L9n5

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L9n4

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L9n6

Contents

L9n5.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9n5 at Knotilus!

L9n5 is 9^2_{44} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9n5's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X15,1,16,4 X5,12,6,13 X3849 X9,16,10,17 X11,18,12,5 X17,10,18,11 X2,14,3,13
Gauss code {1, -9, -5, 3}, {-4, -1, 2, 5, -6, 8, -7, 4, 9, -2, -3, 6, -8, 7}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif
A Morse Link Presentation L9n5 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^2+1\right)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial \frac{3}{q^{9/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{5/2}}-\frac{2}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{11/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -z^3 a^7-3 z a^7-2 a^7 z^{-1} +z^5 a^5+5 z^3 a^5+9 z a^5+5 a^5 z^{-1} -2 z^3 a^3-6 z a^3-3 a^3 z^{-1} (db)
Kauffman polynomial a^{10} z^4-3 a^{10} z^2+a^{10}+a^9 z^5-2 a^9 z^3+a^8 z^6-2 a^8 z^4+a^8 z^2+a^7 z^7-4 a^7 z^5+8 a^7 z^3-5 a^7 z+2 a^7 z^{-1} +2 a^6 z^6-7 a^6 z^4+12 a^6 z^2-5 a^6+a^5 z^7-5 a^5 z^5+13 a^5 z^3-13 a^5 z+5 a^5 z^{-1} +a^4 z^6-4 a^4 z^4+8 a^4 z^2-5 a^4+3 a^3 z^3-8 a^3 z+3 a^3 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
-2       22
-4      121
-6     2  2
-8    11  0
-10   22   0
-12   1    1
-14 12     -1
-16        0
-181       -1
Integral Khovanov Homology

(db, data source)

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