L9n17

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L9n16.gif

L9n16

L9n18.gif

L9n18

Contents

L9n17.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9n17 at Knotilus!

L9n17 is 9^2_{52} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9n17's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X7,14,8,15 X13,18,14,7 X17,1,18,6 X16,11,17,12 X5,12,6,13 X4,16,5,15
Gauss code {1, -2, 3, -9, -8, 6}, {-4, -1, 2, -3, 7, 8, -5, 4, 9, -7, -6, 5}
A Braid Representative
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif
A Morse Link Presentation L9n17 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^2 t(1)^2-t(2) t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-2 t(1)-t(2)+1}{t(1) t(2)} (db)
Jones polynomial \frac{4}{q^{9/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{5/2}}-\frac{2}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{2}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{4}{q^{11/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -z^3 a^7-2 z a^7-a^7 z^{-1} +z^5 a^5+4 z^3 a^5+6 z a^5+3 a^5 z^{-1} -2 z^3 a^3-5 z a^3-2 a^3 z^{-1} (db)
Kauffman polynomial -z^4 a^{10}+2 z^2 a^{10}-2 z^5 a^9+4 z^3 a^9-z a^9-2 z^6 a^8+4 z^4 a^8-3 z^2 a^8+a^8-z^7 a^7+z^5 a^7-2 z^3 a^7+2 z a^7-a^7 z^{-1} -3 z^6 a^6+7 z^4 a^6-9 z^2 a^6+3 a^6-z^7 a^5+3 z^5 a^5-9 z^3 a^5+9 z a^5-3 a^5 z^{-1} -z^6 a^4+2 z^4 a^4-4 z^2 a^4+3 a^4-3 z^3 a^3+6 z a^3-2 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 \
-7-6-5-4-3-2-10χ
-2       22
-4      110
-6     31 2
-8    22  0
-10   22   0
-12  12    1
-14 12     -1
-16 1      1
-181       -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}\oplus{\mathbb Z}_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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L9n16.gif

L9n16

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L9n18