K11a180

K11a179

K11a181

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a180 at Knotilus!

Knot presentations

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

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

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a180/ThurstonBennequinNumber Hyperbolic Volume 12.7428 A-Polynomial See Data:K11a180/A-polynomial

[edit Notes for K11a180's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant 0

[edit Notes for K11a180's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-5 t^3+11 t^2-17 t+21-17 t^{-1} +11 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	{ 89, 0 }
Jones polynomial	[math]\displaystyle{ -q^5+3 q^4-5 q^3+9 q^2-12 q+14-14 q^{-1} +12 q^{-2} -9 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +6 z^6+a^4 z^4-9 a^2 z^4-4 z^4 a^{-2} +13 z^4+3 a^4 z^2-12 a^2 z^2-4 z^2 a^{-2} +11 z^2+2 a^4-4 a^2+3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^2 z^{10}+z^{10}+3 a^3 z^9+6 a z^9+3 z^9 a^{-1} +4 a^4 z^8+5 a^2 z^8+4 z^8 a^{-2} +5 z^8+3 a^5 z^7-5 a^3 z^7-16 a z^7-4 z^7 a^{-1} +4 z^7 a^{-3} +a^6 z^6-11 a^4 z^6-22 a^2 z^6-7 z^6 a^{-2} +3 z^6 a^{-4} -20 z^6-9 a^5 z^5-a^3 z^5+18 a z^5+z^5 a^{-1} -8 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+8 a^4 z^4+31 a^2 z^4+4 z^4 a^{-2} -7 z^4 a^{-4} +31 z^4+6 a^5 z^3+a^3 z^3-6 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} -2 z^3 a^{-5} +2 a^6 z^2-5 a^4 z^2-20 a^2 z^2+3 z^2 a^{-4} -16 z^2-a^5 z-2 z a^{-1} -z a^{-3} +2 a^4+4 a^2+3 }[/math]
The A2 invariant	[math]\displaystyle{ q^{18}+q^{12}-2 q^{10}+2 q^8-q^6-q^4+q^2-3+3 q^{-2} - q^{-4} +2 q^{-6} +2 q^{-8} - q^{-10} + q^{-12} - q^{-14} }[/math]
The G2 invariant	[math]\displaystyle{ q^{94}-2 q^{92}+5 q^{90}-9 q^{88}+10 q^{86}-9 q^{84}+16 q^{80}-32 q^{78}+48 q^{76}-53 q^{74}+38 q^{72}-5 q^{70}-43 q^{68}+94 q^{66}-122 q^{64}+121 q^{62}-76 q^{60}-2 q^{58}+89 q^{56}-155 q^{54}+177 q^{52}-139 q^{50}+54 q^{48}+47 q^{46}-126 q^{44}+151 q^{42}-112 q^{40}+33 q^{38}+56 q^{36}-112 q^{34}+105 q^{32}-46 q^{30}-52 q^{28}+139 q^{26}-177 q^{24}+144 q^{22}-49 q^{20}-78 q^{18}+186 q^{16}-241 q^{14}+217 q^{12}-125 q^{10}-9 q^8+131 q^6-206 q^4+206 q^2-135+32 q^{-2} +67 q^{-4} -120 q^{-6} +113 q^{-8} -52 q^{-10} -25 q^{-12} +91 q^{-14} -105 q^{-16} +68 q^{-18} +8 q^{-20} -85 q^{-22} +140 q^{-24} -141 q^{-26} +99 q^{-28} -27 q^{-30} -53 q^{-32} +108 q^{-34} -132 q^{-36} +119 q^{-38} -74 q^{-40} +21 q^{-42} +27 q^{-44} -62 q^{-46} +75 q^{-48} -70 q^{-50} +50 q^{-52} -23 q^{-54} -3 q^{-56} +22 q^{-58} -31 q^{-60} +30 q^{-62} -21 q^{-64} +13 q^{-66} -2 q^{-68} -5 q^{-70} +6 q^{-72} -7 q^{-74} +4 q^{-76} -2 q^{-78} + q^{-80} }[/math]

Further Quantum Invariants

K11a180 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["K11a180"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-5 t^3+11 t^2-17 t+21-17 t^{-1} +11 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]=

{ 89, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^5+3 q^4-5 q^3+9 q^2-12 q+14-14 q^{-1} +12 q^{-2} -9 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +6 z^6+a^4 z^4-9 a^2 z^4-4 z^4 a^{-2} +13 z^4+3 a^4 z^2-12 a^2 z^2-4 z^2 a^{-2} +11 z^2+2 a^4-4 a^2+3 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^2 z^{10}+z^{10}+3 a^3 z^9+6 a z^9+3 z^9 a^{-1} +4 a^4 z^8+5 a^2 z^8+4 z^8 a^{-2} +5 z^8+3 a^5 z^7-5 a^3 z^7-16 a z^7-4 z^7 a^{-1} +4 z^7 a^{-3} +a^6 z^6-11 a^4 z^6-22 a^2 z^6-7 z^6 a^{-2} +3 z^6 a^{-4} -20 z^6-9 a^5 z^5-a^3 z^5+18 a z^5+z^5 a^{-1} -8 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+8 a^4 z^4+31 a^2 z^4+4 z^4 a^{-2} -7 z^4 a^{-4} +31 z^4+6 a^5 z^3+a^3 z^3-6 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} -2 z^3 a^{-5} +2 a^6 z^2-5 a^4 z^2-20 a^2 z^2+3 z^2 a^{-4} -16 z^2-a^5 z-2 z a^{-1} -z a^{-3} +2 a^4+4 a^2+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["K11a180"];

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+11 t^2-17 t+21-17 t^{-1} +11 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q^5+3 q^4-5 q^3+9 q^2-12 q+14-14 q^{-1} +12 q^{-2} -9 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6} }[/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{68}{3} }[/math] [math]\displaystyle{ \frac{28}{3} }[/math] [math]\displaystyle{ -128 }[/math] [math]\displaystyle{ -\frac{704}{3} }[/math] [math]\displaystyle{ -\frac{320}{3} }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ -\frac{544}{3} }[/math] [math]\displaystyle{ -\frac{224}{3} }[/math] [math]\displaystyle{ \frac{3089}{15} }[/math] [math]\displaystyle{ -\frac{2356}{15} }[/math] [math]\displaystyle{ \frac{12356}{45} }[/math] [math]\displaystyle{ -\frac{65}{9} }[/math] [math]\displaystyle{ \frac{929}{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]0 is the signature of K11a180. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

K11a179 K11a181