L9n8

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L9n7.gif

L9n7

L9n9.gif

L9n9

Contents

L9n8.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9n8 at Knotilus!

L9n8 is 9^2_{56} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9n8's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X8493 X9,14,10,15 X13,10,14,11 X11,18,12,5 X2,16,3,15
Gauss code {1, -9, 5, -3}, {-4, -1, 2, -5, -6, 7, -8, 4, -7, 6, 9, -2, 3, 8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9n8 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(v^2-v+1\right)}{\sqrt{u} v^{3/2}} (db)
Jones polynomial \frac{4}{q^{9/2}}-\frac{5}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{4}{q^{3/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{11/2}}-\sqrt{q}+\frac{2}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^7 z^{-1} -z^3 a^5+2 a^5 z^{-1} +z^5 a^3+3 z^3 a^3+2 z a^3-z^3 a-2 z a-a z^{-1} (db)
Kauffman polynomial -3 z^2 a^8+2 a^8-z^5 a^7-z^3 a^7-a^7 z^{-1} -2 z^6 a^6+4 z^4 a^6-8 z^2 a^6+5 a^6-z^7 a^5-z^5 a^5+4 z^3 a^5-z a^5-2 a^5 z^{-1} -4 z^6 a^4+9 z^4 a^4-6 z^2 a^4+3 a^4-z^7 a^3-z^5 a^3+8 z^3 a^3-4 z a^3-2 z^6 a^2+5 z^4 a^2-z^2 a^2-a^2-z^5 a+3 z^3 a-3 z a+a z^{-1} (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     31 2
-4    23  1
-6   31   2
-8  12    1
-10 23     -1
-12 1      1
-142       -2
Integral Khovanov Homology

(db, data source)

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