K11a129

K11a128

K11a130

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a129 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a129/ThurstonBennequinNumber Hyperbolic Volume 15.5089 A-Polynomial See Data:K11a129/A-polynomial

[edit Notes for K11a129'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 K11a129's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-6 t^3+15 t^2-22 t+25-22 t^{-1} +15 t^{-2} -6 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+2 z^6-z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 113, -4 }
Jones polynomial	[math]\displaystyle{ -q+4-7 q^{-1} +12 q^{-2} -15 q^{-3} +18 q^{-4} -18 q^{-5} +15 q^{-6} -12 q^{-7} +7 q^{-8} -3 q^{-9} + q^{-10} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^4 a^8+3 z^2 a^8+2 a^8-2 z^6 a^6-8 z^4 a^6-10 z^2 a^6-5 a^6+z^8 a^4+5 z^6 a^4+9 z^4 a^4+8 z^2 a^4+3 a^4-z^6 a^2-3 z^4 a^2-z^2 a^2+a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-2 z^3 a^{11}+6 z^6 a^{10}-6 z^4 a^{10}+3 z^2 a^{10}+9 z^7 a^9-15 z^5 a^9+14 z^3 a^9-5 z a^9+9 z^8 a^8-15 z^6 a^8+10 z^4 a^8-4 z^2 a^8+2 a^8+6 z^9 a^7-4 z^7 a^7-15 z^5 a^7+20 z^3 a^7-9 z a^7+2 z^{10} a^6+10 z^8 a^6-41 z^6 a^6+43 z^4 a^6-22 z^2 a^6+5 a^6+11 z^9 a^5-30 z^7 a^5+18 z^5 a^5+z^3 a^5-3 z a^5+2 z^{10} a^4+5 z^8 a^4-35 z^6 a^4+42 z^4 a^4-18 z^2 a^4+3 a^4+5 z^9 a^3-16 z^7 a^3+12 z^5 a^3-z^3 a^3+z a^3+4 z^8 a^2-15 z^6 a^2+16 z^4 a^2-4 z^2 a^2-a^2+z^7 a-3 z^5 a+2 z^3 a }[/math]
The A2 invariant	[math]\displaystyle{ q^{30}+2 q^{24}-3 q^{22}+q^{20}-3 q^{18}-2 q^{16}+2 q^{14}-3 q^{12}+5 q^{10}-q^8+2 q^6+2 q^4-q^2+2- q^{-2} }[/math]
The G2 invariant	[math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+6 q^{154}-5 q^{152}+9 q^{148}-19 q^{146}+29 q^{144}-37 q^{142}+33 q^{140}-20 q^{138}-6 q^{136}+46 q^{134}-80 q^{132}+105 q^{130}-107 q^{128}+75 q^{126}-22 q^{124}-56 q^{122}+137 q^{120}-193 q^{118}+212 q^{116}-171 q^{114}+77 q^{112}+61 q^{110}-193 q^{108}+289 q^{106}-295 q^{104}+195 q^{102}-23 q^{100}-162 q^{98}+283 q^{96}-276 q^{94}+149 q^{92}+57 q^{90}-242 q^{88}+303 q^{86}-219 q^{84}-6 q^{82}+257 q^{80}-431 q^{78}+431 q^{76}-257 q^{74}-33 q^{72}+325 q^{70}-516 q^{68}+523 q^{66}-362 q^{64}+81 q^{62}+206 q^{60}-404 q^{58}+453 q^{56}-328 q^{54}+106 q^{52}+135 q^{50}-300 q^{48}+319 q^{46}-185 q^{44}-38 q^{42}+255 q^{40}-358 q^{38}+300 q^{36}-96 q^{34}-158 q^{32}+364 q^{30}-427 q^{28}+334 q^{26}-126 q^{24}-111 q^{22}+284 q^{20}-335 q^{18}+273 q^{16}-133 q^{14}-9 q^{12}+107 q^{10}-144 q^8+125 q^6-74 q^4+26 q^2+11-26 q^{-2} +23 q^{-4} -17 q^{-6} +8 q^{-8} -3 q^{-10} + q^{-12} }[/math]

Further Quantum Invariants

K11a129 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["K11a129"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-6 t^3+15 t^2-22 t+25-22 t^{-1} +15 t^{-2} -6 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+2 z^6-z^4+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]=

{ 113, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q+4-7 q^{-1} +12 q^{-2} -15 q^{-3} +18 q^{-4} -18 q^{-5} +15 q^{-6} -12 q^{-7} +7 q^{-8} -3 q^{-9} + q^{-10} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ z^4 a^8+3 z^2 a^8+2 a^8-2 z^6 a^6-8 z^4 a^6-10 z^2 a^6-5 a^6+z^8 a^4+5 z^6 a^4+9 z^4 a^4+8 z^2 a^4+3 a^4-z^6 a^2-3 z^4 a^2-z^2 a^2+a^2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-2 z^3 a^{11}+6 z^6 a^{10}-6 z^4 a^{10}+3 z^2 a^{10}+9 z^7 a^9-15 z^5 a^9+14 z^3 a^9-5 z a^9+9 z^8 a^8-15 z^6 a^8+10 z^4 a^8-4 z^2 a^8+2 a^8+6 z^9 a^7-4 z^7 a^7-15 z^5 a^7+20 z^3 a^7-9 z a^7+2 z^{10} a^6+10 z^8 a^6-41 z^6 a^6+43 z^4 a^6-22 z^2 a^6+5 a^6+11 z^9 a^5-30 z^7 a^5+18 z^5 a^5+z^3 a^5-3 z a^5+2 z^{10} a^4+5 z^8 a^4-35 z^6 a^4+42 z^4 a^4-18 z^2 a^4+3 a^4+5 z^9 a^3-16 z^7 a^3+12 z^5 a^3-z^3 a^3+z a^3+4 z^8 a^2-15 z^6 a^2+16 z^4 a^2-4 z^2 a^2-a^2+z^7 a-3 z^5 a+2 z^3 a }[/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["K11a129"];

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+15 t^2-22 t+25-22 t^{-1} +15 t^{-2} -6 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q+4-7 q^{-1} +12 q^{-2} -15 q^{-3} +18 q^{-4} -18 q^{-5} +15 q^{-6} -12 q^{-7} +7 q^{-8} -3 q^{-9} + q^{-10} }[/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:

(0, 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{ 0 }[/math] [math]\displaystyle{ 24 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -96 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 144 }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ -40 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 608 }[/math] [math]\displaystyle{ -\frac{136}{3} }[/math] [math]\displaystyle{ \frac{2296}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ 80 }[/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 K11a129. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 3                       1 -1 1                     3   3 -1                   4 1   -3 -3                 8 3     5 -5               8 5       -3 -7             10 7         3 -9           8 8           0 -11         7 10             -3 -13       5 8               3 -15     2 7                 -5 -17   1 5                   4 -19   2                     -2 -21 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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.

K11a128 K11a130