L10a9

L10a8

L10a10

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a9 at Knotilus!

Link Presentations

[edit Notes on L10a9's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X18,14,19,13 X16,9,17,10 X8,17,9,18 X20,16,5,15 X14,20,15,19 X2536 X4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -6, 5, -2, 10, -3, 4, -8, 7, -5, 6, -4, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)^{3}\left(t(2)^{2}+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$9q^{9/2}-10q^{7/2}+10q^{5/2}-{\frac {1}{q^{5/2}}}-10q^{3/2}+{\frac {2}{q^{3/2}}}-q^{15/2}+3q^{13/2}-6q^{11/2}+7{\sqrt {q}}-{\frac {5}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-5}-3z^{3}a^{-5}-3za^{-5}-a^{-5}z^{-1}+z^{7}a^{-3}+5z^{5}a^{-3}+10z^{3}a^{-3}+10za^{-3}+3a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-8z^{3}a^{-1}+3az-10za^{-1}+2az^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-9}+3z^{4}a^{-8}+6z^{5}a^{-7}-4z^{3}a^{-7}+za^{-7}+9z^{6}a^{-6}-14z^{4}a^{-6}+7z^{2}a^{-6}-a^{-6}+9z^{7}a^{-5}-18z^{5}a^{-5}+10z^{3}a^{-5}-4za^{-5}+a^{-5}z^{-1}+5z^{8}a^{-4}-4z^{6}a^{-4}-15z^{4}a^{-4}+14z^{2}a^{-4}-3a^{-4}+z^{9}a^{-3}+11z^{7}a^{-3}-44z^{5}a^{-3}+45z^{3}a^{-3}-19za^{-3}+3a^{-3}z^{-1}+7z^{8}a^{-2}-21z^{6}a^{-2}+11z^{4}a^{-2}+6z^{2}a^{-2}-3a^{-2}+z^{9}a^{-1}+az^{7}+3z^{7}a^{-1}-5az^{5}-25z^{5}a^{-1}+9az^{3}+39z^{3}a^{-1}-7az-21za^{-1}+2az^{-1}+4a^{-1}z^{-1}+2z^{8}-8z^{6}+9z^{4}-z^{2}-2$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 16                     1 1 14                   2   -2 12                 4 1   3 10               5 2     -3 8             5 4       1 6           5 5         0 4         5 5           0 2       4 7             3 0     1 3               -2 -2   1 4                 3 -4   1                   -1 -6 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L10a8 L10a10