L9a2

L9a1

L9a3

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a2 at Knotilus!
	L9a2 is $9_{31}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a2's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X12,6,13,5 X8493 X16,10,17,9 X18,12,5,11 X10,18,11,17 X2,14,3,13
Gauss code	{1, -9, 5, -3}, {4, -1, 2, -5, 6, -8, 7, -4, 9, -2, 3, -6, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-t(2)^{3}+t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{15/2}+3q^{13/2}-4q^{11/2}+6q^{9/2}-7q^{7/2}+6q^{5/2}-6q^{3/2}+3{\sqrt {q}}-{\frac {3}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}-3z^{3}a^{-1}+7z^{3}a^{-3}-3z^{3}a^{-5}+za^{-3}-za^{-5}+2a^{-1}z^{-1}-3a^{-3}z^{-1}+a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-9}+3z^{4}a^{-8}-2z^{2}a^{-8}+4z^{5}a^{-7}-3z^{3}a^{-7}+4z^{6}a^{-6}-3z^{4}a^{-6}-2z^{2}a^{-6}+a^{-6}+4z^{7}a^{-5}-8z^{5}a^{-5}+6z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+2z^{8}a^{-4}-2z^{6}a^{-4}-3z^{4}a^{-4}+3a^{-4}+7z^{7}a^{-3}-24z^{5}a^{-3}+22z^{3}a^{-3}-3za^{-3}-3a^{-3}z^{-1}+2z^{8}a^{-2}-5z^{6}a^{-2}+z^{2}a^{-2}+3a^{-2}+3z^{7}a^{-1}-12z^{5}a^{-1}+12z^{3}a^{-1}-za^{-1}-2a^{-1}z^{-1}+z^{6}-3z^{4}+z^{2}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 16                   1 1 14                 2   -2 12               2 1   1 10             4 2     -2 8           3 2       1 6         3 4         1 4       3 3           0 2     2 5             3 0   1 1               0 -2   2                 2 -4 1                   -1

Integral Khovanov Homology (db, data source)

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

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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