L9n22

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L9n21.gif

L9n21

L9n23.gif

L9n23

Contents

L9n22.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9n22 at Knotilus!

L9n22 is 9^3_{15} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9n22's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,17,12,16 X15,9,16,18 X17,13,18,12 X2536 X4,9,1,10
Gauss code {1, -8, 2, -9}, {8, -1, 3, -4}, {9, -2, -5, 7, 4, -3, -6, 5, -7, 6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9n22 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v+u w^3-u w^2+u w-u+v w^3-v w^2+v w-v-w^3}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial -2 q+3-3 q^{-1} +4 q^{-2} -2 q^{-3} +4 q^{-4} - q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial 2 a^6 z^{-2} +a^6-3 z^2 a^4-5 a^4 z^{-2} -8 a^4+2 z^4 a^2+8 z^2 a^2+4 a^2 z^{-2} +10 a^2-2 z^2- z^{-2} -3 (db)
Kauffman polynomial a^6 z^6-5 a^6 z^4+9 a^6 z^2+2 a^6 z^{-2} -7 a^6+a^5 z^7-2 a^5 z^5-5 a^5 z^3+11 a^5 z-5 a^5 z^{-1} +5 a^4 z^6-20 a^4 z^4+23 a^4 z^2+5 a^4 z^{-2} -14 a^4+a^3 z^7+2 a^3 z^5-17 a^3 z^3+21 a^3 z-9 a^3 z^{-1} +4 a^2 z^6-14 a^2 z^4+16 a^2 z^2+4 a^2 z^{-2} -10 a^2+4 a z^5-12 a z^3+13 a z+3 z a^{-1} -5 a z^{-1} - a^{-1} z^{-1} +z^4+2 z^2+ z^{-2} -2 (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 \
-6-5-4-3-2-101χ
3       2-2
1      1 1
-1     33 0
-3    111 1
-5   13   2
-7  31    2
-9 14     3
-11        0
-131       1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4} {\mathbb Z}^{3}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z} {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=1 {\mathbb Z}_2^{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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L9n21.gif

L9n21

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L9n23