L9a28

L9a27

L9a29

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a28 at Knotilus!
	L9a28 is $9_{20}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a28's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,16,7,15 X16,12,17,11 X12,18,13,17 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -9, 4, -8}, {8, -1, 2, -3, 6, -7, 9, -4, 5, -6, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{4}-2t(1)^{2}t(2)^{3}+2t(1)t(2)^{3}+2t(1)^{2}t(2)^{2}-3t(1)t(2)^{2}+2t(2)^{2}+2t(1)t(2)-2t(2)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {5}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {5}{q^{5/2}}}+q^{3/2}-{\frac {5}{q^{3/2}}}-{\frac {1}{q^{15/2}}}+{\frac {2}{q^{13/2}}}-{\frac {4}{q^{11/2}}}-2{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{5}+4a^{5}z^{3}+5a^{5}z+2a^{5}z^{-1}-a^{3}z^{7}-6a^{3}z^{5}-13a^{3}z^{3}-12a^{3}z-3a^{3}z^{-1}+az^{5}+4az^{3}+4az+az^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{3}-a^{9}z+2a^{8}z^{4}-a^{8}z^{2}+3a^{7}z^{5}-3a^{7}z^{3}+2a^{7}z+3a^{6}z^{6}-3a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{7}-7a^{5}z^{5}+9a^{5}z^{3}-7a^{5}z+2a^{5}z^{-1}+a^{4}z^{8}+2a^{4}z^{6}-11a^{4}z^{4}+10a^{4}z^{2}-3a^{4}+5a^{3}z^{7}-18a^{3}z^{5}+22a^{3}z^{3}-14a^{3}z+3a^{3}z^{-1}+a^{2}z^{8}-10a^{2}z^{4}+12a^{2}z^{2}-3a^{2}+2az^{7}-8az^{5}+9az^{3}-4az+az^{-1}+z^{6}-4z^{4}+4z^{2}-1$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                   1 -1 2                 1   1 0               2 1   -1 -2             3 1     2 -4           3 3       0 -6         3 2         1 -8       2 3           1 -10     2 3             -1 -12     2               2 -14 1 2                 -1 -16 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L9a27 L9a29