L9a19

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L9a18.gif

L9a18

L9a20.gif

L9a20

Contents

L9a19.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9a19 at Knotilus!

L9a19 is 9^2_{38} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9a19's Link Presentations]

Planar diagram presentation X6172 X2,9,3,10 X12,3,13,4 X10,5,11,6 X18,11,5,12 X4,17,1,18 X16,14,17,13 X14,8,15,7 X8,16,9,15
Gauss code {1, -2, 3, -6}, {4, -1, 8, -9, 2, -4, 5, -3, 7, -8, 9, -7, 6, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9a19 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u v^3-5 u v^2+6 u v-2 u-2 v^3+6 v^2-5 v+2}{\sqrt{u} v^{3/2}} (db)
Jones polynomial q^{5/2}-4 q^{3/2}+7 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^3+a^5 z+a^5 z^{-1} -a^3 z^5-2 a^3 z^3-3 a^3 z-a^3 z^{-1} -a z^5-a z^3+z^3 a^{-1} (db)
Kauffman polynomial a^7 z^5-2 a^7 z^3+a^7 z+3 a^6 z^6-6 a^6 z^4+3 a^6 z^2+4 a^5 z^7-7 a^5 z^5+4 a^5 z^3-2 a^5 z+a^5 z^{-1} +2 a^4 z^8+4 a^4 z^6-13 a^4 z^4+7 a^4 z^2-a^4+10 a^3 z^7-18 a^3 z^5+10 a^3 z^3-3 a^3 z+a^3 z^{-1} +2 a^2 z^8+8 a^2 z^6-17 a^2 z^4+z^4 a^{-2} +6 a^2 z^2+6 a z^7-6 a z^5+4 z^5 a^{-1} +a z^3-3 z^3 a^{-1} +7 z^6-9 z^4+2 z^2 (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 \
-6-5-4-3-2-10123χ
6         1-1
4        3 3
2       41 -3
0      53  2
-2     65   -1
-4    54    1
-6   36     3
-8  35      -2
-10 14       3
-12 2        -2
-141         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\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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L9a18.gif

L9a18

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L9a20