L9a3

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L9a2.gif

L9a2

L9a4.gif

L9a4

Contents

L9a3.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9a3 at Knotilus!

L9a3 is 9^2_{33} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9a3's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X8493 X14,10,15,9 X10,14,11,13 X18,12,5,11 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.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9a3 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2-3 t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -6 q^{9/2}+8 q^{7/2}-10 q^{5/2}+\frac{1}{q^{5/2}}+9 q^{3/2}-\frac{3}{q^{3/2}}-q^{13/2}+4 q^{11/2}-9 \sqrt{q}+\frac{5}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-1} +z^5 a^{-3} -a z^3+2 z^3 a^{-1} +z^3 a^{-3} -z^3 a^{-5} -a z+3 z a^{-1} -2 z a^{-3} +2 a^{-1} z^{-1} -3 a^{-3} z^{-1} + a^{-5} z^{-1} (db)
Kauffman polynomial -2 z^8 a^{-2} -2 z^8 a^{-4} -4 z^7 a^{-1} -9 z^7 a^{-3} -5 z^7 a^{-5} -4 z^6 a^{-2} -4 z^6 a^{-4} -4 z^6 a^{-6} -4 z^6-3 a z^5+2 z^5 a^{-1} +16 z^5 a^{-3} +10 z^5 a^{-5} -z^5 a^{-7} -a^2 z^4+9 z^4 a^{-2} +13 z^4 a^{-4} +8 z^4 a^{-6} +3 z^4+4 a z^3+3 z^3 a^{-1} -6 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} +a^2 z^2-2 z^2 a^{-2} -3 z^2 a^{-4} -2 z^2 a^{-6} -2 a z-5 z a^{-1} -3 z a^{-3} -3 a^{-2} -3 a^{-4} - a^{-6} +2 a^{-1} z^{-1} +3 a^{-3} z^{-1} + a^{-5} 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 \
-3-2-10123456χ
14         11
12        3 -3
10       31 2
8      53  -2
6     53   2
4    45    1
2   55     0
0  26      4
-2 13       -2
-4 2        2
-61         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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L9a2.gif

L9a2

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L9a4