K11a174

K11a173

K11a175

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a174 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a174/ThurstonBennequinNumber Hyperbolic Volume 12.1343 A-Polynomial See Data:K11a174/A-polynomial

[edit Notes for K11a174'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 K11a174's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+5 t^3-11 t^2+15 t-15+15 t^{-1} -11 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-3 z^6-z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 79, -2 }
Jones polynomial	[math]\displaystyle{ -q^4+3 q^3-5 q^2+8 q-10+12 q^{-1} -12 q^{-2} +11 q^{-3} -8 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^2 z^8+a^4 z^6-6 a^2 z^6+2 z^6+4 a^4 z^4-13 a^2 z^4-z^4 a^{-2} +9 z^4+4 a^4 z^2-12 a^2 z^2-3 z^2 a^{-2} +11 z^2+a^4-3 a^2- a^{-2} +4 }[/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+3 z^8 a^{-2} +4 z^8+4 a^5 z^7-4 a^3 z^7-19 a z^7-10 z^7 a^{-1} +z^7 a^{-3} +4 a^6 z^6-5 a^4 z^6-24 a^2 z^6-13 z^6 a^{-2} -28 z^6+3 a^7 z^5-2 a^5 z^5+a^3 z^5+15 a z^5+5 z^5 a^{-1} -4 z^5 a^{-3} +a^8 z^4-4 a^6 z^4+3 a^4 z^4+31 a^2 z^4+16 z^4 a^{-2} +39 z^4-4 a^7 z^3-3 a^5 z^3-a^3 z^3-2 a z^3+4 z^3 a^{-1} +4 z^3 a^{-3} -a^8 z^2+a^6 z^2-2 a^4 z^2-17 a^2 z^2-7 z^2 a^{-2} -20 z^2+a^7 z+2 a^5 z+a^3 z-a z-2 z a^{-1} -z a^{-3} +a^4+3 a^2+ a^{-2} +4 }[/math]
The A2 invariant	[math]\displaystyle{ q^{20}-q^{18}+q^{16}-q^{14}-q^{12}+q^{10}-2 q^8+3 q^6-q^4+q^2+1- q^{-2} +2 q^{-4} + q^{-8} - q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-2 q^{104}-4 q^{102}+12 q^{100}-17 q^{98}+22 q^{96}-21 q^{94}+10 q^{92}+5 q^{90}-22 q^{88}+36 q^{86}-43 q^{84}+41 q^{82}-27 q^{80}+7 q^{78}+20 q^{76}-40 q^{74}+56 q^{72}-55 q^{70}+45 q^{68}-29 q^{66}-q^{64}+29 q^{62}-52 q^{60}+63 q^{58}-55 q^{56}+33 q^{54}-4 q^{52}-30 q^{50}+47 q^{48}-45 q^{46}+21 q^{44}+11 q^{42}-42 q^{40}+48 q^{38}-24 q^{36}-19 q^{34}+67 q^{32}-95 q^{30}+89 q^{28}-45 q^{26}-24 q^{24}+92 q^{22}-133 q^{20}+133 q^{18}-86 q^{16}+15 q^{14}+58 q^{12}-105 q^{10}+117 q^8-88 q^6+32 q^4+25 q^2-67+72 q^{-2} -39 q^{-4} -7 q^{-6} +56 q^{-8} -76 q^{-10} +62 q^{-12} -15 q^{-14} -46 q^{-16} +98 q^{-18} -115 q^{-20} +92 q^{-22} -34 q^{-24} -32 q^{-26} +86 q^{-28} -105 q^{-30} +93 q^{-32} -53 q^{-34} +4 q^{-36} +32 q^{-38} -53 q^{-40} +49 q^{-42} -34 q^{-44} +16 q^{-46} + q^{-48} -10 q^{-50} +10 q^{-52} -9 q^{-54} +5 q^{-56} -2 q^{-58} + q^{-60} }[/math]

Further Quantum Invariants

K11a174 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["K11a174"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+5 t^3-11 t^2+15 t-15+15 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+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]=

{ 79, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^4+3 q^3-5 q^2+8 q-10+12 q^{-1} -12 q^{-2} +11 q^{-3} -8 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7} }[/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+a^4 z^6-6 a^2 z^6+2 z^6+4 a^4 z^4-13 a^2 z^4-z^4 a^{-2} +9 z^4+4 a^4 z^2-12 a^2 z^2-3 z^2 a^{-2} +11 z^2+a^4-3 a^2- a^{-2} +4 }[/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+3 z^8 a^{-2} +4 z^8+4 a^5 z^7-4 a^3 z^7-19 a z^7-10 z^7 a^{-1} +z^7 a^{-3} +4 a^6 z^6-5 a^4 z^6-24 a^2 z^6-13 z^6 a^{-2} -28 z^6+3 a^7 z^5-2 a^5 z^5+a^3 z^5+15 a z^5+5 z^5 a^{-1} -4 z^5 a^{-3} +a^8 z^4-4 a^6 z^4+3 a^4 z^4+31 a^2 z^4+16 z^4 a^{-2} +39 z^4-4 a^7 z^3-3 a^5 z^3-a^3 z^3-2 a z^3+4 z^3 a^{-1} +4 z^3 a^{-3} -a^8 z^2+a^6 z^2-2 a^4 z^2-17 a^2 z^2-7 z^2 a^{-2} -20 z^2+a^7 z+2 a^5 z+a^3 z-a z-2 z a^{-1} -z a^{-3} +a^4+3 a^2+ a^{-2} +4 }[/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["K11a174"];

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+15 t-15+15 t^{-1} -11 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ -q^4+3 q^3-5 q^2+8 q-10+12 q^{-1} -12 q^{-2} +11 q^{-3} -8 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7} }[/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, 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{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{208}{3} }[/math] [math]\displaystyle{ -\frac{160}{3} }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 208 }[/math] [math]\displaystyle{ \frac{536}{3} }[/math] [math]\displaystyle{ -\frac{104}{3} }[/math] [math]\displaystyle{ \frac{112}{3} }[/math] [math]\displaystyle{ -64 }[/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 K11a174. 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 9                       1 -1 7                     2   2 5                   3 1   -2 3                 5 2     3 1               5 3       -2 -1             7 5         2 -3           6 6           0 -5         5 6             -1 -7       3 6               3 -9     2 5                 -3 -11   1 3                   2 -13   2                     -2 -15 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=-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}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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.

K11a173 K11a175