K11a73

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K11a72.gif

K11a72

K11a74.gif

K11a74

K11a73.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a73 at Knotilus!

[edit K11a73 Quick Notes]


[edit K11a73 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X16,10,17,9 X2,11,3,12 X18,13,19,14 X20,16,21,15 X22,17,1,18 X6,20,7,19 X8,21,9,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -11, 5, -2, 6, -3, 7, -4, 8, -5, 9, -7, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 10 12 14 16 2 18 20 22 6 8
A Braid Representative
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A Morse Link Presentation K11a73 ML.gif

Four dimensional invariants

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

[edit Notes for K11a73's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^4-7 t^3+21 t^2-37 t+45-37 t^{-1} +21 t^{-2} -7 t^{-3} + t^{-4} }[/math]
Conway polynomial [math]\displaystyle{ z^8+z^6-z^4+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 177, 0 }
Jones polynomial [math]\displaystyle{ -q^5+5 q^4-11 q^3+18 q^2-25 q+29-28 q^{-1} +25 q^{-2} -18 q^{-3} +11 q^{-4} -5 q^{-5} + q^{-6} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +4 z^6+a^4 z^4-5 a^2 z^4-2 z^4 a^{-2} +5 z^4+a^4 z^2-a^2 z^2-a^4+3 a^2+ a^{-2} -2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 3 a^2 z^{10}+3 z^{10}+9 a^3 z^9+19 a z^9+10 z^9 a^{-1} +10 a^4 z^8+21 a^2 z^8+14 z^8 a^{-2} +25 z^8+5 a^5 z^7-8 a^3 z^7-23 a z^7+z^7 a^{-1} +11 z^7 a^{-3} +a^6 z^6-21 a^4 z^6-60 a^2 z^6-18 z^6 a^{-2} +5 z^6 a^{-4} -61 z^6-9 a^5 z^5-14 a^3 z^5-16 a z^5-26 z^5 a^{-1} -14 z^5 a^{-3} +z^5 a^{-5} -a^6 z^4+13 a^4 z^4+42 a^2 z^4+6 z^4 a^{-2} -4 z^4 a^{-4} +38 z^4+4 a^5 z^3+14 a^3 z^3+23 a z^3+18 z^3 a^{-1} +5 z^3 a^{-3} -2 a^4 z^2-5 a^2 z^2-3 z^2-2 a^3 z-4 a z-2 z a^{-1} -a^4-3 a^2- a^{-2} -2 }[/math]
The A2 invariant [math]\displaystyle{ q^{18}-2 q^{16}-q^{14}+2 q^{12}-4 q^{10}+6 q^8+4 q^2-6+5 q^{-2} -5 q^{-4} + q^{-6} +3 q^{-8} -3 q^{-10} +3 q^{-12} - q^{-14} }[/math]
The G2 invariant [math]\displaystyle{ q^{94}-4 q^{92}+11 q^{90}-24 q^{88}+36 q^{86}-43 q^{84}+30 q^{82}+20 q^{80}-101 q^{78}+212 q^{76}-304 q^{74}+312 q^{72}-190 q^{70}-98 q^{68}+491 q^{66}-854 q^{64}+1023 q^{62}-847 q^{60}+283 q^{58}+525 q^{56}-1305 q^{54}+1734 q^{52}-1579 q^{50}+834 q^{48}+250 q^{46}-1270 q^{44}+1787 q^{42}-1587 q^{40}+759 q^{38}+346 q^{36}-1218 q^{34}+1474 q^{32}-989 q^{30}-18 q^{28}+1115 q^{26}-1794 q^{24}+1733 q^{22}-901 q^{20}-408 q^{18}+1698 q^{16}-2465 q^{14}+2411 q^{12}-1509 q^{10}+86 q^8+1352 q^6-2298 q^4+2402 q^2-1665+400 q^{-2} +860 q^{-4} -1617 q^{-6} +1594 q^{-8} -856 q^{-10} -216 q^{-12} +1119 q^{-14} -1449 q^{-16} +1056 q^{-18} -149 q^{-20} -885 q^{-22} +1595 q^{-24} -1680 q^{-26} +1162 q^{-28} -252 q^{-30} -678 q^{-32} +1298 q^{-34} -1452 q^{-36} +1156 q^{-38} -583 q^{-40} -34 q^{-42} +503 q^{-44} -720 q^{-46} +692 q^{-48} -493 q^{-50} +241 q^{-52} -8 q^{-54} -149 q^{-56} +206 q^{-58} -199 q^{-60} +140 q^{-62} -72 q^{-64} +21 q^{-66} +16 q^{-68} -28 q^{-70} +27 q^{-72} -20 q^{-74} +10 q^{-76} -4 q^{-78} + q^{-80} }[/math]
Further Quantum Invariants
K11a73 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["K11a73"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^4-7 t^3+21 t^2-37 t+45-37 t^{-1} +21 t^{-2} -7 t^{-3} + t^{-4} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^8+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]= { 177, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^5+5 q^4-11 q^3+18 q^2-25 q+29-28 q^{-1} +25 q^{-2} -18 q^{-3} +11 q^{-4} -5 q^{-5} + q^{-6} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +4 z^6+a^4 z^4-5 a^2 z^4-2 z^4 a^{-2} +5 z^4+a^4 z^2-a^2 z^2-a^4+3 a^2+ a^{-2} -2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 3 a^2 z^{10}+3 z^{10}+9 a^3 z^9+19 a z^9+10 z^9 a^{-1} +10 a^4 z^8+21 a^2 z^8+14 z^8 a^{-2} +25 z^8+5 a^5 z^7-8 a^3 z^7-23 a z^7+z^7 a^{-1} +11 z^7 a^{-3} +a^6 z^6-21 a^4 z^6-60 a^2 z^6-18 z^6 a^{-2} +5 z^6 a^{-4} -61 z^6-9 a^5 z^5-14 a^3 z^5-16 a z^5-26 z^5 a^{-1} -14 z^5 a^{-3} +z^5 a^{-5} -a^6 z^4+13 a^4 z^4+42 a^2 z^4+6 z^4 a^{-2} -4 z^4 a^{-4} +38 z^4+4 a^5 z^3+14 a^3 z^3+23 a z^3+18 z^3 a^{-1} +5 z^3 a^{-3} -2 a^4 z^2-5 a^2 z^2-3 z^2-2 a^3 z-4 a z-2 z a^{-1} -a^4-3 a^2- a^{-2} -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["K11a73"];
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+21 t^2-37 t+45-37 t^{-1} +21 t^{-2} -7 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ -q^5+5 q^4-11 q^3+18 q^2-25 q+29-28 q^{-1} +25 q^{-2} -18 q^{-3} +11 q^{-4} -5 q^{-5} + q^{-6} }[/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{ 16 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{80}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 88 }[/math] [math]\displaystyle{ -56 }[/math] [math]\displaystyle{ \frac{328}{3} }[/math] [math]\displaystyle{ -\frac{8}{3} }[/math] [math]\displaystyle{ 24 }[/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]0 is the signature of K11a73. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -6-5-4-3-2-1012345χ
11           1-1
9          4 4
7         71 -6
5        114  7
3       147   -7
1      1511    4
-1     1415     1
-3    1114      -3
-5   714       7
-7  411        -7
-9 17         6
-11 4          -4
-131           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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/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}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

K11a72.gif

K11a72

K11a74.gif

K11a74

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