K11a9

K11a8

K11a10

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a9 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X16,7,17,8 X2,9,3,10 X18,12,19,11 X20,14,21,13 X22,16,1,15 X6,17,7,18 X12,20,13,19 X14,22,15,21
Gauss code	1, -5, 2, -1, 3, -9, 4, -2, 5, -3, 6, -10, 7, -11, 8, -4, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 8 10 16 2 18 20 22 6 12 14

A Braid Representative	{{{braid_table}}}
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:K11a9/ThurstonBennequinNumber Hyperbolic Volume 11.385 A-Polynomial See Data:K11a9/A-polynomial

[edit Notes for K11a9'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 K11a9's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-5 t^3+10 t^2-11 t+11-11 t^{-1} +10 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+3 z^6+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 65, 4 }
Jones polynomial	[math]\displaystyle{ -q^9+3 q^8-5 q^7+7 q^6-9 q^5+10 q^4-9 q^3+8 q^2-6 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} +13 z^4 a^{-4} -4 z^4 a^{-6} +z^4-14 z^2 a^{-2} +14 z^2 a^{-4} -4 z^2 a^{-6} +4 z^2-6 a^{-2} +6 a^{-4} -2 a^{-6} +3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +6 z^9 a^{-3} +4 z^9 a^{-5} +5 z^8 a^{-4} +6 z^8 a^{-6} +z^8-11 z^7 a^{-1} -28 z^7 a^{-3} -11 z^7 a^{-5} +6 z^7 a^{-7} -20 z^6 a^{-2} -36 z^6 a^{-4} -16 z^6 a^{-6} +6 z^6 a^{-8} -6 z^6+19 z^5 a^{-1} +38 z^5 a^{-3} +5 z^5 a^{-5} -9 z^5 a^{-7} +5 z^5 a^{-9} +44 z^4 a^{-2} +54 z^4 a^{-4} +12 z^4 a^{-6} -7 z^4 a^{-8} +3 z^4 a^{-10} +12 z^4-12 z^3 a^{-1} -17 z^3 a^{-3} -z^3 a^{-5} -z^3 a^{-7} -4 z^3 a^{-9} +z^3 a^{-11} -30 z^2 a^{-2} -29 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} -z^2 a^{-10} -10 z^2+2 z a^{-1} +3 z a^{-3} +z a^{-5} +z a^{-7} +z a^{-9} +6 a^{-2} +6 a^{-4} +2 a^{-6} +3 }[/math]
The A2 invariant	[math]\displaystyle{ q^6+q^4+1- q^{-2} - q^{-6} - q^{-8} +2 q^{-10} - q^{-12} +3 q^{-14} - q^{-22} + q^{-24} - q^{-26} }[/math]
The G2 invariant	[math]\displaystyle{ q^{26}-q^{24}+4 q^{22}-5 q^{20}+6 q^{18}-5 q^{16}+q^{14}+10 q^{12}-19 q^{10}+30 q^8-29 q^6+19 q^4+4 q^2-31+54 q^{-2} -58 q^{-4} +43 q^{-6} -13 q^{-8} -24 q^{-10} +50 q^{-12} -58 q^{-14} +42 q^{-16} -14 q^{-18} -18 q^{-20} +34 q^{-22} -35 q^{-24} +12 q^{-26} +13 q^{-28} -30 q^{-30} +36 q^{-32} -26 q^{-34} + q^{-36} +26 q^{-38} -50 q^{-40} +57 q^{-42} -47 q^{-44} +21 q^{-46} +16 q^{-48} -43 q^{-50} +60 q^{-52} -55 q^{-54} +38 q^{-56} -5 q^{-58} -24 q^{-60} +39 q^{-62} -35 q^{-64} +19 q^{-66} +5 q^{-68} -15 q^{-70} +19 q^{-72} -7 q^{-74} -8 q^{-76} +18 q^{-78} -21 q^{-80} +16 q^{-82} -6 q^{-84} -9 q^{-86} +17 q^{-88} -19 q^{-90} +18 q^{-92} -15 q^{-94} +9 q^{-96} -5 q^{-98} -5 q^{-100} +12 q^{-102} -22 q^{-104} +25 q^{-106} -20 q^{-108} +14 q^{-110} - q^{-112} -13 q^{-114} +21 q^{-116} -25 q^{-118} +21 q^{-120} -12 q^{-122} +2 q^{-124} +6 q^{-126} -12 q^{-128} +14 q^{-130} -11 q^{-132} +8 q^{-134} -2 q^{-136} - q^{-138} +2 q^{-140} -4 q^{-142} +3 q^{-144} -2 q^{-146} + q^{-148} }[/math]

Further Quantum Invariants

K11a9 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["K11a9"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-5 t^3+10 t^2-11 t+11-11 t^{-1} +10 t^{-2} -5 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 65, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^9+3 q^8-5 q^7+7 q^6-9 q^5+10 q^4-9 q^3+8 q^2-6 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} +13 z^4 a^{-4} -4 z^4 a^{-6} +z^4-14 z^2 a^{-2} +14 z^2 a^{-4} -4 z^2 a^{-6} +4 z^2-6 a^{-2} +6 a^{-4} -2 a^{-6} +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} +6 z^9 a^{-3} +4 z^9 a^{-5} +5 z^8 a^{-4} +6 z^8 a^{-6} +z^8-11 z^7 a^{-1} -28 z^7 a^{-3} -11 z^7 a^{-5} +6 z^7 a^{-7} -20 z^6 a^{-2} -36 z^6 a^{-4} -16 z^6 a^{-6} +6 z^6 a^{-8} -6 z^6+19 z^5 a^{-1} +38 z^5 a^{-3} +5 z^5 a^{-5} -9 z^5 a^{-7} +5 z^5 a^{-9} +44 z^4 a^{-2} +54 z^4 a^{-4} +12 z^4 a^{-6} -7 z^4 a^{-8} +3 z^4 a^{-10} +12 z^4-12 z^3 a^{-1} -17 z^3 a^{-3} -z^3 a^{-5} -z^3 a^{-7} -4 z^3 a^{-9} +z^3 a^{-11} -30 z^2 a^{-2} -29 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} -z^2 a^{-10} -10 z^2+2 z a^{-1} +3 z a^{-3} +z a^{-5} +z a^{-7} +z a^{-9} +6 a^{-2} +6 a^{-4} +2 a^{-6} +3 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a140,}

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["K11a9"];

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

{K11a140,}

Vassiliev invariants

V2 and V3:

(0, 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{ 0 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ -\frac{224}{3} }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 88 }[/math] [math]\displaystyle{ -352 }[/math] [math]\displaystyle{ 240 }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ 72 }[/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 K11a9. 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                   3 1   -2 13                 4 2     2 11               5 3       -2 9             5 4         1 7           4 5           1 5         4 5             -1 3       3 5               2 1     1 3                 -2 -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}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

K11a8 K11a10