L9n11

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L9n10.gif

L9n10

L9n12.gif

L9n12

Contents

L9n11.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9n11 at Knotilus!

L9n11 is 9^2_{57} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9n11's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X14,8,15,7 X15,18,16,5 X11,16,12,17 X17,12,18,13 X8,14,9,13 X2536 X4,9,1,10
Gauss code {1, -8, 2, -9}, {8, -1, 3, -7, 9, -2, -5, 6, 7, -3, -4, 5, -6, 4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9n11 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^3-t(2)^3-2 t(1) t(2)^2+t(2)^2+t(1) t(2)-2 t(2)-t(1)+1}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{3}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{1}{q^{13/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^5 \left(-z^3\right)-a^5 z+a^5 z^{-1} +a^3 z^5+3 a^3 z^3+a^3 z-a^3 z^{-1} -a z^3-2 a z (db)
Kauffman polynomial -z^2 a^8-3 z^3 a^7+z a^7-z^6 a^6+z^4 a^6-2 z^2 a^6-z^7 a^5+2 z^5 a^5-4 z^3 a^5+4 z a^5-a^5 z^{-1} -3 z^6 a^4+7 z^4 a^4-4 z^2 a^4+a^4-z^7 a^3+z^5 a^3+2 z^3 a^3+z a^3-a^3 z^{-1} -2 z^6 a^2+6 z^4 a^2-3 z^2 a^2-z^5 a+3 z^3 a-2 z a (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 \
-5-4-3-2-1012χ
2       11
0      1 -1
-2     21 1
-4    22  0
-6   21   1
-8  12    1
-10 22     0
-12 2      2
-141       -1
Integral Khovanov Homology

(db, data source)

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

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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L9n10

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L9n12