L9n19

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L9n18.gif

L9n18

L9n20.gif

L9n20

Contents

L9n19.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9n19 at Knotilus!

L9n19 is 9^2_{61} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9n19's Link Presentations]

Planar diagram presentation X10,1,11,2 X3,12,4,13 X18,5,9,6 X6,9,7,10 X16,12,17,11 X7,14,8,15 X13,4,14,5 X15,8,16,1 X2,17,3,18
Gauss code {1, -9, -2, 7, 3, -4, -6, 8}, {4, -1, 5, 2, -7, 6, -8, -5, 9, -3}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gif
BraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L9n19 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{\left(t(1)^2+t(2)\right) \left(t(1) t(2)^2+1\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -\frac{1}{q^{5/2}}-\frac{1}{q^{19/2}} (db)
Signature -4 (db)
HOMFLY-PT polynomial z a^9+a^9 z^{-1} -a^7 z^{-1} -z^5 a^5-5 z^3 a^5-5 z a^5 (db)
Kauffman polynomial a^{11} z^5-5 a^{11} z^3+5 a^{11} z-a^9 z+a^9 z^{-1} -a^8-a^7 z+a^7 z^{-1} +a^5 z^5-5 a^5 z^3+5 a^5 z (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 \
-8-7-6-5-4-3-2-10χ
-4        11
-6        11
-8     11  0
-10         0
-12   121   0
-14         0
-16  11     0
-181        1
-201        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4 i=-2
r=-8 {\mathbb Z} {\mathbb Z}
r=-7
r=-6 {\mathbb Z}
r=-5 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}_2 {\mathbb Z}^{2}
r=-3 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}_2 {\mathbb Z}
r=-1
r=0 {\mathbb Z} {\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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L9n18

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