K11a40

K11a39

K11a41

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a40 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,5,15,6 X2837 X18,10,19,9 X20,12,21,11 X16,13,17,14 X6,15,7,16 X22,18,1,17 X10,20,11,19 X12,22,13,21
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, 5, -10, 6, -11, 7, -3, 8, -7, 9, -5, 10, -6, 11, -9
Dowker-Thistlethwaite code	4 8 14 2 18 20 16 6 22 10 12

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a40/ThurstonBennequinNumber Hyperbolic Volume 12.8867 A-Polynomial See Data:K11a40/A-polynomial

[edit Notes for K11a40'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 K11a40's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-5 t^3+12 t^2-17 t+19-17 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+3 z^6+2 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 89, 4 }
Jones polynomial	[math]\displaystyle{ q^{10}-3 q^9+6 q^8-10 q^7+12 q^6-14 q^5+14 q^4-11 q^3+9 q^2-5 q+3- q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-4} -z^6 a^{-2} +6 z^6 a^{-4} -2 z^6 a^{-6} -4 z^4 a^{-2} +14 z^4 a^{-4} -9 z^4 a^{-6} +z^4 a^{-8} -4 z^2 a^{-2} +16 z^2 a^{-4} -13 z^2 a^{-6} +3 z^2 a^{-8} - a^{-2} +7 a^{-4} -7 a^{-6} +2 a^{-8} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-4} +z^{10} a^{-6} +3 z^9 a^{-3} +6 z^9 a^{-5} +3 z^9 a^{-7} +3 z^8 a^{-2} +5 z^8 a^{-4} +7 z^8 a^{-6} +5 z^8 a^{-8} +z^7 a^{-1} -9 z^7 a^{-3} -14 z^7 a^{-5} +2 z^7 a^{-7} +6 z^7 a^{-9} -13 z^6 a^{-2} -30 z^6 a^{-4} -26 z^6 a^{-6} -4 z^6 a^{-8} +5 z^6 a^{-10} -4 z^5 a^{-1} +2 z^5 a^{-3} -16 z^5 a^{-7} -7 z^5 a^{-9} +3 z^5 a^{-11} +17 z^4 a^{-2} +42 z^4 a^{-4} +30 z^4 a^{-6} -z^4 a^{-8} -5 z^4 a^{-10} +z^4 a^{-12} +4 z^3 a^{-1} +8 z^3 a^{-3} +14 z^3 a^{-5} +17 z^3 a^{-7} +4 z^3 a^{-9} -3 z^3 a^{-11} -8 z^2 a^{-2} -25 z^2 a^{-4} -21 z^2 a^{-6} -z^2 a^{-8} +2 z^2 a^{-10} -z^2 a^{-12} -z a^{-1} -3 z a^{-3} -8 z a^{-5} -8 z a^{-7} -z a^{-9} +z a^{-11} + a^{-2} +7 a^{-4} +7 a^{-6} +2 a^{-8} }[/math]
The A2 invariant	[math]\displaystyle{ -q^2+1- q^{-2} + q^{-4} +2 q^{-6} +5 q^{-10} - q^{-12} +2 q^{-14} - q^{-16} -3 q^{-18} -3 q^{-22} + q^{-24} + q^{-30} }[/math]
The G2 invariant	[math]\displaystyle{ q^{12}-2 q^{10}+5 q^8-9 q^6+10 q^4-11 q^2+3+14 q^{-2} -36 q^{-4} +57 q^{-6} -65 q^{-8} +46 q^{-10} -4 q^{-12} -58 q^{-14} +118 q^{-16} -146 q^{-18} +129 q^{-20} -62 q^{-22} -37 q^{-24} +130 q^{-26} -180 q^{-28} +171 q^{-30} -98 q^{-32} + q^{-34} +95 q^{-36} -139 q^{-38} +127 q^{-40} -58 q^{-42} -22 q^{-44} +94 q^{-46} -112 q^{-48} +76 q^{-50} +5 q^{-52} -92 q^{-54} +161 q^{-56} -166 q^{-58} +111 q^{-60} -6 q^{-62} -115 q^{-64} +201 q^{-66} -231 q^{-68} +181 q^{-70} -73 q^{-72} -59 q^{-74} +157 q^{-76} -198 q^{-78} +162 q^{-80} -78 q^{-82} -24 q^{-84} +87 q^{-86} -106 q^{-88} +69 q^{-90} -6 q^{-92} -55 q^{-94} +86 q^{-96} -73 q^{-98} +23 q^{-100} +37 q^{-102} -92 q^{-104} +116 q^{-106} -103 q^{-108} +64 q^{-110} -6 q^{-112} -52 q^{-114} +97 q^{-116} -115 q^{-118} +106 q^{-120} -67 q^{-122} +20 q^{-124} +27 q^{-126} -63 q^{-128} +77 q^{-130} -70 q^{-132} +51 q^{-134} -20 q^{-136} -6 q^{-138} +23 q^{-140} -31 q^{-142} +28 q^{-144} -20 q^{-146} +11 q^{-148} -2 q^{-150} -4 q^{-152} +5 q^{-154} -6 q^{-156} +4 q^{-158} -2 q^{-160} + q^{-162} }[/math]

