L9n23

From Knot Atlas
Jump to: navigation, search

L9n22.gif

L9n22

L9n24.gif

L9n24

Contents

L9n23.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9n23 at Knotilus!

L9n23 is 9^3_{13} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9n23's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X11,16,12,17 X9,11,10,18 X17,5,18,10 X4,15,1,16
Gauss code {1, 4, -3, -9}, {-2, -1, 5, 3, -7, 8}, {-6, 2, -4, -5, 9, 6, -8, 7}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation L9n23 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(1) t(3)^2 t(2)^2+t(1) t(3) t(2)^2+t(1) t(3)^2 t(2)-t(2)-t(3)+1}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial 1- q^{-1} +2 q^{-2} - q^{-3} +3 q^{-4} - q^{-5} +2 q^{-6} - q^{-7} (db)
Signature -4 (db)
HOMFLY-PT polynomial -a^8+z^4 a^6+4 z^2 a^6+a^6 z^{-2} +3 a^6-z^6 a^4-5 z^4 a^4-7 z^2 a^4-2 a^4 z^{-2} -5 a^4+z^4 a^2+4 z^2 a^2+a^2 z^{-2} +3 a^2 (db)
Kauffman polynomial z a^9+2 z^2 a^8-2 a^8+z^5 a^7-3 z^3 a^7+3 z a^7+2 z^6 a^6-9 z^4 a^6+13 z^2 a^6+a^6 z^{-2} -8 a^6+z^7 a^5-3 z^5 a^5-z^3 a^5+5 z a^5-2 a^5 z^{-1} +3 z^6 a^4-14 z^4 a^4+18 z^2 a^4+2 a^4 z^{-2} -9 a^4+z^7 a^3-4 z^5 a^3+2 z^3 a^3+3 z a^3-2 a^3 z^{-1} +z^6 a^2-5 z^4 a^2+7 z^2 a^2+a^2 z^{-2} -4 a^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 \
-5-4-3-2-1012χ
1       11
-1        0
-3     21 1
-5   111  1
-7   31   2
-9 112    2
-11 21     1
-13 1      1
-151       -1
Integral Khovanov Homology

(db, data source)

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

Back to the top.

L9n22.gif

L9n22

L9n24.gif

L9n24