L9n1

From Knot Atlas
Jump to: navigation, search

L9a55.gif

L9a55

L9n2.gif

L9n2

Contents

L9n1.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9n1 at Knotilus!

L9n1 is 9^2_{45} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9n1's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X11,18,12,5 X17,12,18,13 X9,16,10,17 X13,2,14,3
Gauss code {1, 9, -5, -3}, {-4, -1, 2, 5, -8, 4, -6, 7, -9, -2, 3, 8, -7, 6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9n1 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1) t(2)^3-t(1) t(2)^2-t(2)+2}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}+\frac{1}{q^{19/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -a^{11} z^{-1} +z^3 a^9+4 z a^9+2 a^9 z^{-1} -z^5 a^7-4 z^3 a^7-3 z a^7-z^5 a^5-4 z^3 a^5-3 z a^5-a^5 z^{-1} (db)
Kauffman polynomial a^{12} z^2-2 a^{12}+a^{11} z^3-a^{11} z+a^{11} z^{-1} +a^{10} z^6-5 a^{10} z^4+10 a^{10} z^2-5 a^{10}+a^9 z^7-5 a^9 z^5+9 a^9 z^3-5 a^9 z+2 a^9 z^{-1} +2 a^8 z^6-8 a^8 z^4+9 a^8 z^2-3 a^8+a^7 z^7-4 a^7 z^5+4 a^7 z^3-a^7 z+a^6 z^6-3 a^6 z^4+a^6+a^5 z^5-4 a^5 z^3+3 a^5 z-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 \
-7-6-5-4-3-2-10χ
-4       11
-6      110
-8     1  1
-10   111  1
-12   31   2
-14   1    1
-16 12     -1
-18        0
-201       -1
Integral Khovanov Homology

(db, data source)

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

Back to the top.

L9a55.gif

L9a55

L9n2.gif

L9n2