K11a53

K11a52

K11a54

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a53 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X14,6,15,5 X16,7,17,8 X2,9,3,10 X18,11,19,12 X20,13,21,14 X22,16,1,15 X10,17,11,18 X12,19,13,20 X6,21,7,22
Gauss code	1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -10, 7, -3, 8, -4, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 8 14 16 2 18 20 22 10 12 6

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

Three dimensional invariants

Symmetry type Chiral Unknotting number [math]\displaystyle{ \{2,3\} }[/math] 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a53/ThurstonBennequinNumber Hyperbolic Volume 14.416 A-Polynomial See Data:K11a53/A-polynomial

[edit Notes for K11a53'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 K11a53's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-6 t^3+14 t^2-18 t+19-18 t^{-1} +14 t^{-2} -6 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+2 z^6-2 z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 97, -4 }
Jones polynomial	[math]\displaystyle{ -q+4-6 q^{-1} +10 q^{-2} -13 q^{-3} +15 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+a^8-2 z^6 a^6-8 z^4 a^6-8 z^2 a^6-2 a^6+z^8 a^4+5 z^6 a^4+8 z^4 a^4+5 z^2 a^4-z^6 a^2-3 z^4 a^2+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}+z a^{11}+5 z^6 a^{10}-5 z^4 a^{10}+2 z^2 a^{10}+6 z^7 a^9-6 z^5 a^9+z^3 a^9+z a^9+6 z^8 a^8-8 z^6 a^8+4 z^4 a^8-3 z^2 a^8+a^8+5 z^9 a^7-9 z^7 a^7+7 z^5 a^7-6 z^3 a^7+z a^7+2 z^{10} a^6+4 z^8 a^6-26 z^6 a^6+32 z^4 a^6-16 z^2 a^6+2 a^6+10 z^9 a^5-35 z^7 a^5+38 z^5 a^5-17 z^3 a^5+3 z a^5+2 z^{10} a^4+2 z^8 a^4-29 z^6 a^4+38 z^4 a^4-12 z^2 a^4+5 z^9 a^3-19 z^7 a^3+19 z^5 a^3-6 z^3 a^3+2 z a^3+4 z^8 a^2-16 z^6 a^2+16 z^4 a^2-2 z^2 a^2-2 a^2+z^7 a-3 z^5 a+z^3 a }[/math]
The A2 invariant	[math]\displaystyle{ q^{30}-q^{26}+q^{24}-2 q^{22}+2 q^{20}-q^{18}-q^{16}+q^{14}-4 q^{12}+3 q^{10}-q^8+2 q^6+2 q^4+2- q^{-2} }[/math]
The G2 invariant	[math]\displaystyle{ q^{162}-2 q^{160}+4 q^{158}-6 q^{156}+5 q^{154}-4 q^{152}-2 q^{150}+11 q^{148}-20 q^{146}+28 q^{144}-30 q^{142}+22 q^{140}-4 q^{138}-21 q^{136}+50 q^{134}-68 q^{132}+71 q^{130}-56 q^{128}+20 q^{126}+22 q^{124}-60 q^{122}+91 q^{120}-96 q^{118}+87 q^{116}-61 q^{114}+18 q^{112}+33 q^{110}-87 q^{108}+125 q^{106}-133 q^{104}+99 q^{102}-29 q^{100}-57 q^{98}+125 q^{96}-140 q^{94}+94 q^{92}-100 q^{88}+143 q^{86}-110 q^{84}+9 q^{82}+119 q^{80}-205 q^{78}+213 q^{76}-129 q^{74}-20 q^{72}+170 q^{70}-272 q^{68}+278 q^{66}-195 q^{64}+47 q^{62}+107 q^{60}-213 q^{58}+249 q^{56}-202 q^{54}+87 q^{52}+43 q^{50}-151 q^{48}+181 q^{46}-129 q^{44}+16 q^{42}+113 q^{40}-183 q^{38}+167 q^{36}-68 q^{34}-73 q^{32}+199 q^{30}-249 q^{28}+203 q^{26}-79 q^{24}-66 q^{22}+181 q^{20}-216 q^{18}+181 q^{16}-90 q^{14}-4 q^{12}+72 q^{10}-101 q^8+91 q^6-55 q^4+21 q^2+7-18 q^{-2} +17 q^{-4} -14 q^{-6} +7 q^{-8} -3 q^{-10} + q^{-12} }[/math]

Further Quantum Invariants

K11a53 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["K11a53"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-6 t^3+14 t^2-18 t+19-18 t^{-1} +14 t^{-2} -6 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 97, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q+4-6 q^{-1} +10 q^{-2} -13 q^{-3} +15 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+a^8-2 z^6 a^6-8 z^4 a^6-8 z^2 a^6-2 a^6+z^8 a^4+5 z^6 a^4+8 z^4 a^4+5 z^2 a^4-z^6 a^2-3 z^4 a^2+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}+z a^{11}+5 z^6 a^{10}-5 z^4 a^{10}+2 z^2 a^{10}+6 z^7 a^9-6 z^5 a^9+z^3 a^9+z a^9+6 z^8 a^8-8 z^6 a^8+4 z^4 a^8-3 z^2 a^8+a^8+5 z^9 a^7-9 z^7 a^7+7 z^5 a^7-6 z^3 a^7+z a^7+2 z^{10} a^6+4 z^8 a^6-26 z^6 a^6+32 z^4 a^6-16 z^2 a^6+2 a^6+10 z^9 a^5-35 z^7 a^5+38 z^5 a^5-17 z^3 a^5+3 z a^5+2 z^{10} a^4+2 z^8 a^4-29 z^6 a^4+38 z^4 a^4-12 z^2 a^4+5 z^9 a^3-19 z^7 a^3+19 z^5 a^3-6 z^3 a^3+2 z a^3+4 z^8 a^2-16 z^6 a^2+16 z^4 a^2-2 z^2 a^2-2 a^2+z^7 a-3 z^5 a+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["K11a53"];

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+14 t^2-18 t+19-18 t^{-1} +14 t^{-2} -6 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q+4-6 q^{-1} +10 q^{-2} -13 q^{-3} +15 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]=

{}

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, 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{ 0 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -48 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ -\frac{224}{3} }[/math] [math]\displaystyle{ -80 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 680 }[/math] [math]\displaystyle{ -\frac{608}{3} }[/math] [math]\displaystyle{ \frac{2816}{3} }[/math] [math]\displaystyle{ \frac{152}{3} }[/math] [math]\displaystyle{ 104 }[/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 K11a53. 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                   3 1   -2 -3                 7 3     4 -5               7 4       -3 -7             8 6         2 -9           7 7           0 -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}^{7}\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^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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}^{3}\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^{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.

K11a52 K11a54