K11a179

K11a178

K11a180

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a179 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a179/ThurstonBennequinNumber Hyperbolic Volume 10.0674 A-Polynomial See Data:K11a179/A-polynomial

[edit Notes for K11a179'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 K11a179's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-5 t^3+9 t^2-9 t+9-9 t^{-1} +9 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+3 z^6-z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 57, 4 }
Jones polynomial	[math]\displaystyle{ -q^9+3 q^8-4 q^7+6 q^6-8 q^5+8 q^4-8 q^3+7 q^2-5 q+4-2 q^{-1} + q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-4} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -10 z^4 a^{-2} +12 z^4 a^{-4} -4 z^4 a^{-6} +z^4-13 z^2 a^{-2} +10 z^2 a^{-4} -3 z^2 a^{-6} +4 z^2-4 a^{-2} +2 a^{-4} +3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +5 z^9 a^{-3} +3 z^9 a^{-5} -z^8 a^{-2} +2 z^8 a^{-4} +4 z^8 a^{-6} +z^8-11 z^7 a^{-1} -23 z^7 a^{-3} -8 z^7 a^{-5} +4 z^7 a^{-7} -15 z^6 a^{-2} -22 z^6 a^{-4} -9 z^6 a^{-6} +4 z^6 a^{-8} -6 z^6+18 z^5 a^{-1} +28 z^5 a^{-3} +2 z^5 a^{-5} -4 z^5 a^{-7} +4 z^5 a^{-9} +35 z^4 a^{-2} +31 z^4 a^{-4} +3 z^4 a^{-6} -2 z^4 a^{-8} +3 z^4 a^{-10} +12 z^4-9 z^3 a^{-1} -7 z^3 a^{-3} +z^3 a^{-5} -5 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} -23 z^2 a^{-2} -13 z^2 a^{-4} -z^2 a^{-6} -3 z^2 a^{-8} -2 z^2 a^{-10} -10 z^2-z a^{-3} +z a^{-5} +3 z a^{-7} +z a^{-9} +4 a^{-2} +2 a^{-4} +3 }[/math]
The A2 invariant	[math]\displaystyle{ q^6+q^4+1+ q^{-4} - q^{-8} -3 q^{-12} + q^{-14} + q^{-18} + q^{-20} + q^{-24} - q^{-26} }[/math]
The G2 invariant	[math]\displaystyle{ q^{26}-q^{24}+4 q^{22}-5 q^{20}+6 q^{18}-4 q^{16}+11 q^{12}-18 q^{10}+24 q^8-21 q^6+10 q^4+8 q^2-25+39 q^{-2} -35 q^{-4} +23 q^{-6} -2 q^{-8} -17 q^{-10} +27 q^{-12} -29 q^{-14} +18 q^{-16} -4 q^{-18} -9 q^{-20} +15 q^{-22} -10 q^{-24} +12 q^{-28} -18 q^{-30} +17 q^{-32} -11 q^{-34} -6 q^{-36} +18 q^{-38} -32 q^{-40} +34 q^{-42} -25 q^{-44} +8 q^{-46} +9 q^{-48} -26 q^{-50} +31 q^{-52} -29 q^{-54} +15 q^{-56} -2 q^{-58} -10 q^{-60} +14 q^{-62} -10 q^{-64} +5 q^{-66} +3 q^{-68} -2 q^{-70} +2 q^{-72} +4 q^{-74} -6 q^{-76} +10 q^{-78} -6 q^{-80} +3 q^{-82} +3 q^{-84} -3 q^{-86} +6 q^{-88} -7 q^{-90} +7 q^{-92} -7 q^{-94} +9 q^{-96} -8 q^{-98} -3 q^{-100} +6 q^{-102} -11 q^{-104} +16 q^{-106} -16 q^{-108} +11 q^{-110} -2 q^{-112} -7 q^{-114} +13 q^{-116} -18 q^{-118} +14 q^{-120} -10 q^{-122} +5 q^{-124} +2 q^{-126} -8 q^{-128} +12 q^{-130} -9 q^{-132} +8 q^{-134} -3 q^{-136} + q^{-140} -4 q^{-142} +3 q^{-144} -2 q^{-146} + q^{-148} }[/math]

Further Quantum Invariants

K11a179 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["K11a179"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-5 t^3+9 t^2-9 t+9-9 t^{-1} +9 t^{-2} -5 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 57, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^9+3 q^8-4 q^7+6 q^6-8 q^5+8 q^4-8 q^3+7 q^2-5 q+4-2 q^{-1} + q^{-2} }[/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} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -10 z^4 a^{-2} +12 z^4 a^{-4} -4 z^4 a^{-6} +z^4-13 z^2 a^{-2} +10 z^2 a^{-4} -3 z^2 a^{-6} +4 z^2-4 a^{-2} +2 a^{-4} +3 }[/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} +2 z^9 a^{-1} +5 z^9 a^{-3} +3 z^9 a^{-5} -z^8 a^{-2} +2 z^8 a^{-4} +4 z^8 a^{-6} +z^8-11 z^7 a^{-1} -23 z^7 a^{-3} -8 z^7 a^{-5} +4 z^7 a^{-7} -15 z^6 a^{-2} -22 z^6 a^{-4} -9 z^6 a^{-6} +4 z^6 a^{-8} -6 z^6+18 z^5 a^{-1} +28 z^5 a^{-3} +2 z^5 a^{-5} -4 z^5 a^{-7} +4 z^5 a^{-9} +35 z^4 a^{-2} +31 z^4 a^{-4} +3 z^4 a^{-6} -2 z^4 a^{-8} +3 z^4 a^{-10} +12 z^4-9 z^3 a^{-1} -7 z^3 a^{-3} +z^3 a^{-5} -5 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} -23 z^2 a^{-2} -13 z^2 a^{-4} -z^2 a^{-6} -3 z^2 a^{-8} -2 z^2 a^{-10} -10 z^2-z a^{-3} +z a^{-5} +3 z a^{-7} +z a^{-9} +4 a^{-2} +2 a^{-4} +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["K11a179"];

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+9 t^2-9 t+9-9 t^{-1} +9 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q^9+3 q^8-4 q^7+6 q^6-8 q^5+8 q^4-8 q^3+7 q^2-5 q+4-2 q^{-1} + q^{-2} }[/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:

(-2, -2)

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{ -16 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{116}{3} }[/math] [math]\displaystyle{ \frac{76}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{512}{3} }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ -\frac{928}{3} }[/math] [math]\displaystyle{ -\frac{608}{3} }[/math] [math]\displaystyle{ -\frac{2551}{15} }[/math] [math]\displaystyle{ -\frac{572}{5} }[/math] [math]\displaystyle{ -\frac{8764}{45} }[/math] [math]\displaystyle{ \frac{391}{9} }[/math] [math]\displaystyle{ -\frac{391}{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 K11a179. 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 19                       1 -1 17                     2   2 15                   2 1   -1 13                 4 2     2 11               4 2       -2 9             4 4         0 7           4 4           0 5         3 4             -1 3       3 5               2 1     1 2                 -1 -1   1 3                   2 -3   1                     -1 -5 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

K11a178 K11a180