Further Quantum Invariants

K11a40 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["K11a40"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-5 t^3+12 t^2-17 t+19-17 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+3 z^6+2 z^4+2 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]=

{ 89, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{10}-3 q^9+6 q^8-10 q^7+12 q^6-14 q^5+14 q^4-11 q^3+9 q^2-5 q+3- 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} +6 z^6 a^{-4} -2 z^6 a^{-6} -4 z^4 a^{-2} +14 z^4 a^{-4} -9 z^4 a^{-6} +z^4 a^{-8} -4 z^2 a^{-2} +16 z^2 a^{-4} -13 z^2 a^{-6} +3 z^2 a^{-8} - a^{-2} +7 a^{-4} -7 a^{-6} +2 a^{-8} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{10} a^{-4} +z^{10} a^{-6} +3 z^9 a^{-3} +6 z^9 a^{-5} +3 z^9 a^{-7} +3 z^8 a^{-2} +5 z^8 a^{-4} +7 z^8 a^{-6} +5 z^8 a^{-8} +z^7 a^{-1} -9 z^7 a^{-3} -14 z^7 a^{-5} +2 z^7 a^{-7} +6 z^7 a^{-9} -13 z^6 a^{-2} -30 z^6 a^{-4} -26 z^6 a^{-6} -4 z^6 a^{-8} +5 z^6 a^{-10} -4 z^5 a^{-1} +2 z^5 a^{-3} -16 z^5 a^{-7} -7 z^5 a^{-9} +3 z^5 a^{-11} +17 z^4 a^{-2} +42 z^4 a^{-4} +30 z^4 a^{-6} -z^4 a^{-8} -5 z^4 a^{-10} +z^4 a^{-12} +4 z^3 a^{-1} +8 z^3 a^{-3} +14 z^3 a^{-5} +17 z^3 a^{-7} +4 z^3 a^{-9} -3 z^3 a^{-11} -8 z^2 a^{-2} -25 z^2 a^{-4} -21 z^2 a^{-6} -z^2 a^{-8} +2 z^2 a^{-10} -z^2 a^{-12} -z a^{-1} -3 z a^{-3} -8 z a^{-5} -8 z a^{-7} -z a^{-9} +z a^{-11} + a^{-2} +7 a^{-4} +7 a^{-6} +2 a^{-8} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a330,}

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["K11a40"];

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+12 t^2-17 t+19-17 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^{10}-3 q^9+6 q^8-10 q^7+12 q^6-14 q^5+14 q^4-11 q^3+9 q^2-5 q+3- 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]=

{K11a330,}

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:

(2, 1)

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{ 8 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{20}{3} }[/math] [math]\displaystyle{ -\frac{4}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ -\frac{208}{3} }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{160}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ -\frac{5129}{15} }[/math] [math]\displaystyle{ -\frac{2564}{15} }[/math] [math]\displaystyle{ \frac{11284}{45} }[/math] [math]\displaystyle{ -\frac{103}{9} }[/math] [math]\displaystyle{ \frac{631}{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]4 is the signature of K11a40. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 21                       1 1 19                     2   -2 17                   4 1   3 15                 6 2     -4 13               6 4       2 11             8 6         -2 9           6 6           0 7         5 8             3 5       4 6               -2 3     2 6                 4 1   1 3                   -2 -1   2                     2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/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}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=8 }[/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.

K11a39 K11a41