K11a147

K11a146

K11a148

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a147 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a147/ThurstonBennequinNumber Hyperbolic Volume 17.2353 A-Polynomial See Data:K11a147/A-polynomial

[edit Notes for K11a147's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a147's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+6 t^3-17 t^2+32 t-39+32 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-2 z^6-z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 151, 2 }
Jones polynomial	[math]\displaystyle{ -q^8+4 q^7-10 q^6+16 q^5-21 q^4+25 q^3-24 q^2+21 q-15+9 q^{-1} -4 q^{-2} + q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^8 a^{-2} -5 z^6 a^{-2} +2 z^6 a^{-4} +z^6-10 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +3 z^4-8 z^2 a^{-2} +9 z^2 a^{-4} -2 z^2 a^{-6} +3 z^2-2 a^{-2} +4 a^{-4} -2 a^{-6} +1 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10} a^{-4} +6 z^9 a^{-1} +13 z^9 a^{-3} +7 z^9 a^{-5} +15 z^8 a^{-2} +19 z^8 a^{-4} +11 z^8 a^{-6} +7 z^8+4 a z^7-4 z^7 a^{-1} -13 z^7 a^{-3} +4 z^7 a^{-5} +9 z^7 a^{-7} +a^2 z^6-41 z^6 a^{-2} -45 z^6 a^{-4} -16 z^6 a^{-6} +4 z^6 a^{-8} -15 z^6-9 a z^5-12 z^5 a^{-1} -15 z^5 a^{-3} -27 z^5 a^{-5} -14 z^5 a^{-7} +z^5 a^{-9} -2 a^2 z^4+32 z^4 a^{-2} +34 z^4 a^{-4} +10 z^4 a^{-6} -4 z^4 a^{-8} +10 z^4+6 a z^3+12 z^3 a^{-1} +19 z^3 a^{-3} +24 z^3 a^{-5} +10 z^3 a^{-7} -z^3 a^{-9} +a^2 z^2-12 z^2 a^{-2} -14 z^2 a^{-4} -5 z^2 a^{-6} +z^2 a^{-8} -3 z^2-a z-3 z a^{-1} -5 z a^{-3} -7 z a^{-5} -4 z a^{-7} +2 a^{-2} +4 a^{-4} +2 a^{-6} +1 }[/math]
The A2 invariant	[math]\displaystyle{ q^8-2 q^6+3 q^4-2 q^2-1+4 q^{-2} -5 q^{-4} +5 q^{-6} -2 q^{-8} +2 q^{-10} +3 q^{-12} -3 q^{-14} +4 q^{-16} -3 q^{-18} - q^{-20} + q^{-22} - q^{-24} }[/math]
The G2 invariant	[math]\displaystyle{ q^{46}-3 q^{44}+8 q^{42}-16 q^{40}+22 q^{38}-25 q^{36}+14 q^{34}+18 q^{32}-66 q^{30}+128 q^{28}-176 q^{26}+175 q^{24}-101 q^{22}-63 q^{20}+289 q^{18}-494 q^{16}+595 q^{14}-503 q^{12}+187 q^{10}+276 q^8-746 q^6+1033 q^4-982 q^2+583+52 q^{-2} -689 q^{-4} +1072 q^{-6} -1041 q^{-8} +605 q^{-10} +49 q^{-12} -639 q^{-14} +889 q^{-16} -697 q^{-18} +143 q^{-20} +532 q^{-22} -1009 q^{-24} +1072 q^{-26} -654 q^{-28} -117 q^{-30} +932 q^{-32} -1485 q^{-34} +1542 q^{-36} -1054 q^{-38} +200 q^{-40} +732 q^{-42} -1392 q^{-44} +1563 q^{-46} -1188 q^{-48} +435 q^{-50} +387 q^{-52} -955 q^{-54} +1059 q^{-56} -686 q^{-58} +53 q^{-60} +563 q^{-62} -860 q^{-64} +723 q^{-66} -231 q^{-68} -416 q^{-70} +919 q^{-72} -1075 q^{-74} +828 q^{-76} -285 q^{-78} -335 q^{-80} +801 q^{-82} -973 q^{-84} +831 q^{-86} -473 q^{-88} +42 q^{-90} +307 q^{-92} -501 q^{-94} +504 q^{-96} -370 q^{-98} +187 q^{-100} -9 q^{-102} -105 q^{-104} +149 q^{-106} -143 q^{-108} +99 q^{-110} -49 q^{-112} +12 q^{-114} +12 q^{-116} -19 q^{-118} +17 q^{-120} -13 q^{-122} +7 q^{-124} -3 q^{-126} + q^{-128} }[/math]

