K11a170

K11a169

K11a171

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a170 at Knotilus!

Knot presentations

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

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:K11a170/ThurstonBennequinNumber Hyperbolic Volume 19.2051 A-Polynomial See Data:K11a170/A-polynomial

[edit Notes for K11a170'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 K11a170's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-6 t^3+20 t^2-40 t+51-40 t^{-1} +20 t^{-2} -6 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+2 z^6+4 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 185, 0 }
Jones polynomial	[math]\displaystyle{ q^6-6 q^5+13 q^4-20 q^3+27 q^2-30 q+30-25 q^{-1} +18 q^{-2} -10 q^{-3} +4 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +5 z^6-3 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} +11 z^4-4 a^2 z^2-3 z^2 a^{-2} +9 z^2-a^2+ a^{-2} - a^{-4} +2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 4 z^{10} a^{-2} +4 z^{10}+11 a z^9+23 z^9 a^{-1} +12 z^9 a^{-3} +13 a^2 z^8+22 z^8 a^{-2} +13 z^8 a^{-4} +22 z^8+9 a^3 z^7-7 a z^7-37 z^7 a^{-1} -15 z^7 a^{-3} +6 z^7 a^{-5} +4 a^4 z^6-20 a^2 z^6-65 z^6 a^{-2} -26 z^6 a^{-4} +z^6 a^{-6} -62 z^6+a^5 z^5-12 a^3 z^5-11 a z^5+3 z^5 a^{-1} -7 z^5 a^{-3} -8 z^5 a^{-5} -4 a^4 z^4+17 a^2 z^4+41 z^4 a^{-2} +11 z^4 a^{-4} +51 z^4-a^5 z^3+8 a^3 z^3+15 a z^3+10 z^3 a^{-1} +4 z^3 a^{-3} +a^4 z^2-7 a^2 z^2-6 z^2 a^{-2} +2 z^2 a^{-4} -16 z^2-2 a^3 z-4 a z-2 z a^{-1} +2 z a^{-3} +2 z a^{-5} +a^2- a^{-2} - a^{-4} +2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{14}+2 q^{12}-4 q^{10}+3 q^8+2 q^6-5 q^4+6 q^2-5+4 q^{-2} + q^{-4} - q^{-6} +6 q^{-8} -5 q^{-10} +2 q^{-12} -3 q^{-16} + q^{-18} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+17 q^{72}-19 q^{70}+12 q^{68}+10 q^{66}-43 q^{64}+89 q^{62}-133 q^{60}+152 q^{58}-131 q^{56}+45 q^{54}+111 q^{52}-308 q^{50}+508 q^{48}-630 q^{46}+572 q^{44}-291 q^{42}-223 q^{40}+849 q^{38}-1362 q^{36}+1527 q^{34}-1168 q^{32}+303 q^{30}+809 q^{28}-1756 q^{26}+2130 q^{24}-1712 q^{22}+614 q^{20}+723 q^{18}-1724 q^{16}+1928 q^{14}-1209 q^{12}-84 q^{10}+1365 q^8-2013 q^6+1679 q^4-493 q^2-1082+2370 q^{-2} -2805 q^{-4} +2187 q^{-6} -705 q^{-8} -1085 q^{-10} +2534 q^{-12} -3117 q^{-14} +2634 q^{-16} -1294 q^{-18} -417 q^{-20} +1865 q^{-22} -2504 q^{-24} +2151 q^{-26} -954 q^{-28} -523 q^{-30} +1645 q^{-32} -1927 q^{-34} +1239 q^{-36} +66 q^{-38} -1391 q^{-40} +2141 q^{-42} -1960 q^{-44} +941 q^{-46} +475 q^{-48} -1691 q^{-50} +2230 q^{-52} -1946 q^{-54} +1022 q^{-56} +97 q^{-58} -1000 q^{-60} +1400 q^{-62} -1274 q^{-64} +806 q^{-66} -224 q^{-68} -236 q^{-70} +457 q^{-72} -468 q^{-74} +331 q^{-76} -160 q^{-78} +29 q^{-80} +50 q^{-82} -69 q^{-84} +57 q^{-86} -35 q^{-88} +15 q^{-90} -5 q^{-92} + q^{-94} }[/math]

Further Quantum Invariants

K11a170 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["K11a170"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-6 t^3+20 t^2-40 t+51-40 t^{-1} +20 t^{-2} -6 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+2 z^6+4 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]=

{ 185, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^6-6 q^5+13 q^4-20 q^3+27 q^2-30 q+30-25 q^{-1} +18 q^{-2} -10 q^{-3} +4 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} +5 z^6-3 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} +11 z^4-4 a^2 z^2-3 z^2 a^{-2} +9 z^2-a^2+ a^{-2} - a^{-4} +2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 4 z^{10} a^{-2} +4 z^{10}+11 a z^9+23 z^9 a^{-1} +12 z^9 a^{-3} +13 a^2 z^8+22 z^8 a^{-2} +13 z^8 a^{-4} +22 z^8+9 a^3 z^7-7 a z^7-37 z^7 a^{-1} -15 z^7 a^{-3} +6 z^7 a^{-5} +4 a^4 z^6-20 a^2 z^6-65 z^6 a^{-2} -26 z^6 a^{-4} +z^6 a^{-6} -62 z^6+a^5 z^5-12 a^3 z^5-11 a z^5+3 z^5 a^{-1} -7 z^5 a^{-3} -8 z^5 a^{-5} -4 a^4 z^4+17 a^2 z^4+41 z^4 a^{-2} +11 z^4 a^{-4} +51 z^4-a^5 z^3+8 a^3 z^3+15 a z^3+10 z^3 a^{-1} +4 z^3 a^{-3} +a^4 z^2-7 a^2 z^2-6 z^2 a^{-2} +2 z^2 a^{-4} -16 z^2-2 a^3 z-4 a z-2 z a^{-1} +2 z a^{-3} +2 z a^{-5} +a^2- a^{-2} - a^{-4} +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["K11a170"];

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+20 t^2-40 t+51-40 t^{-1} +20 t^{-2} -6 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^6-6 q^5+13 q^4-20 q^3+27 q^2-30 q+30-25 q^{-1} +18 q^{-2} -10 q^{-3} +4 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]=

{}

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, 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{ 8 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{76}{3} }[/math] [math]\displaystyle{ -\frac{52}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{272}{3} }[/math] [math]\displaystyle{ \frac{128}{3} }[/math] [math]\displaystyle{ -24 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{608}{3} }[/math] [math]\displaystyle{ -\frac{416}{3} }[/math] [math]\displaystyle{ \frac{4231}{15} }[/math] [math]\displaystyle{ \frac{612}{5} }[/math] [math]\displaystyle{ -\frac{7196}{45} }[/math] [math]\displaystyle{ \frac{233}{9} }[/math] [math]\displaystyle{ -\frac{1049}{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 K11a170. 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                     5   -5 9                   8 1   7 7                 12 5     -7 5               15 8       7 3             15 12         -3 1           15 15           0 -1         11 16             5 -3       7 14               -7 -5     3 11                 8 -7   1 7                   -6 -9   3                     3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{15} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{15} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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.

K11a169 K11a171