K11a92

K11a91

K11a93

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a92 at Knotilus!

Knot presentations

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

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

Three dimensional invariants

Symmetry type Reversible 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:K11a92/ThurstonBennequinNumber Hyperbolic Volume 14.2422 A-Polynomial See Data:K11a92/A-polynomial

[edit Notes for K11a92'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 K11a92's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+5 t^3-12 t^2+21 t-25+21 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-3 z^6-2 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 103, -2 }
Jones polynomial	[math]\displaystyle{ q^3-3 q^2+6 q-10+14 q^{-1} -16 q^{-2} +17 q^{-3} -14 q^{-4} +11 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^2 z^8+2 a^4 z^6-6 a^2 z^6+z^6-a^6 z^4+9 a^4 z^4-14 a^2 z^4+4 z^4-3 a^6 z^2+14 a^4 z^2-14 a^2 z^2+5 z^2-3 a^6+7 a^4-5 a^2+2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^4 z^{10}+a^2 z^{10}+4 a^5 z^9+7 a^3 z^9+3 a z^9+6 a^6 z^8+10 a^4 z^8+8 a^2 z^8+4 z^8+5 a^7 z^7-3 a^5 z^7-13 a^3 z^7-2 a z^7+3 z^7 a^{-1} +3 a^8 z^6-12 a^6 z^6-34 a^4 z^6-30 a^2 z^6+z^6 a^{-2} -10 z^6+a^9 z^5-8 a^7 z^5-5 a^5 z^5+3 a^3 z^5-10 a z^5-9 z^5 a^{-1} -5 a^8 z^4+13 a^6 z^4+47 a^4 z^4+39 a^2 z^4-3 z^4 a^{-2} +7 z^4-2 a^9 z^3+3 a^7 z^3+12 a^5 z^3+12 a^3 z^3+12 a z^3+7 z^3 a^{-1} +a^8 z^2-10 a^6 z^2-27 a^4 z^2-23 a^2 z^2+2 z^2 a^{-2} -5 z^2+a^9 z-2 a^7 z-6 a^5 z-6 a^3 z-5 a z-2 z a^{-1} +3 a^6+7 a^4+5 a^2+2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{24}-q^{20}-2 q^{18}+3 q^{16}-q^{14}+3 q^{12}+2 q^{10}-q^8+3 q^6-4 q^4+2 q^2-1- q^{-2} +2 q^{-4} - q^{-6} + q^{-8} }[/math]
The G2 invariant	[math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+9 q^{120}-7 q^{118}+13 q^{114}-28 q^{112}+43 q^{110}-53 q^{108}+43 q^{106}-19 q^{104}-24 q^{102}+81 q^{100}-127 q^{98}+153 q^{96}-137 q^{94}+64 q^{92}+41 q^{90}-162 q^{88}+249 q^{86}-269 q^{84}+202 q^{82}-65 q^{80}-105 q^{78}+239 q^{76}-286 q^{74}+225 q^{72}-84 q^{70}-83 q^{68}+194 q^{66}-206 q^{64}+112 q^{62}+54 q^{60}-202 q^{58}+279 q^{56}-224 q^{54}+60 q^{52}+159 q^{50}-339 q^{48}+423 q^{46}-355 q^{44}+170 q^{42}+81 q^{40}-299 q^{38}+419 q^{36}-394 q^{34}+241 q^{32}-23 q^{30}-180 q^{28}+282 q^{26}-261 q^{24}+132 q^{22}+42 q^{20}-183 q^{18}+218 q^{16}-148 q^{14}-7 q^{12}+172 q^{10}-275 q^8+271 q^6-166 q^4-2 q^2+163-265 q^{-2} +280 q^{-4} -203 q^{-6} +83 q^{-8} +43 q^{-10} -134 q^{-12} +170 q^{-14} -152 q^{-16} +102 q^{-18} -37 q^{-20} -18 q^{-22} +51 q^{-24} -61 q^{-26} +52 q^{-28} -32 q^{-30} +15 q^{-32} + q^{-34} -10 q^{-36} +10 q^{-38} -9 q^{-40} +5 q^{-42} -2 q^{-44} + q^{-46} }[/math]

Further Quantum Invariants

K11a92 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["K11a92"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+5 t^3-12 t^2+21 t-25+21 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^8-3 z^6-2 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]=

{ 103, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^3-3 q^2+6 q-10+14 q^{-1} -16 q^{-2} +17 q^{-3} -14 q^{-4} +11 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -a^2 z^8+2 a^4 z^6-6 a^2 z^6+z^6-a^6 z^4+9 a^4 z^4-14 a^2 z^4+4 z^4-3 a^6 z^2+14 a^4 z^2-14 a^2 z^2+5 z^2-3 a^6+7 a^4-5 a^2+2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ a^4 z^{10}+a^2 z^{10}+4 a^5 z^9+7 a^3 z^9+3 a z^9+6 a^6 z^8+10 a^4 z^8+8 a^2 z^8+4 z^8+5 a^7 z^7-3 a^5 z^7-13 a^3 z^7-2 a z^7+3 z^7 a^{-1} +3 a^8 z^6-12 a^6 z^6-34 a^4 z^6-30 a^2 z^6+z^6 a^{-2} -10 z^6+a^9 z^5-8 a^7 z^5-5 a^5 z^5+3 a^3 z^5-10 a z^5-9 z^5 a^{-1} -5 a^8 z^4+13 a^6 z^4+47 a^4 z^4+39 a^2 z^4-3 z^4 a^{-2} +7 z^4-2 a^9 z^3+3 a^7 z^3+12 a^5 z^3+12 a^3 z^3+12 a z^3+7 z^3 a^{-1} +a^8 z^2-10 a^6 z^2-27 a^4 z^2-23 a^2 z^2+2 z^2 a^{-2} -5 z^2+a^9 z-2 a^7 z-6 a^5 z-6 a^3 z-5 a z-2 z a^{-1} +3 a^6+7 a^4+5 a^2+2 }[/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["K11a92"];

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^3-12 t^2+21 t-25+21 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^3-3 q^2+6 q-10+14 q^{-1} -16 q^{-2} +17 q^{-3} -14 q^{-4} +11 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} }[/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, -5)

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{ -40 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{460}{3} }[/math] [math]\displaystyle{ \frac{92}{3} }[/math] [math]\displaystyle{ -320 }[/math] [math]\displaystyle{ -\frac{1936}{3} }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ -104 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 800 }[/math] [math]\displaystyle{ \frac{3680}{3} }[/math] [math]\displaystyle{ \frac{736}{3} }[/math] [math]\displaystyle{ \frac{48391}{15} }[/math] [math]\displaystyle{ \frac{1036}{15} }[/math] [math]\displaystyle{ \frac{56884}{45} }[/math] [math]\displaystyle{ \frac{377}{9} }[/math] [math]\displaystyle{ \frac{2071}{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 K11a92. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     2   -2 3                   4 1   3 1                 6 2     -4 -1               8 4       4 -3             9 7         -2 -5           8 7           1 -7         6 9             3 -9       5 8               -3 -11     2 6                 4 -13   1 5                   -4 -15   2                     2 -17 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=-1 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/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}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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.

K11a91 K11a93