L9a5

L9a4

L9a6

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a5 at Knotilus!
	L9a5 is $9_{30}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a5's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X18,16,5,15 X16,12,17,11 X12,18,13,17 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -7, 9, -2, 5, -6, 7, -3, 4, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{3}-3t(2)^{3}-4t(1)t(2)^{2}+5t(2)^{2}+5t(1)t(2)-4t(2)-3t(1)+1}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+6q^{7/2}-{\frac {1}{q^{7/2}}}-8q^{5/2}+{\frac {2}{q^{5/2}}}+9q^{3/2}-{\frac {6}{q^{3/2}}}+q^{11/2}-9{\sqrt {q}}+{\frac {7}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}-2az^{3}+z^{3}a^{-1}-2z^{3}a^{-3}+a^{3}z-2az-za^{-1}-za^{-3}+za^{-5}+a^{3}z^{-1}-2a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-2}-z^{8}-3az^{7}-7z^{7}a^{-1}-4z^{7}a^{-3}-2a^{2}z^{6}-10z^{6}a^{-2}-5z^{6}a^{-4}-7z^{6}-a^{3}z^{5}+5az^{5}+11z^{5}a^{-1}+2z^{5}a^{-3}-3z^{5}a^{-5}+3a^{2}z^{4}+24z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}+19z^{4}+3a^{3}z^{3}-3az^{3}-7z^{3}a^{-1}+2z^{3}a^{-3}+3z^{3}a^{-5}-19z^{2}a^{-2}-5z^{2}a^{-4}+z^{2}a^{-6}-13z^{2}-3a^{3}z+2az+6za^{-1}-za^{-5}-a^{2}+5a^{-2}+2a^{-4}+3+a^{3}z^{-1}-2a^{-1}z^{-1}-a^{-3}z^{-1}$ (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               4 1   -3 6             4 2     2 4           5 4       -1 2         4 4         0 0       4 6           2 -2     2 3             -1 -4     4               4 -6 1 2                 -1 -8 1                   1

Integral Khovanov Homology (db, data source)

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

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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