K11a194

K11a193

K11a195

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a194 at Knotilus!

Knot presentations

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

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:K11a194/ThurstonBennequinNumber Hyperbolic Volume 13.7968 A-Polynomial See Data:K11a194/A-polynomial

[edit Notes for K11a194'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 K11a194's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-5 t^3+12 t^2-18 t+21-18 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+3 z^6+2 z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 93, -4 }
Jones polynomial	[math]\displaystyle{ -q+3-5 q^{-1} +10 q^{-2} -12 q^{-3} +14 q^{-4} -15 q^{-5} +13 q^{-6} -10 q^{-7} +6 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^6-9 z^4 a^6-13 z^2 a^6-6 a^6+z^8 a^4+6 z^6 a^4+14 z^4 a^4+15 z^2 a^4+5 a^4-z^6 a^2-4 z^4 a^2-4 z^2 a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^4 a^{12}-z^2 a^{12}+3 z^5 a^{11}-3 z^3 a^{11}+5 z^6 a^{10}-5 z^4 a^{10}+z^2 a^{10}+7 z^7 a^9-12 z^5 a^9+11 z^3 a^9-4 z a^9+7 z^8 a^8-14 z^6 a^8+13 z^4 a^8-5 z^2 a^8+2 a^8+4 z^9 a^7-z^7 a^7-17 z^5 a^7+23 z^3 a^7-8 z a^7+z^{10} a^6+10 z^8 a^6-39 z^6 a^6+46 z^4 a^6-25 z^2 a^6+6 a^6+7 z^9 a^5-18 z^7 a^5+7 z^5 a^5+5 z^3 a^5-3 z a^5+z^{10} a^4+6 z^8 a^4-33 z^6 a^4+45 z^4 a^4-27 z^2 a^4+5 a^4+3 z^9 a^3-9 z^7 a^3+5 z^5 a^3+z a^3+3 z^8 a^2-13 z^6 a^2+18 z^4 a^2-9 z^2 a^2+z^7 a-4 z^5 a+4 z^3 a }[/math]
The A2 invariant	[math]\displaystyle{ q^{30}+q^{24}-3 q^{22}+q^{20}-2 q^{18}-q^{16}+q^{14}-3 q^{12}+4 q^{10}+3 q^6+2 q^4-q^2+1- q^{-2} }[/math]
The G2 invariant	[math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+5 q^{154}-3 q^{152}-2 q^{150}+10 q^{148}-17 q^{146}+24 q^{144}-27 q^{142}+19 q^{140}-7 q^{138}-13 q^{136}+36 q^{134}-55 q^{132}+68 q^{130}-65 q^{128}+41 q^{126}+q^{124}-47 q^{122}+96 q^{120}-124 q^{118}+125 q^{116}-90 q^{114}+17 q^{112}+68 q^{110}-135 q^{108}+168 q^{106}-138 q^{104}+66 q^{102}+28 q^{100}-112 q^{98}+143 q^{96}-110 q^{94}+17 q^{92}+82 q^{90}-147 q^{88}+138 q^{86}-54 q^{84}-71 q^{82}+185 q^{80}-241 q^{78}+202 q^{76}-92 q^{74}-74 q^{72}+215 q^{70}-286 q^{68}+263 q^{66}-150 q^{64}-5 q^{62}+143 q^{60}-221 q^{58}+215 q^{56}-136 q^{54}+6 q^{52}+107 q^{50}-157 q^{48}+135 q^{46}-36 q^{44}-76 q^{42}+165 q^{40}-176 q^{38}+112 q^{36}-q^{34}-123 q^{32}+211 q^{30}-210 q^{28}+146 q^{26}-34 q^{24}-75 q^{22}+150 q^{20}-163 q^{18}+126 q^{16}-62 q^{14}-6 q^{12}+53 q^{10}-70 q^8+61 q^6-37 q^4+16 q^2+3-13 q^{-2} +11 q^{-4} -10 q^{-6} +5 q^{-8} -2 q^{-10} + q^{-12} }[/math]

Further Quantum Invariants

K11a194 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["K11a194"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-5 t^3+12 t^2-18 t+21-18 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+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]=

{ 93, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q+3-5 q^{-1} +10 q^{-2} -12 q^{-3} +14 q^{-4} -15 q^{-5} +13 q^{-6} -10 q^{-7} +6 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^6-9 z^4 a^6-13 z^2 a^6-6 a^6+z^8 a^4+6 z^6 a^4+14 z^4 a^4+15 z^2 a^4+5 a^4-z^6 a^2-4 z^4 a^2-4 z^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}-3 z^3 a^{11}+5 z^6 a^{10}-5 z^4 a^{10}+z^2 a^{10}+7 z^7 a^9-12 z^5 a^9+11 z^3 a^9-4 z a^9+7 z^8 a^8-14 z^6 a^8+13 z^4 a^8-5 z^2 a^8+2 a^8+4 z^9 a^7-z^7 a^7-17 z^5 a^7+23 z^3 a^7-8 z a^7+z^{10} a^6+10 z^8 a^6-39 z^6 a^6+46 z^4 a^6-25 z^2 a^6+6 a^6+7 z^9 a^5-18 z^7 a^5+7 z^5 a^5+5 z^3 a^5-3 z a^5+z^{10} a^4+6 z^8 a^4-33 z^6 a^4+45 z^4 a^4-27 z^2 a^4+5 a^4+3 z^9 a^3-9 z^7 a^3+5 z^5 a^3+z a^3+3 z^8 a^2-13 z^6 a^2+18 z^4 a^2-9 z^2 a^2+z^7 a-4 z^5 a+4 z^3 a }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a106, K11a346,}

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["K11a194"];

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^2-18 t+21-18 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q+3-5 q^{-1} +10 q^{-2} -12 q^{-3} +14 q^{-4} -15 q^{-5} +13 q^{-6} -10 q^{-7} +6 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]=

{K11a106, K11a346,}

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:

(1, 1)

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{ 4 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{226}{3} }[/math] [math]\displaystyle{ -\frac{38}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{944}{3} }[/math] [math]\displaystyle{ -\frac{64}{3} }[/math] [math]\displaystyle{ 104 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{904}{3} }[/math] [math]\displaystyle{ -\frac{152}{3} }[/math] [math]\displaystyle{ -\frac{26129}{30} }[/math] [math]\displaystyle{ \frac{5738}{15} }[/math] [math]\displaystyle{ -\frac{24058}{45} }[/math] [math]\displaystyle{ -\frac{2191}{18} }[/math] [math]\displaystyle{ -\frac{2129}{30} }[/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 K11a194. 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                     2   2 -1                   3 1   -2 -3                 7 2     5 -5               6 4       -2 -7             8 6         2 -9           7 6           -1 -11         6 8             -2 -13       4 7               3 -15     2 6                 -4 -17   1 4                   3 -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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

K11a193 K11a195