L11a420

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L11a419.gif

L11a419

L11a421.gif

L11a421

L11a420.gif
(Knotscape image) 		See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a420's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X20,11,21,12 X22,15,11,16 X14,21,15,22 X8,18,9,17 X16,8,17,7 X18,10,19,9 X10,20,5,19 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {10, -1, 7, -6, 8, -9}, {3, -2, 11, -5, 4, -7, 6, -8, 9, -3, 5, -4}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L11a420 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...) [math]\displaystyle{ -\frac{t(1) t(2)^2 t(3)^4-t(2)^2 t(3)^4-t(1) t(2) t(3)^4-2 t(1) t(2)^2 t(3)^3+3 t(2)^2 t(3)^3-t(1) t(3)^3+3 t(1) t(2) t(3)^3-2 t(2) t(3)^3+2 t(1) t(2)^2 t(3)^2-3 t(2)^2 t(3)^2+3 t(1) t(3)^2-4 t(1) t(2) t(3)^2+4 t(2) t(3)^2-2 t(3)^2+t(2)^2 t(3)-3 t(1) t(3)+2 t(1) t(2) t(3)-3 t(2) t(3)+2 t(3)+t(1)+t(2)-1}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial [math]\displaystyle{ q^7-3 q^6+7 q^5-10 q^4- q^{-4} +14 q^3+2 q^{-3} -14 q^2-5 q^{-2} +15 q+9 q^{-1} -11 }[/math] (db)
Signature 2 (db)
HOMFLY-PT polynomial [math]\displaystyle{ -z^8 a^{-2} -6 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-15 z^4 a^{-2} +4 z^4 a^{-4} +10 z^4-4 a^2 z^2-21 z^2 a^{-2} +6 z^2 a^{-4} +18 z^2-4 a^2-16 a^{-2} +5 a^{-4} +15-a^2 z^{-2} -5 a^{-2} z^{-2} +2 a^{-4} z^{-2} +4 z^{-2} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ z^{10} a^{-2} +z^{10}+2 a z^9+6 z^9 a^{-1} +4 z^9 a^{-3} +2 a^2 z^8+9 z^8 a^{-2} +7 z^8 a^{-4} +4 z^8+a^3 z^7-3 a z^7-12 z^7 a^{-1} -z^7 a^{-3} +7 z^7 a^{-5} -8 a^2 z^6-35 z^6 a^{-2} -12 z^6 a^{-4} +6 z^6 a^{-6} -25 z^6-5 a^3 z^5-12 a z^5-11 z^5 a^{-1} -15 z^5 a^{-3} -8 z^5 a^{-5} +3 z^5 a^{-7} +10 a^2 z^4+47 z^4 a^{-2} +11 z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} +37 z^4+8 a^3 z^3+28 a z^3+39 z^3 a^{-1} +23 z^3 a^{-3} +2 z^3 a^{-5} -2 z^3 a^{-7} -7 a^2 z^2-39 z^2 a^{-2} -9 z^2 a^{-4} +6 z^2 a^{-6} -z^2 a^{-8} -30 z^2-5 a^3 z-21 a z-33 z a^{-1} -16 z a^{-3} +z a^{-5} +4 a^2+20 a^{-2} +6 a^{-4} -2 a^{-6} +17+a^3 z^{-1} +5 a z^{-1} +9 a^{-1} z^{-1} +5 a^{-3} z^{-1} -a^2 z^{-2} -5 a^{-2} z^{-2} -2 a^{-4} z^{-2} -4 z^{-2} }[/math] (db)

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]).   
\ r
  \  
j \
 -5-4-3-2-10123456χ
15           11
13          31-2
11         4  4
9        63  -3
7       84   4
5      77    0
3     87     1
1    59      4
-1   46       -2
-3  15        4
-5 14         -3
-7 1          1
-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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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 Link Page master template (intermediate).

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L11a419.gif

L11a419

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L11a421

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