L9a5

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L9a4.gif

L9a4

L9a6.gif

L9a6

Contents

L9a5.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9a5 at Knotilus!

L9a5 is 9^2_{30} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9a5's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X14,8,15,7 X18,16,5,15 X16,12,17,11 X12,18,13,17 X8,14,9,13 X2536 X4,9,1,10
Gauss code {1, -8, 2, -9}, {8, -1, 3, -7, 9, -2, 5, -6, 7, -3, 4, -5, 6, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9a5 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^3-3 t(2)^3-4 t(1) t(2)^2+5 t(2)^2+5 t(1) t(2)-4 t(2)-3 t(1)+1}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -3 q^{9/2}+6 q^{7/2}-\frac{1}{q^{7/2}}-8 q^{5/2}+\frac{2}{q^{5/2}}+9 q^{3/2}-\frac{6}{q^{3/2}}+q^{11/2}-9 \sqrt{q}+\frac{7}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-1} -2 a z^3+z^3 a^{-1} -2 z^3 a^{-3} +a^3 z-2 a z-z a^{-1} -z a^{-3} +z a^{-5} +a^3 z^{-1} -2 a^{-1} z^{-1} + a^{-3} z^{-1} (db)
Kauffman polynomial -z^8 a^{-2} -z^8-3 a z^7-7 z^7 a^{-1} -4 z^7 a^{-3} -2 a^2 z^6-10 z^6 a^{-2} -5 z^6 a^{-4} -7 z^6-a^3 z^5+5 a z^5+11 z^5 a^{-1} +2 z^5 a^{-3} -3 z^5 a^{-5} +3 a^2 z^4+24 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +19 z^4+3 a^3 z^3-3 a z^3-7 z^3 a^{-1} +2 z^3 a^{-3} +3 z^3 a^{-5} -19 z^2 a^{-2} -5 z^2 a^{-4} +z^2 a^{-6} -13 z^2-3 a^3 z+2 a z+6 z a^{-1} -z a^{-5} -a^2+5 a^{-2} +2 a^{-4} +3+a^3 z^{-1} -2 a^{-1} z^{-1} - 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 \
-4-3-2-1012345χ
12         1-1
10        2 2
8       41 -3
6      42  2
4     54   -1
2    44    0
0   46     2
-2  23      -1
-4  4       4
-612        -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} {\mathbb Z}
r=-3 {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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L9a4

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