L9n2

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L9n1.gif

L9n1

L9n3.gif

L9n3

Contents

L9n2.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9n2 at Knotilus!

L9n2 is 9^2_{46} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9n2's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X15,1,16,4 X5,10,6,11 X3849 X11,18,12,5 X17,12,18,13 X9,16,10,17 X2,14,3,13
Gauss code {1, -9, -5, 3}, {-4, -1, 2, 5, -8, 4, -6, 7, 9, -2, -3, 8, -7, 6}
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.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9n2 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 (t(1)-1) (t(2)-1)}{\sqrt{t(1)} \sqrt{t(2)}} (db)
Jones polynomial -\frac{2}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 (-z)-a^7 z^{-1} +a^5 z^3+2 a^5 z+2 a^5 z^{-1} +a^3 z^3+a^3 z-2 a z-a z^{-1} (db)
Kauffman polynomial -z^6 a^8+5 z^4 a^8-7 z^2 a^8+2 a^8-z^7 a^7+4 z^5 a^7-4 z^3 a^7+2 z a^7-a^7 z^{-1} -3 z^6 a^6+12 z^4 a^6-13 z^2 a^6+5 a^6-z^7 a^5+2 z^5 a^5+3 z a^5-2 a^5 z^{-1} -2 z^6 a^4+6 z^4 a^4-6 z^2 a^4+3 a^4-2 z^5 a^3+4 z^3 a^3-2 z a^3-z^4 a^2-a^2-3 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 \
-7-6-5-4-3-2-10χ
0       22
-2      220
-4     1  1
-6    12  1
-8   21   1
-10   1    1
-12 12     -1
-14        0
-161       -1
Integral Khovanov Homology

(db, data source)

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

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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