K11a99

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K11a98.gif

K11a98

K11a100.gif

K11a100

K11a99.gif
(Knotscape image) 		See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a99 at Knotilus!

[edit K11a99 Quick Notes]


[edit K11a99 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X20,8,21,7 X16,10,17,9 X2,11,3,12 X18,13,19,14 X8,16,9,15 X22,17,1,18 X6,20,7,19 X14,21,15,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -11, 8, -5, 9, -7, 10, -4, 11, -9
Dowker-Thistlethwaite code 4 10 12 20 16 2 18 8 22 6 14
A Braid Representative
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A Morse Link Presentation K11a99 ML.gif

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:K11a99/ThurstonBennequinNumber
Hyperbolic Volume 16.6086
A-Polynomial See Data:K11a99/A-polynomial

[edit Notes for K11a99'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 K11a99's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^4+6 t^3-17 t^2+28 t-31+28 t^{-1} -17 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 { 135, 2 }
Jones polynomial [math]\displaystyle{ q^7-4 q^6+9 q^5-14 q^4+19 q^3-22 q^2+21 q-18+14 q^{-1} -8 q^{-2} +4 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^4-10 z^4 a^{-2} +3 z^4 a^{-4} +7 z^4-2 a^2 z^2-10 z^2 a^{-2} +3 z^2 a^{-4} +7 z^2-4 a^{-2} +2 a^{-4} +3 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+5 a z^9+13 z^9 a^{-1} +8 z^9 a^{-3} +4 a^2 z^8+19 z^8 a^{-2} +13 z^8 a^{-4} +10 z^8+a^3 z^7-12 a z^7-27 z^7 a^{-1} -z^7 a^{-3} +13 z^7 a^{-5} -14 a^2 z^6-62 z^6 a^{-2} -18 z^6 a^{-4} +9 z^6 a^{-6} -49 z^6-3 a^3 z^5+a z^5-25 z^5 a^{-3} -17 z^5 a^{-5} +4 z^5 a^{-7} +16 a^2 z^4+55 z^4 a^{-2} +7 z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} +55 z^4+3 a^3 z^3+10 a z^3+15 z^3 a^{-1} +18 z^3 a^{-3} +9 z^3 a^{-5} -z^3 a^{-7} -6 a^2 z^2-23 z^2 a^{-2} -5 z^2 a^{-4} +3 z^2 a^{-6} -21 z^2-a^3 z-3 a z-3 z a^{-1} -3 z a^{-3} -2 z a^{-5} +4 a^{-2} +2 a^{-4} +3 }[/math]
The A2 invariant [math]\displaystyle{ -q^{12}+q^{10}+q^8-q^6+4 q^4-2 q^2+2+ q^{-2} -4 q^{-4} +3 q^{-6} -5 q^{-8} +3 q^{-10} - q^{-14} +3 q^{-16} -2 q^{-18} + q^{-20} }[/math]
The G2 invariant [math]\displaystyle{ q^{60}-3 q^{58}+9 q^{56}-19 q^{54}+28 q^{52}-33 q^{50}+17 q^{48}+26 q^{46}-91 q^{44}+165 q^{42}-204 q^{40}+167 q^{38}-37 q^{36}-174 q^{34}+394 q^{32}-522 q^{30}+480 q^{28}-242 q^{26}-135 q^{24}+516 q^{22}-741 q^{20}+713 q^{18}-405 q^{16}-48 q^{14}+467 q^{12}-683 q^{10}+599 q^8-269 q^6-156 q^4+498 q^2-586+391 q^{-2} +18 q^{-4} -460 q^{-6} +748 q^{-8} -756 q^{-10} +454 q^{-12} +43 q^{-14} -573 q^{-16} +935 q^{-18} -987 q^{-20} +703 q^{-22} -172 q^{-24} -401 q^{-26} +798 q^{-28} -894 q^{-30} +643 q^{-32} -188 q^{-34} -279 q^{-36} +566 q^{-38} -560 q^{-40} +292 q^{-42} +105 q^{-44} -431 q^{-46} +544 q^{-48} -402 q^{-50} +68 q^{-52} +293 q^{-54} -544 q^{-56} +602 q^{-58} -444 q^{-60} +168 q^{-62} +140 q^{-64} -372 q^{-66} +462 q^{-68} -420 q^{-70} +276 q^{-72} -96 q^{-74} -65 q^{-76} +179 q^{-78} -227 q^{-80} +213 q^{-82} -152 q^{-84} +78 q^{-86} -5 q^{-88} -46 q^{-90} +68 q^{-92} -74 q^{-94} +57 q^{-96} -33 q^{-98} +13 q^{-100} +4 q^{-102} -10 q^{-104} +11 q^{-106} -10 q^{-108} +6 q^{-110} -3 q^{-112} + q^{-114} }[/math]
Further Quantum Invariants
K11a99 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["K11a99"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^4+6 t^3-17 t^2+28 t-31+28 t^{-1} -17 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]= { 135, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^7-4 q^6+9 q^5-14 q^4+19 q^3-22 q^2+21 q-18+14 q^{-1} -8 q^{-2} +4 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^4-10 z^4 a^{-2} +3 z^4 a^{-4} +7 z^4-2 a^2 z^2-10 z^2 a^{-2} +3 z^2 a^{-4} +7 z^2-4 a^{-2} +2 a^{-4} +3 }[/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}+5 a z^9+13 z^9 a^{-1} +8 z^9 a^{-3} +4 a^2 z^8+19 z^8 a^{-2} +13 z^8 a^{-4} +10 z^8+a^3 z^7-12 a z^7-27 z^7 a^{-1} -z^7 a^{-3} +13 z^7 a^{-5} -14 a^2 z^6-62 z^6 a^{-2} -18 z^6 a^{-4} +9 z^6 a^{-6} -49 z^6-3 a^3 z^5+a z^5-25 z^5 a^{-3} -17 z^5 a^{-5} +4 z^5 a^{-7} +16 a^2 z^4+55 z^4 a^{-2} +7 z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} +55 z^4+3 a^3 z^3+10 a z^3+15 z^3 a^{-1} +18 z^3 a^{-3} +9 z^3 a^{-5} -z^3 a^{-7} -6 a^2 z^2-23 z^2 a^{-2} -5 z^2 a^{-4} +3 z^2 a^{-6} -21 z^2-a^3 z-3 a z-3 z a^{-1} -3 z a^{-3} -2 z a^{-5} +4 a^{-2} +2 a^{-4} +3 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a277,}

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["K11a99"];
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^3-17 t^2+28 t-31+28 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^7-4 q^6+9 q^5-14 q^4+19 q^3-22 q^2+21 q-18+14 q^{-1} -8 q^{-2} +4 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]= {K11a277,}
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, -2)
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{ -16 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{116}{3} }[/math] [math]\displaystyle{ \frac{76}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{800}{3} }[/math] [math]\displaystyle{ \frac{224}{3} }[/math] [math]\displaystyle{ 80 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ -\frac{928}{3} }[/math] [math]\displaystyle{ -\frac{608}{3} }[/math] [math]\displaystyle{ \frac{1769}{15} }[/math] [math]\displaystyle{ \frac{948}{5} }[/math] [math]\displaystyle{ -\frac{13084}{45} }[/math] [math]\displaystyle{ \frac{1111}{9} }[/math] [math]\displaystyle{ -\frac{1591}{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 K11a99. 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         61 5
9        83  -5
7       116   5
5      118    -3
3     1011     -1
1    912      3
-1   59       -4
-3  39        6
-5 15         -4
-7 3          3
-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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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.

K11a98.gif

K11a98

K11a100.gif

K11a100

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