K11a83

K11a82

K11a84

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a83 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible 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:K11a83/ThurstonBennequinNumber Hyperbolic Volume 14.7703 A-Polynomial See Data:K11a83/A-polynomial

[edit Notes for K11a83'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 K11a83's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-5 t^3+14 t^2-23 t+27-23 t^{-1} +14 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+3 z^6+4 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 113, 4 }
Jones polynomial	[math]\displaystyle{ q^{10}-4 q^9+8 q^8-13 q^7+16 q^6-18 q^5+18 q^4-14 q^3+11 q^2-6 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} +15 z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} -5 z^2 a^{-2} +18 z^2 a^{-4} -11 z^2 a^{-6} +2 z^2 a^{-8} -2 a^{-2} +8 a^{-4} -6 a^{-6} + a^{-8} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-4} +z^{10} a^{-6} +3 z^9 a^{-3} +8 z^9 a^{-5} +5 z^9 a^{-7} +3 z^8 a^{-2} +9 z^8 a^{-4} +16 z^8 a^{-6} +10 z^8 a^{-8} +z^7 a^{-1} -6 z^7 a^{-3} -13 z^7 a^{-5} +5 z^7 a^{-7} +11 z^7 a^{-9} -12 z^6 a^{-2} -41 z^6 a^{-4} -51 z^6 a^{-6} -14 z^6 a^{-8} +8 z^6 a^{-10} -4 z^5 a^{-1} -6 z^5 a^{-3} -15 z^5 a^{-5} -32 z^5 a^{-7} -15 z^5 a^{-9} +4 z^5 a^{-11} +16 z^4 a^{-2} +52 z^4 a^{-4} +50 z^4 a^{-6} +6 z^4 a^{-8} -7 z^4 a^{-10} +z^4 a^{-12} +5 z^3 a^{-1} +16 z^3 a^{-3} +31 z^3 a^{-5} +29 z^3 a^{-7} +7 z^3 a^{-9} -2 z^3 a^{-11} -9 z^2 a^{-2} -29 z^2 a^{-4} -24 z^2 a^{-6} -3 z^2 a^{-8} +z^2 a^{-10} -2 z a^{-1} -7 z a^{-3} -13 z a^{-5} -9 z a^{-7} -z a^{-9} +2 a^{-2} +8 a^{-4} +6 a^{-6} + a^{-8} }[/math]
The A2 invariant	[math]\displaystyle{ -q^2+1-2 q^{-2} + q^{-4} +2 q^{-6} - q^{-8} +6 q^{-10} - q^{-12} +3 q^{-14} -3 q^{-18} + q^{-20} -4 q^{-22} + q^{-24} - q^{-28} + q^{-30} }[/math]
The G2 invariant	[math]\displaystyle{ q^{12}-2 q^{10}+6 q^8-11 q^6+14 q^4-16 q^2+5+17 q^{-2} -48 q^{-4} +80 q^{-6} -97 q^{-8} +78 q^{-10} -21 q^{-12} -76 q^{-14} +181 q^{-16} -251 q^{-18} +249 q^{-20} -156 q^{-22} -17 q^{-24} +210 q^{-26} -357 q^{-28} +399 q^{-30} -297 q^{-32} +94 q^{-34} +147 q^{-36} -324 q^{-38} +373 q^{-40} -270 q^{-42} +74 q^{-44} +147 q^{-46} -278 q^{-48} +274 q^{-50} -123 q^{-52} -99 q^{-54} +309 q^{-56} -392 q^{-58} +321 q^{-60} -100 q^{-62} -190 q^{-64} +437 q^{-66} -552 q^{-68} +482 q^{-70} -243 q^{-72} -79 q^{-74} +363 q^{-76} -518 q^{-78} +481 q^{-80} -289 q^{-82} +13 q^{-84} +217 q^{-86} -334 q^{-88} +287 q^{-90} -121 q^{-92} -84 q^{-94} +234 q^{-96} -259 q^{-98} +151 q^{-100} +31 q^{-102} -219 q^{-104} +327 q^{-106} -315 q^{-108} +199 q^{-110} -15 q^{-112} -164 q^{-114} +283 q^{-116} -310 q^{-118} +249 q^{-120} -130 q^{-122} - q^{-124} +106 q^{-126} -168 q^{-128} +173 q^{-130} -136 q^{-132} +81 q^{-134} -17 q^{-136} -29 q^{-138} +53 q^{-140} -62 q^{-142} +51 q^{-144} -32 q^{-146} +15 q^{-148} + q^{-150} -8 q^{-152} +10 q^{-154} -10 q^{-156} +6 q^{-158} -3 q^{-160} + q^{-162} }[/math]

