L9a27

L9a26

L9a28

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a27 at Knotilus!
	L9a27 is $9_{12}^{2}$ 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
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(uv-u-2v+1)(uv-2u-v+1)}{uv}}$ (db)
Jones polynomial	$q^{11/2}-3q^{9/2}+5q^{7/2}-8q^{5/2}+8q^{3/2}-9{\sqrt {q}}+{\frac {7}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-5}-2z^{3}a^{-3}+a^{3}z-2za^{-3}-a^{-3}z^{-1}+z^{5}a^{-1}-2az^{3}+2z^{3}a^{-1}-2az+3za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-2}-z^{8}-3az^{7}-6z^{7}a^{-1}-3z^{7}a^{-3}-3a^{2}z^{6}-7z^{6}a^{-2}-4z^{6}a^{-4}-6z^{6}-a^{3}z^{5}+4az^{5}+7z^{5}a^{-1}-z^{5}a^{-3}-3z^{5}a^{-5}+7a^{2}z^{4}+13z^{4}a^{-2}+4z^{4}a^{-4}-z^{4}a^{-6}+15z^{4}+2a^{3}z^{3}+az^{3}+z^{3}a^{-1}+6z^{3}a^{-3}+4z^{3}a^{-5}-4a^{2}z^{2}-5z^{2}a^{-2}-z^{2}a^{-4}+z^{2}a^{-6}-7z^{2}-a^{3}z-az-2za^{-1}-4za^{-3}-2za^{-5}-a^{-2}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 12                   1 -1 10                 2   2 8               3 1   -2 6             5 2     3 4           4 4       0 2         5 4         1 0       3 5           2 -2     2 4             -2 -4   1 3               2 -6   2                 -2 -8 1                   1

Integral Khovanov Homology (db, data source)

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

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.

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