Further Quantum Invariants

K11a147 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["K11a147"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+6 t^3-17 t^2+32 t-39+32 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^8-2 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]=

{ 151, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^8+4 q^7-10 q^6+16 q^5-21 q^4+25 q^3-24 q^2+21 q-15+9 q^{-1} -4 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} -5 z^6 a^{-2} +2 z^6 a^{-4} +z^6-10 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +3 z^4-8 z^2 a^{-2} +9 z^2 a^{-4} -2 z^2 a^{-6} +3 z^2-2 a^{-2} +4 a^{-4} -2 a^{-6} +1 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10} a^{-4} +6 z^9 a^{-1} +13 z^9 a^{-3} +7 z^9 a^{-5} +15 z^8 a^{-2} +19 z^8 a^{-4} +11 z^8 a^{-6} +7 z^8+4 a z^7-4 z^7 a^{-1} -13 z^7 a^{-3} +4 z^7 a^{-5} +9 z^7 a^{-7} +a^2 z^6-41 z^6 a^{-2} -45 z^6 a^{-4} -16 z^6 a^{-6} +4 z^6 a^{-8} -15 z^6-9 a z^5-12 z^5 a^{-1} -15 z^5 a^{-3} -27 z^5 a^{-5} -14 z^5 a^{-7} +z^5 a^{-9} -2 a^2 z^4+32 z^4 a^{-2} +34 z^4 a^{-4} +10 z^4 a^{-6} -4 z^4 a^{-8} +10 z^4+6 a z^3+12 z^3 a^{-1} +19 z^3 a^{-3} +24 z^3 a^{-5} +10 z^3 a^{-7} -z^3 a^{-9} +a^2 z^2-12 z^2 a^{-2} -14 z^2 a^{-4} -5 z^2 a^{-6} +z^2 a^{-8} -3 z^2-a z-3 z a^{-1} -5 z a^{-3} -7 z a^{-5} -4 z a^{-7} +2 a^{-2} +4 a^{-4} +2 a^{-6} +1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a322,}

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["K11a147"];

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+6 t^3-17 t^2+32 t-39+32 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^8+4 q^7-10 q^6+16 q^5-21 q^4+25 q^3-24 q^2+21 q-15+9 q^{-1} -4 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]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11a322,}

Vassiliev invariants

V2 and V3:

(2, 4)

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{ 32 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{364}{3} }[/math] [math]\displaystyle{ \frac{68}{3} }[/math] [math]\displaystyle{ 256 }[/math] [math]\displaystyle{ \frac{1568}{3} }[/math] [math]\displaystyle{ \frac{128}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 512 }[/math] [math]\displaystyle{ \frac{2912}{3} }[/math] [math]\displaystyle{ \frac{544}{3} }[/math] [math]\displaystyle{ \frac{35911}{15} }[/math] [math]\displaystyle{ -\frac{3964}{15} }[/math] [math]\displaystyle{ \frac{55564}{45} }[/math] [math]\displaystyle{ \frac{905}{9} }[/math] [math]\displaystyle{ \frac{2551}{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]2 is the signature of K11a147. 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                     3   3 13                   7 1   -6 11                 9 3     6 9               12 7       -5 7             13 9         4 5           11 12           1 3         10 13             -3 1       6 12               6 -1     3 9                 -6 -3   1 6                   5 -5   3                     -3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

K11a146 K11a148