Further Quantum Invariants

K11a83 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["K11a83"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-5 t^3+14 t^2-23 t+27-23 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+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]=

{ 113, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^{10}-4 q^9+8 q^8-13 q^7+16 q^6-18 q^5+18 q^4-14 q^3+11 q^2-6 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} +15 z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} -5 z^2 a^{-2} +18 z^2 a^{-4} -11 z^2 a^{-6} +2 z^2 a^{-8} -2 a^{-2} +8 a^{-4} -6 a^{-6} + 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} +8 z^9 a^{-5} +5 z^9 a^{-7} +3 z^8 a^{-2} +9 z^8 a^{-4} +16 z^8 a^{-6} +10 z^8 a^{-8} +z^7 a^{-1} -6 z^7 a^{-3} -13 z^7 a^{-5} +5 z^7 a^{-7} +11 z^7 a^{-9} -12 z^6 a^{-2} -41 z^6 a^{-4} -51 z^6 a^{-6} -14 z^6 a^{-8} +8 z^6 a^{-10} -4 z^5 a^{-1} -6 z^5 a^{-3} -15 z^5 a^{-5} -32 z^5 a^{-7} -15 z^5 a^{-9} +4 z^5 a^{-11} +16 z^4 a^{-2} +52 z^4 a^{-4} +50 z^4 a^{-6} +6 z^4 a^{-8} -7 z^4 a^{-10} +z^4 a^{-12} +5 z^3 a^{-1} +16 z^3 a^{-3} +31 z^3 a^{-5} +29 z^3 a^{-7} +7 z^3 a^{-9} -2 z^3 a^{-11} -9 z^2 a^{-2} -29 z^2 a^{-4} -24 z^2 a^{-6} -3 z^2 a^{-8} +z^2 a^{-10} -2 z a^{-1} -7 z a^{-3} -13 z a^{-5} -9 z a^{-7} -z a^{-9} +2 a^{-2} +8 a^{-4} +6 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["K11a83"];

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^2-23 t+27-23 t^{-1} +14 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^{10}-4 q^9+8 q^8-13 q^7+16 q^6-18 q^5+18 q^4-14 q^3+11 q^2-6 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]=

{}

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, 6)

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{ 48 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{680}{3} }[/math] [math]\displaystyle{ \frac{88}{3} }[/math] [math]\displaystyle{ 768 }[/math] [math]\displaystyle{ 1152 }[/math] [math]\displaystyle{ 224 }[/math] [math]\displaystyle{ 112 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 1152 }[/math] [math]\displaystyle{ \frac{10880}{3} }[/math] [math]\displaystyle{ \frac{1408}{3} }[/math] [math]\displaystyle{ \frac{90422}{15} }[/math] [math]\displaystyle{ \frac{2344}{5} }[/math] [math]\displaystyle{ \frac{87608}{45} }[/math] [math]\displaystyle{ \frac{58}{9} }[/math] [math]\displaystyle{ \frac{3782}{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 K11a83. 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                     3   -3 17                   5 1   4 15                 8 3     -5 13               8 5       3 11             10 8         -2 9           8 8           0 7         6 10             4 5       5 8               -3 3     2 7                 5 1   1 4                   -3 -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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

K11a82 K11a84