L9n12

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L9n11.gif

L9n11

L9n13.gif

L9n13

Contents

L9n12.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9n12 at Knotilus!

L9n12 is 9^2_{59} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9n12's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X11,17,12,16 X7,15,8,14 X15,9,16,8 X13,5,14,18 X17,13,18,12 X2536 X4,9,1,10
Gauss code {1, -8, 2, -9}, {8, -1, -4, 5, 9, -2, -3, 7, -6, 4, -5, 3, -7, 6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9n12 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(2)^5-t(1) t(2)^3+t(2)^3+t(1) t(2)^2-t(2)^2-t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial q^{11/2}-q^{9/2}+q^{7/2}-q^{5/2}+q^{3/2}-\sqrt{q}-\frac{1}{\sqrt{q}}-\frac{1}{q^{5/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z a^{-5} +2 z a^{-3} +2 a^{-3} z^{-1} -z^5 a^{-1} +a z^3-6 z^3 a^{-1} +4 a z-9 z a^{-1} +3 a z^{-1} -5 a^{-1} z^{-1} (db)
Kauffman polynomial z^4 a^{-6} -3 z^2 a^{-6} + a^{-6} +z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} +z^4 a^{-4} -2 z^2 a^{-4} +z^3 a^{-3} -3 z a^{-3} +2 a^{-3} z^{-1} +z^6 a^{-2} -7 z^4 a^{-2} +13 z^2 a^{-2} -5 a^{-2} +a z^7+z^7 a^{-1} -7 a z^5-8 z^5 a^{-1} +15 a z^3+19 z^3 a^{-1} -12 a z-16 z a^{-1} +3 a z^{-1} +5 a^{-1} z^{-1} +z^6-7 z^4+12 z^2-5 (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          0
8       11 0
6     11   0
4     11   0
2   121    0
0    2     2
-2  1       1
-41         1
-61         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2 i=4
r=-4 {\mathbb Z} {\mathbb Z}
r=-3
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2} {\mathbb Z}^{2}
r=1 {\mathbb Z} {\mathbb Z} {\mathbb Z}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}
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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L9n11.gif

L9n11

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L9n13