K11a163

From Knot Atlas
Jump to navigationJump to search

K11a162.gif

K11a162

K11a164.gif

K11a164

K11a163.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a163 at Knotilus!

[edit K11a163 Quick Notes]


[edit K11a163 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X18,5,19,6 X14,8,15,7 X16,10,17,9 X2,11,3,12 X20,14,21,13 X8,16,9,15 X22,17,1,18 X12,20,13,19 X6,21,7,22
Gauss code 1, -6, 2, -1, 3, -11, 4, -8, 5, -2, 6, -10, 7, -4, 8, -5, 9, -3, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 10 18 14 16 2 20 8 22 12 6
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11a163 ML.gif

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:K11a163/ThurstonBennequinNumber
Hyperbolic Volume 15.9193
A-Polynomial See Data:K11a163/A-polynomial

[edit Notes for K11a163'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 K11a163's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+6 t^3-15 t^2+24 t-27+24 t^{-1} -15 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ -z^8-2 z^6+z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 119, 2 }
Jones polynomial [math]\displaystyle{ q^7-4 q^6+8 q^5-13 q^4+17 q^3-19 q^2+19 q-15+12 q^{-1} -7 q^{-2} +3 q^{-3} - q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-9 z^4 a^{-2} +3 z^4 a^{-4} +8 z^4-3 a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} +10 z^2-2 a^2-2 a^{-2} +5 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+4 a z^9+11 z^9 a^{-1} +7 z^9 a^{-3} +3 a^2 z^8+12 z^8 a^{-2} +11 z^8 a^{-4} +4 z^8+a^3 z^7-11 a z^7-28 z^7 a^{-1} -5 z^7 a^{-3} +11 z^7 a^{-5} -11 a^2 z^6-44 z^6 a^{-2} -17 z^6 a^{-4} +8 z^6 a^{-6} -30 z^6-4 a^3 z^5+4 a z^5+13 z^5 a^{-1} -13 z^5 a^{-3} -14 z^5 a^{-5} +4 z^5 a^{-7} +13 a^2 z^4+40 z^4 a^{-2} +8 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +37 z^4+5 a^3 z^3+6 a z^3+6 z^3 a^{-1} +12 z^3 a^{-3} +5 z^3 a^{-5} -2 z^3 a^{-7} -7 a^2 z^2-14 z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} -18 z^2-2 a^3 z-4 a z-4 z a^{-1} -2 z a^{-3} +2 a^2+2 a^{-2} +5 }[/math]
The A2 invariant [math]\displaystyle{ -q^{12}-2 q^6+3 q^4-q^2+3+3 q^{-2} -2 q^{-4} +4 q^{-6} -4 q^{-8} +2 q^{-10} - q^{-12} -2 q^{-14} +2 q^{-16} -2 q^{-18} + q^{-20} }[/math]
The G2 invariant [math]\displaystyle{ q^{60}-2 q^{58}+6 q^{56}-11 q^{54}+15 q^{52}-18 q^{50}+9 q^{48}+12 q^{46}-48 q^{44}+91 q^{42}-123 q^{40}+113 q^{38}-47 q^{36}-78 q^{34}+225 q^{32}-330 q^{30}+336 q^{28}-214 q^{26}-25 q^{24}+288 q^{22}-482 q^{20}+515 q^{18}-346 q^{16}+48 q^{14}+262 q^{12}-454 q^{10}+448 q^8-255 q^6-33 q^4+292 q^2-393+310 q^{-2} -59 q^{-4} -236 q^{-6} +464 q^{-8} -507 q^{-10} +354 q^{-12} -51 q^{-14} -304 q^{-16} +578 q^{-18} -666 q^{-20} +535 q^{-22} -213 q^{-24} -177 q^{-26} +493 q^{-28} -622 q^{-30} +507 q^{-32} -224 q^{-34} -113 q^{-36} +354 q^{-38} -407 q^{-40} +265 q^{-42} -5 q^{-44} -237 q^{-46} +357 q^{-48} -301 q^{-50} +97 q^{-52} +145 q^{-54} -336 q^{-56} +400 q^{-58} -320 q^{-60} +149 q^{-62} +55 q^{-64} -222 q^{-66} +302 q^{-68} -296 q^{-70} +215 q^{-72} -99 q^{-74} -16 q^{-76} +106 q^{-78} -157 q^{-80} +161 q^{-82} -127 q^{-84} +77 q^{-86} -19 q^{-88} -26 q^{-90} +51 q^{-92} -61 q^{-94} +51 q^{-96} -32 q^{-98} +15 q^{-100} + q^{-102} -8 q^{-104} +10 q^{-106} -10 q^{-108} +6 q^{-110} -3 q^{-112} + q^{-114} }[/math]
Further Quantum Invariants
K11a163 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["K11a163"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+6 t^3-15 t^2+24 t-27+24 t^{-1} -15 t^{-2} +6 t^{-3} - t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^8-2 z^6+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]= { 119, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^7-4 q^6+8 q^5-13 q^4+17 q^3-19 q^2+19 q-15+12 q^{-1} -7 q^{-2} +3 q^{-3} - q^{-4} }[/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} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-9 z^4 a^{-2} +3 z^4 a^{-4} +8 z^4-3 a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} +10 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{ 2 z^{10} a^{-2} +2 z^{10}+4 a z^9+11 z^9 a^{-1} +7 z^9 a^{-3} +3 a^2 z^8+12 z^8 a^{-2} +11 z^8 a^{-4} +4 z^8+a^3 z^7-11 a z^7-28 z^7 a^{-1} -5 z^7 a^{-3} +11 z^7 a^{-5} -11 a^2 z^6-44 z^6 a^{-2} -17 z^6 a^{-4} +8 z^6 a^{-6} -30 z^6-4 a^3 z^5+4 a z^5+13 z^5 a^{-1} -13 z^5 a^{-3} -14 z^5 a^{-5} +4 z^5 a^{-7} +13 a^2 z^4+40 z^4 a^{-2} +8 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +37 z^4+5 a^3 z^3+6 a z^3+6 z^3 a^{-1} +12 z^3 a^{-3} +5 z^3 a^{-5} -2 z^3 a^{-7} -7 a^2 z^2-14 z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} -18 z^2-2 a^3 z-4 a z-4 z a^{-1} -2 z a^{-3} +2 a^2+2 a^{-2} +5 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a66,}

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["K11a163"];
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^3-15 t^2+24 t-27+24 t^{-1} -15 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^7-4 q^6+8 q^5-13 q^4+17 q^3-19 q^2+19 q-15+12 q^{-1} -7 q^{-2} +3 q^{-3} - q^{-4} }[/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]= {K11a66,}
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{124}{3} }[/math] [math]\displaystyle{ \frac{20}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ -64 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{992}{3} }[/math] [math]\displaystyle{ \frac{160}{3} }[/math] [math]\displaystyle{ \frac{5071}{15} }[/math] [math]\displaystyle{ \frac{2516}{15} }[/math] [math]\displaystyle{ -\frac{2516}{45} }[/math] [math]\displaystyle{ \frac{305}{9} }[/math] [math]\displaystyle{ -\frac{209}{15} }[/math]

V2,1 through V6,9 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 K11a163. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -5-4-3-2-10123456χ
15           11
13          3 -3
11         51 4
9        83  -5
7       95   4
5      108    -2
3     99     0
1    711      4
-1   58       -3
-3  27        5
-5 15         -4
-7 2          2
-91           -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=3 }[/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}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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.

K11a162.gif

K11a162

K11a164.gif

K11a164

Retrieved from "https://katlas.org/index.php?title=K11a163&oldid=110572"