K11a175

K11a174

K11a176

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a175 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 X20,18,21,17 X10,20,11,19 X8,21,9,22
Gauss code	1, -6, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, -9, 11, -7
Dowker-Thistlethwaite code	4 12 14 16 18 2 22 6 20 10 8

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:K11a175/ThurstonBennequinNumber Hyperbolic Volume 13.9605 A-Polynomial See Data:K11a175/A-polynomial

[edit Notes for K11a175's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant 0

[edit Notes for K11a175's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-5 t^3+13 t^2-21 t+25-21 t^{-1} +13 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+3 z^6+3 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 105, 0 }
Jones polynomial	[math]\displaystyle{ q^6-4 q^5+7 q^4-11 q^3+15 q^2-16 q+17-14 q^{-1} +10 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +14 z^4-5 a^2 z^2-9 z^2 a^{-2} +2 z^2 a^{-4} +14 z^2-2 a^2-2 a^{-2} +5 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-2} +z^{10}+3 a z^9+7 z^9 a^{-1} +4 z^9 a^{-3} +4 a^2 z^8+10 z^8 a^{-2} +6 z^8 a^{-4} +8 z^8+4 a^3 z^7-12 z^7 a^{-1} -4 z^7 a^{-3} +4 z^7 a^{-5} +3 a^4 z^6-3 a^2 z^6-35 z^6 a^{-2} -17 z^6 a^{-4} +z^6 a^{-6} -23 z^6+a^5 z^5-5 a^3 z^5-3 a z^5+3 z^5 a^{-1} -11 z^5 a^{-3} -11 z^5 a^{-5} -6 a^4 z^4+39 z^4 a^{-2} +13 z^4 a^{-4} -2 z^4 a^{-6} +30 z^4-2 a^5 z^3+2 a z^3+7 z^3 a^{-1} +13 z^3 a^{-3} +6 z^3 a^{-5} +3 a^4 z^2-3 a^2 z^2-17 z^2 a^{-2} -4 z^2 a^{-4} -19 z^2+a^5 z+a^3 z-a z-3 z a^{-1} -2 z a^{-3} +2 a^2+2 a^{-2} +5 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{14}+q^{12}-2 q^{10}+q^8+q^6-2 q^4+4 q^2-2+3 q^{-2} + q^{-4} +3 q^{-8} -3 q^{-10} - q^{-14} - q^{-16} + q^{-18} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-6 q^{70}-2 q^{68}+16 q^{66}-28 q^{64}+40 q^{62}-44 q^{60}+31 q^{58}-7 q^{56}-33 q^{54}+73 q^{52}-105 q^{50}+116 q^{48}-99 q^{46}+48 q^{44}+28 q^{42}-114 q^{40}+187 q^{38}-217 q^{36}+188 q^{34}-106 q^{32}-21 q^{30}+149 q^{28}-237 q^{26}+260 q^{24}-186 q^{22}+57 q^{20}+82 q^{18}-179 q^{16}+186 q^{14}-106 q^{12}-28 q^{10}+153 q^8-206 q^6+154 q^4-3 q^2-182+332 q^{-2} -366 q^{-4} +266 q^{-6} -62 q^{-8} -177 q^{-10} +371 q^{-12} -440 q^{-14} +374 q^{-16} -184 q^{-18} -46 q^{-20} +244 q^{-22} -332 q^{-24} +289 q^{-26} -145 q^{-28} -33 q^{-30} +168 q^{-32} -213 q^{-34} +144 q^{-36} +5 q^{-38} -159 q^{-40} +256 q^{-42} -247 q^{-44} +122 q^{-46} +54 q^{-48} -226 q^{-50} +318 q^{-52} -304 q^{-54} +192 q^{-56} -27 q^{-58} -131 q^{-60} +228 q^{-62} -241 q^{-64} +183 q^{-66} -85 q^{-68} -13 q^{-70} +77 q^{-72} -103 q^{-74} +90 q^{-76} -55 q^{-78} +24 q^{-80} +5 q^{-82} -17 q^{-84} +17 q^{-86} -14 q^{-88} +7 q^{-90} -3 q^{-92} + q^{-94} }[/math]

Further Quantum Invariants

K11a175 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["K11a175"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-5 t^3+13 t^2-21 t+25-21 t^{-1} +13 t^{-2} -5 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 105, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^6-4 q^5+7 q^4-11 q^3+15 q^2-16 q+17-14 q^{-1} +10 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^{10} a^{-2} +z^{10}+3 a z^9+7 z^9 a^{-1} +4 z^9 a^{-3} +4 a^2 z^8+10 z^8 a^{-2} +6 z^8 a^{-4} +8 z^8+4 a^3 z^7-12 z^7 a^{-1} -4 z^7 a^{-3} +4 z^7 a^{-5} +3 a^4 z^6-3 a^2 z^6-35 z^6 a^{-2} -17 z^6 a^{-4} +z^6 a^{-6} -23 z^6+a^5 z^5-5 a^3 z^5-3 a z^5+3 z^5 a^{-1} -11 z^5 a^{-3} -11 z^5 a^{-5} -6 a^4 z^4+39 z^4 a^{-2} +13 z^4 a^{-4} -2 z^4 a^{-6} +30 z^4-2 a^5 z^3+2 a z^3+7 z^3 a^{-1} +13 z^3 a^{-3} +6 z^3 a^{-5} +3 a^4 z^2-3 a^2 z^2-17 z^2 a^{-2} -4 z^2 a^{-4} -19 z^2+a^5 z+a^3 z-a z-3 z a^{-1} -2 z a^{-3} +2 a^2+2 a^{-2} +5 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a306,}

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["K11a175"];

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+13 t^2-21 t+25-21 t^{-1} +13 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^6-4 q^5+7 q^4-11 q^3+15 q^2-16 q+17-14 q^{-1} +10 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} }[/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]=

{K11a306,}

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, 0)

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{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{76}{3} }[/math] [math]\displaystyle{ -\frac{28}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{608}{3} }[/math] [math]\displaystyle{ -\frac{224}{3} }[/math] [math]\displaystyle{ \frac{2551}{15} }[/math] [math]\displaystyle{ \frac{676}{15} }[/math] [math]\displaystyle{ -\frac{2996}{45} }[/math] [math]\displaystyle{ -\frac{7}{9} }[/math] [math]\displaystyle{ -\frac{89}{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]0 is the signature of K11a175. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   4 1   3 7                 7 3     -4 5               8 4       4 3             8 7         -1 1           9 8           1 -1         6 9             3 -3       4 8               -4 -5     2 6                 4 -7   1 4                   -3 -9   2                     2 -11 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/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.

K11a174 K11a176