L9a27

From Knot Atlas
Jump to: navigation, search

L9a26.gif

L9a26

L9a28.gif

L9a28

Contents

L9a27.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9a27 at Knotilus!

L9a27 is 9^2_{12} in the Rolfsen table of links.


Link Presentations

[edit Notes on L9a27's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X18,10,7,9 X2738 X16,13,17,14 X6,12,1,11 X4,16,5,15 X14,6,15,5 X12,17,13,18
Gauss code {1, -4, 2, -7, 8, -6}, {4, -1, 3, -2, 6, -9, 5, -8, 7, -5, 9, -3}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L9a27 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u v-u-2 v+1) (u v-2 u-v+1)}{u v} (db)
Jones polynomial q^{11/2}-3 q^{9/2}+5 q^{7/2}-8 q^{5/2}+8 q^{3/2}-9 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z a^{-5} -2 z^3 a^{-3} +a^3 z-2 z a^{-3} - a^{-3} z^{-1} +z^5 a^{-1} -2 a z^3+2 z^3 a^{-1} -2 a z+3 z a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -z^8 a^{-2} -z^8-3 a z^7-6 z^7 a^{-1} -3 z^7 a^{-3} -3 a^2 z^6-7 z^6 a^{-2} -4 z^6 a^{-4} -6 z^6-a^3 z^5+4 a z^5+7 z^5 a^{-1} -z^5 a^{-3} -3 z^5 a^{-5} +7 a^2 z^4+13 z^4 a^{-2} +4 z^4 a^{-4} -z^4 a^{-6} +15 z^4+2 a^3 z^3+a z^3+z^3 a^{-1} +6 z^3 a^{-3} +4 z^3 a^{-5} -4 a^2 z^2-5 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} -7 z^2-a^3 z-a z-2 z a^{-1} -4 z a^{-3} -2 z a^{-5} - a^{-2} + a^{-1} z^{-1} + 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 \
-4-3-2-1012345χ
12         1-1
10        2 2
8       31 -2
6      52  3
4     44   0
2    54    1
0   35     2
-2  24      -2
-4 13       2
-6 2        -2
-81         1
Integral Khovanov Homology

(db, data source)

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

Back to the top.

L9a26.gif

L9a26

L9a28.gif

L9a28