18} -2 q^{-20} -2 q^{-22} - q^{-26} + q^{-28} }[/math]
The G2 invariant	Data:K11n23/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n23 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["K11n23"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+5 z^6+7 z^4+5 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]=

{ 29, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -2 q^7+3 q^6-4 q^5+5 q^4-4 q^3+5 q^2-3 q+2- q^{-1} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^8 a^{-4} -z^6 a^{-2} +7 z^6 a^{-4} -z^6 a^{-6} -5 z^4 a^{-2} +18 z^4 a^{-4} -6 z^4 a^{-6} -7 z^2 a^{-2} +22 z^2 a^{-4} -11 z^2 a^{-6} +z^2 a^{-8} -3 a^{-2} +10 a^{-4} -7 a^{-6} + a^{-8} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +5 z^8 a^{-4} +3 z^8 a^{-6} +z^7 a^{-1} -z^7 a^{-3} +2 z^7 a^{-7} -10 z^6 a^{-2} -25 z^6 a^{-4} -15 z^6 a^{-6} -5 z^5 a^{-1} -12 z^5 a^{-3} -16 z^5 a^{-5} -9 z^5 a^{-7} +15 z^4 a^{-2} +39 z^4 a^{-4} +25 z^4 a^{-6} +z^4 a^{-8} +7 z^3 a^{-1} +21 z^3 a^{-3} +27 z^3 a^{-5} +13 z^3 a^{-7} -10 z^2 a^{-2} -29 z^2 a^{-4} -20 z^2 a^{-6} -z^2 a^{-8} -3 z a^{-1} -9 z a^{-3} -13 z a^{-5} -6 z a^{-7} +z a^{-9} +3 a^{-2} +10 a^{-4} +7 a^{-6} + a^{-8} }[/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["K11n23"];

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^4-3 t^3+5 t^2-4 t+3-4 t^{-1} +5 t^{-2} -3 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -2 q^7+3 q^6-4 q^5+5 q^4-4 q^3+5 q^2-3 q+2- q^{-1} }[/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:

(5, 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{ 20 }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ \frac{1030}{3} }[/math] [math]\displaystyle{ \frac{122}{3} }[/math] [math]\displaystyle{ 1280 }[/math] [math]\displaystyle{ \frac{5728}{3} }[/math] [math]\displaystyle{ \frac{928}{3} }[/math] [math]\displaystyle{ 224 }[/math] [math]\displaystyle{ \frac{4000}{3} }[/math] [math]\displaystyle{ 2048 }[/math] [math]\displaystyle{ \frac{20600}{3} }[/math] [math]\displaystyle{ \frac{2440}{3} }[/math] [math]\displaystyle{ \frac{67135}{6} }[/math] [math]\displaystyle{ \frac{1666}{3} }[/math] [math]\displaystyle{ \frac{33662}{9} }[/math] [math]\displaystyle{ \frac{1669}{18} }[/math] [math]\displaystyle{ \frac{2623}{6} }[/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]4 is the signature of K11n23. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

K11n22 K11n24