L9a8

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L9a7.gif

L9a7

L9a9.gif

L9a9

Contents

L9a8.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9a8 at Knotilus!

L9a8 is 9^2_{25} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9a8's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X16,8,17,7 X18,13,5,14 X14,17,15,18 X12,10,13,9 X8,16,9,15 X2536 X4,12,1,11
Gauss code {1, -8, 2, -9}, {8, -1, 3, -7, 6, -2, 9, -6, 4, -5, 7, -3, 5, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L9a8 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^2-4 t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial q^{11/2}-3 q^{9/2}+5 q^{7/2}-8 q^{5/2}+8 q^{3/2}-8 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-1} -2 a z^3+2 z^3 a^{-1} -2 z^3 a^{-3} +a^3 z-3 a z+3 z a^{-1} -2 z a^{-3} +z a^{-5} +a^3 z^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} - a^{-3} z^{-1} (db)
Kauffman polynomial z^4 a^{-6} -z^2 a^{-6} +3 z^5 a^{-5} -4 z^3 a^{-5} +2 z a^{-5} +4 z^6 a^{-4} -4 z^4 a^{-4} +z^2 a^{-4} +3 z^7 a^{-3} +a^3 z^5+z^5 a^{-3} -3 a^3 z^3-7 z^3 a^{-3} +3 a^3 z+5 z a^{-3} -a^3 z^{-1} - a^{-3} z^{-1} +z^8 a^{-2} +2 a^2 z^6+6 z^6 a^{-2} -4 a^2 z^4-10 z^4 a^{-2} +2 a^2 z^2+3 z^2 a^{-2} +2 a z^7+5 z^7 a^{-1} -3 z^5 a^{-1} -8 a z^3-8 z^3 a^{-1} +8 a z+8 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +z^8+4 z^6-9 z^4+3 z^2-1 (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 \
-4-3-2-1012345χ
12         1-1
10        2 2
8       31 -2
6      52  3
4     33   0
2    55    0
0   45     1
-2  13      -2
-4 14       3
-6 1        -1
-81         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\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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L9a7

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L9a9