L11n260

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L11n259.gif

L11n259

L11n261.gif

L11n261

Contents

L11n260.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n260's Link Presentations]

Planar diagram presentation X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,12,19,11 X19,22,20,9 X15,20,16,21 X21,16,22,17 X12,18,13,17 X2536 X9,1,10,4
Gauss code {1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -9, -4, 3, -7, 8, 9, -5, -6, 7, -8, 6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation L11n260 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{-t(1) t(3)^3+t(1) t(2) t(3)^3-t(2) t(3)^3+t(3)^3+2 t(1) t(3)^2-2 t(1) t(2) t(3)^2+2 t(2) t(3)^2-3 t(3)^2-2 t(1) t(3)+3 t(1) t(2) t(3)-2 t(2) t(3)+2 t(3)+t(1)-t(1) t(2)+t(2)-1}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial 2 q^4-4 q^3+8 q^2-8 q+10-8 q^{-1} +7 q^{-2} -4 q^{-3} + q^{-4} (db)
Signature 0 (db)
HOMFLY-PT polynomial  a^{-4} z^{-2} + a^{-4} +a^2 z^4+z^4 a^{-2} +a^2 z^2-2 a^{-2} z^{-2} -3 a^{-2} -z^6-3 z^4-2 z^2+ z^{-2} +2 (db)
Kauffman polynomial a^4 z^4+3 z^4 a^{-4} -5 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +z^7 a^{-3} +4 a^3 z^5+z^5 a^{-3} -3 a^3 z^3-3 z a^{-3} +2 a^{-3} z^{-1} +z^8 a^{-2} +7 a^2 z^6+3 z^6 a^{-2} -10 a^2 z^4-4 z^4 a^{-2} +3 a^2 z^2-3 z^2 a^{-2} -2 a^{-2} z^{-2} +5 a^{-2} +5 a z^7+6 z^7 a^{-1} -3 a z^5-6 z^5 a^{-1} -3 a z^3-3 z a^{-1} +2 a^{-1} z^{-1} +z^8+10 z^6-18 z^4+5 z^2- z^{-2} +3 (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
  \  
j \
-4-3-2-101234χ
9        22
7       31-2
5      51 4
3     33  0
1    75   2
-1   46    2
-3  34     -1
-5 14      3
-7 3       -3
-91        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

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See/edit the Link Page master template (intermediate).

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L11n259.gif

L11n259

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L11n261