K11a162

K11a161

K11a163

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a162 at Knotilus!

Knot presentations

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

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

Three dimensional invariants

Symmetry type Chiral 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:K11a162/ThurstonBennequinNumber Hyperbolic Volume 18.245 A-Polynomial See Data:K11a162/A-polynomial

[edit Notes for K11a162'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 K11a162's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+7 t^3-20 t^2+35 t-41+35 t^{-1} -20 t^{-2} +7 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-z^6+2 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 167, -2 }
Jones polynomial	[math]\displaystyle{ q^3-5 q^2+11 q-17+24 q^{-1} -27 q^{-2} +27 q^{-3} -23 q^{-4} +17 q^{-5} -10 q^{-6} +4 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^2 z^8+2 a^4 z^6-4 a^2 z^6+z^6-a^6 z^4+6 a^4 z^4-5 a^2 z^4+2 z^4-2 a^6 z^2+5 a^4 z^2-a^2 z^2-a^6+a^4+a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 3 a^4 z^{10}+3 a^2 z^{10}+9 a^5 z^9+18 a^3 z^9+9 a z^9+12 a^6 z^8+20 a^4 z^8+18 a^2 z^8+10 z^8+9 a^7 z^7-2 a^5 z^7-27 a^3 z^7-11 a z^7+5 z^7 a^{-1} +4 a^8 z^6-18 a^6 z^6-53 a^4 z^6-54 a^2 z^6+z^6 a^{-2} -22 z^6+a^9 z^5-13 a^7 z^5-18 a^5 z^5-2 a^3 z^5-7 a z^5-9 z^5 a^{-1} -4 a^8 z^4+13 a^6 z^4+42 a^4 z^4+39 a^2 z^4-z^4 a^{-2} +13 z^4-a^9 z^3+9 a^7 z^3+18 a^5 z^3+13 a^3 z^3+8 a z^3+3 z^3 a^{-1} +a^8 z^2-6 a^6 z^2-13 a^4 z^2-7 a^2 z^2-z^2-3 a^7 z-5 a^5 z-3 a^3 z-a z+a^6+a^4-a^2 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{24}+q^{22}-3 q^{18}+4 q^{16}-4 q^{14}+2 q^{12}+2 q^{10}-3 q^8+6 q^6-5 q^4+5 q^2-2 q^{-2} +3 q^{-4} -3 q^{-6} + q^{-8} }[/math]
The G2 invariant	[math]\displaystyle{ q^{128}-3 q^{126}+7 q^{124}-13 q^{122}+17 q^{120}-19 q^{118}+12 q^{116}+11 q^{114}-46 q^{112}+94 q^{110}-138 q^{108}+151 q^{106}-118 q^{104}+17 q^{102}+152 q^{100}-344 q^{98}+509 q^{96}-563 q^{94}+427 q^{92}-103 q^{90}-372 q^{88}+858 q^{86}-1159 q^{84}+1122 q^{82}-679 q^{80}-70 q^{78}+863 q^{76}-1399 q^{74}+1437 q^{72}-936 q^{70}+69 q^{68}+791 q^{66}-1265 q^{64}+1134 q^{62}-429 q^{60}-505 q^{58}+1243 q^{56}-1419 q^{54}+900 q^{52}+100 q^{50}-1198 q^{48}+1938 q^{46}-1983 q^{44}+1306 q^{42}-122 q^{40}-1126 q^{38}+1989 q^{36}-2169 q^{34}+1607 q^{32}-550 q^{30}-614 q^{28}+1451 q^{26}-1645 q^{24}+1181 q^{22}-254 q^{20}-695 q^{18}+1248 q^{16}-1177 q^{14}+500 q^{12}+447 q^{10}-1235 q^8+1534 q^6-1190 q^4+367 q^2+595-1310 q^{-2} +1523 q^{-4} -1208 q^{-6} +535 q^{-8} +190 q^{-10} -721 q^{-12} +919 q^{-14} -794 q^{-16} +480 q^{-18} -114 q^{-20} -162 q^{-22} +286 q^{-24} -290 q^{-26} +206 q^{-28} -102 q^{-30} +23 q^{-32} +28 q^{-34} -42 q^{-36} +36 q^{-38} -24 q^{-40} +11 q^{-42} -4 q^{-44} + q^{-46} }[/math]

Further Quantum Invariants

K11a162 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["K11a162"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^4+7 t^3-20 t^2+35 t-41+35 t^{-1} -20 t^{-2} +7 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

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

{ 167, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^3-5 q^2+11 q-17+24 q^{-1} -27 q^{-2} +27 q^{-3} -23 q^{-4} +17 q^{-5} -10 q^{-6} +4 q^{-7} - q^{-8} }[/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+2 a^4 z^6-4 a^2 z^6+z^6-a^6 z^4+6 a^4 z^4-5 a^2 z^4+2 z^4-2 a^6 z^2+5 a^4 z^2-a^2 z^2-a^6+a^4+a^2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ 3 a^4 z^{10}+3 a^2 z^{10}+9 a^5 z^9+18 a^3 z^9+9 a z^9+12 a^6 z^8+20 a^4 z^8+18 a^2 z^8+10 z^8+9 a^7 z^7-2 a^5 z^7-27 a^3 z^7-11 a z^7+5 z^7 a^{-1} +4 a^8 z^6-18 a^6 z^6-53 a^4 z^6-54 a^2 z^6+z^6 a^{-2} -22 z^6+a^9 z^5-13 a^7 z^5-18 a^5 z^5-2 a^3 z^5-7 a z^5-9 z^5 a^{-1} -4 a^8 z^4+13 a^6 z^4+42 a^4 z^4+39 a^2 z^4-z^4 a^{-2} +13 z^4-a^9 z^3+9 a^7 z^3+18 a^5 z^3+13 a^3 z^3+8 a z^3+3 z^3 a^{-1} +a^8 z^2-6 a^6 z^2-13 a^4 z^2-7 a^2 z^2-z^2-3 a^7 z-5 a^5 z-3 a^3 z-a z+a^6+a^4-a^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["K11a162"];

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+7 t^3-20 t^2+35 t-41+35 t^{-1} -20 t^{-2} +7 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^3-5 q^2+11 q-17+24 q^{-1} -27 q^{-2} +27 q^{-3} -23 q^{-4} +17 q^{-5} -10 q^{-6} +4 q^{-7} - q^{-8} }[/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, -3)

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{ -24 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{220}{3} }[/math] [math]\displaystyle{ -\frac{4}{3} }[/math] [math]\displaystyle{ -192 }[/math] [math]\displaystyle{ -304 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -56 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ \frac{1760}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ \frac{18991}{15} }[/math] [math]\displaystyle{ -\frac{724}{15} }[/math] [math]\displaystyle{ \frac{16324}{45} }[/math] [math]\displaystyle{ \frac{833}{9} }[/math] [math]\displaystyle{ \frac{271}{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]-2 is the signature of K11a162. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     4   -4 3                   7 1   6 1                 10 4     -6 -1               14 7       7 -3             14 11         -3 -5           13 13           0 -7         10 14             4 -9       7 13               -6 -11     3 10                 7 -13   1 7                   -6 -15   3                     3 -17 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/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.

K11a161 K11a163