L11a274

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L11a273.gif

L11a273

L11a275.gif

L11a275

L11a274.gif
(Knotscape image) 		See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a274's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X6,9,7,10 X18,8,19,7 X20,16,21,15 X22,18,9,17 X16,22,17,21 X8,20,1,19 X4,13,5,14
Gauss code {1, -2, 3, -11, 4, -5, 6, -10}, {5, -1, 2, -3, 11, -4, 7, -9, 8, -6, 10, -7, 9, -8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L11a274 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)^3 t(2)^5-t(1)^2 t(2)^5-t(1)^3 t(2)^4+3 t(1)^2 t(2)^4-t(1) t(2)^4+t(1)^3 t(2)^3-3 t(1)^2 t(2)^3+3 t(1) t(2)^3-t(2)^3-t(1)^3 t(2)^2+3 t(1)^2 t(2)^2-3 t(1) t(2)^2+t(2)^2-t(1)^2 t(2)+3 t(1) t(2)-t(2)-t(1)+1}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial [math]\displaystyle{ q^{9/2}-2 q^{7/2}+4 q^{5/2}-6 q^{3/2}+7 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature -1 (db)
HOMFLY-PT polynomial [math]\displaystyle{ -a z^9+a^3 z^7-8 a z^7+z^7 a^{-1} +6 a^3 z^5-24 a z^5+6 z^5 a^{-1} +12 a^3 z^3-33 a z^3+12 z^3 a^{-1} +9 a^3 z-20 a z+9 z a^{-1} +2 a^3 z^{-1} -3 a z^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial [math]\displaystyle{ -a^2 z^{10}-z^{10}-2 a^3 z^9-5 a z^9-3 z^9 a^{-1} -2 a^4 z^8+a^2 z^8-3 z^8 a^{-2} -2 a^5 z^7+7 a^3 z^7+25 a z^7+14 z^7 a^{-1} -2 z^7 a^{-3} -2 a^6 z^6+4 a^4 z^6+6 a^2 z^6+12 z^6 a^{-2} -z^6 a^{-4} +13 z^6-a^7 z^5+3 a^5 z^5-15 a^3 z^5-52 a z^5-26 z^5 a^{-1} +7 z^5 a^{-3} +5 a^6 z^4-4 a^4 z^4-20 a^2 z^4-14 z^4 a^{-2} +4 z^4 a^{-4} -29 z^4+3 a^7 z^3+a^5 z^3+16 a^3 z^3+48 a z^3+26 z^3 a^{-1} -4 z^3 a^{-3} -2 a^6 z^2+3 a^4 z^2+14 a^2 z^2+8 z^2 a^{-2} -3 z^2 a^{-4} +20 z^2-2 a^7 z-10 a^3 z-23 a z-11 z a^{-1} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} z^{-1} }[/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 \
 -6-5-4-3-2-1012345χ
10           1-1
8          1 1
6         31 -2
4        31  2
2       43   -1
0      63    3
-2     45     1
-4    45      -1
-6   24       2
-8  24        -2
-10 13         2
-12 1          -1
-141           1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}^{2} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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 Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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L11a273

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L11a275

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