K11a139

K11a138

K11a140

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a139 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a139/ThurstonBennequinNumber Hyperbolic Volume 13.7305 A-Polynomial See Data:K11a139/A-polynomial

[edit Notes for K11a139's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 2 }[/math] Rasmussen s-Invariant -2

[edit Notes for K11a139's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+5 t^3-12 t^2+20 t-23+20 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-3 z^6-2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 99, 2 }
Jones polynomial	[math]\displaystyle{ -q^8+3 q^7-6 q^6+10 q^5-14 q^4+16 q^3-15 q^2+14 q-10+6 q^{-1} -3 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^8 a^{-2} -6 z^6 a^{-2} +2 z^6 a^{-4} +z^6-14 z^4 a^{-2} +9 z^4 a^{-4} -z^4 a^{-6} +4 z^4-14 z^2 a^{-2} +13 z^2 a^{-4} -3 z^2 a^{-6} +5 z^2-4 a^{-2} +5 a^{-4} -2 a^{-6} +2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +6 z^9 a^{-3} +3 z^9 a^{-5} +6 z^8 a^{-2} +7 z^8 a^{-4} +5 z^8 a^{-6} +4 z^8+3 a z^7-3 z^7 a^{-1} -10 z^7 a^{-3} +z^7 a^{-5} +5 z^7 a^{-7} +a^2 z^6-20 z^6 a^{-2} -21 z^6 a^{-4} -9 z^6 a^{-6} +3 z^6 a^{-8} -10 z^6-9 a z^5-5 z^5 a^{-1} +4 z^5 a^{-3} -12 z^5 a^{-5} -11 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+25 z^4 a^{-2} +29 z^4 a^{-4} +8 z^4 a^{-6} -6 z^4 a^{-8} +7 z^4+7 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} +17 z^3 a^{-5} +9 z^3 a^{-7} -2 z^3 a^{-9} +2 a^2 z^2-19 z^2 a^{-2} -19 z^2 a^{-4} -4 z^2 a^{-6} +2 z^2 a^{-8} -4 z^2-a z-2 z a^{-1} -3 z a^{-3} -5 z a^{-5} -3 z a^{-7} +4 a^{-2} +5 a^{-4} +2 a^{-6} +2 }[/math]
The A2 invariant	[math]\displaystyle{ q^8-q^6+2 q^4-q^2-1+2 q^{-2} -3 q^{-4} +4 q^{-6} - q^{-8} + q^{-10} + q^{-12} -2 q^{-14} +3 q^{-16} - q^{-18} - q^{-24} }[/math]
The G2 invariant	[math]\displaystyle{ q^{46}-2 q^{44}+5 q^{42}-9 q^{40}+10 q^{38}-10 q^{36}+2 q^{34}+15 q^{32}-34 q^{30}+55 q^{28}-63 q^{26}+50 q^{24}-15 q^{22}-44 q^{20}+112 q^{18}-162 q^{16}+172 q^{14}-120 q^{12}+15 q^{10}+115 q^8-220 q^6+271 q^4-231 q^2+111+47 q^{-2} -192 q^{-4} +253 q^{-6} -214 q^{-8} +98 q^{-10} +53 q^{-12} -158 q^{-14} +179 q^{-16} -114 q^{-18} -25 q^{-20} +168 q^{-22} -258 q^{-24} +232 q^{-26} -109 q^{-28} -77 q^{-30} +265 q^{-32} -369 q^{-34} +357 q^{-36} -226 q^{-38} +23 q^{-40} +189 q^{-42} -339 q^{-44} +362 q^{-46} -257 q^{-48} +90 q^{-50} +97 q^{-52} -207 q^{-54} +215 q^{-56} -124 q^{-58} -9 q^{-60} +126 q^{-62} -179 q^{-64} +133 q^{-66} -18 q^{-68} -116 q^{-70} +219 q^{-72} -235 q^{-74} +174 q^{-76} -60 q^{-78} -77 q^{-80} +175 q^{-82} -225 q^{-84} +203 q^{-86} -128 q^{-88} +34 q^{-90} +55 q^{-92} -112 q^{-94} +129 q^{-96} -112 q^{-98} +72 q^{-100} -24 q^{-102} -18 q^{-104} +39 q^{-106} -46 q^{-108} +39 q^{-110} -24 q^{-112} +12 q^{-114} + q^{-116} -7 q^{-118} +7 q^{-120} -7 q^{-122} +4 q^{-124} -2 q^{-126} + q^{-128} }[/math]

Further Quantum Invariants

K11a139 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["K11a139"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+5 t^3-12 t^2+20 t-23+20 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+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]=

{ 99, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^8+3 q^7-6 q^6+10 q^5-14 q^4+16 q^3-15 q^2+14 q-10+6 q^{-1} -3 q^{-2} + q^{-3} }[/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} -6 z^6 a^{-2} +2 z^6 a^{-4} +z^6-14 z^4 a^{-2} +9 z^4 a^{-4} -z^4 a^{-6} +4 z^4-14 z^2 a^{-2} +13 z^2 a^{-4} -3 z^2 a^{-6} +5 z^2-4 a^{-2} +5 a^{-4} -2 a^{-6} +2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +6 z^9 a^{-3} +3 z^9 a^{-5} +6 z^8 a^{-2} +7 z^8 a^{-4} +5 z^8 a^{-6} +4 z^8+3 a z^7-3 z^7 a^{-1} -10 z^7 a^{-3} +z^7 a^{-5} +5 z^7 a^{-7} +a^2 z^6-20 z^6 a^{-2} -21 z^6 a^{-4} -9 z^6 a^{-6} +3 z^6 a^{-8} -10 z^6-9 a z^5-5 z^5 a^{-1} +4 z^5 a^{-3} -12 z^5 a^{-5} -11 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+25 z^4 a^{-2} +29 z^4 a^{-4} +8 z^4 a^{-6} -6 z^4 a^{-8} +7 z^4+7 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} +17 z^3 a^{-5} +9 z^3 a^{-7} -2 z^3 a^{-9} +2 a^2 z^2-19 z^2 a^{-2} -19 z^2 a^{-4} -4 z^2 a^{-6} +2 z^2 a^{-8} -4 z^2-a z-2 z a^{-1} -3 z a^{-3} -5 z a^{-5} -3 z a^{-7} +4 a^{-2} +5 a^{-4} +2 a^{-6} +2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a57, K11a108, K11a231,}

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["K11a139"];

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+20 t-23+20 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^8+3 q^7-6 q^6+10 q^5-14 q^4+16 q^3-15 q^2+14 q-10+6 q^{-1} -3 q^{-2} + q^{-3} }[/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]=

{K11a57, K11a108, K11a231,}

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, 3)

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{ 24 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{254}{3} }[/math] [math]\displaystyle{ \frac{58}{3} }[/math] [math]\displaystyle{ 96 }[/math] [math]\displaystyle{ 272 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ 120 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ \frac{1016}{3} }[/math] [math]\displaystyle{ \frac{232}{3} }[/math] [math]\displaystyle{ \frac{34831}{30} }[/math] [math]\displaystyle{ -\frac{5062}{15} }[/math] [math]\displaystyle{ \frac{33062}{45} }[/math] [math]\displaystyle{ \frac{2225}{18} }[/math] [math]\displaystyle{ \frac{2671}{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 K11a139. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

K11a138 K11a140