L9n26

L9n25

L9n27

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n26 at Knotilus!
	L9n26 is $9_{19}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n26's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,17,12,16 X7,14,8,15 X13,8,14,9 X15,13,16,18 X17,5,18,12 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {-5, 4, -6, 3, -7, 6}, {8, -1, -4, 5, 9, -2, -3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(2)-1)(t(3)-1)\left(t(3)^{2}-t(1)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{-5}+3q^{-4}-2q^{-3}+3q^{-2}-q-2q^{-1}+3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-2a^{4}z^{2}-2a^{4}z^{-2}-4a^{4}+a^{2}z^{4}+3a^{2}z^{2}+a^{2}z^{-2}+3a^{2}-z^{2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-5a^{6}z^{4}+8a^{6}z^{2}+a^{6}z^{-2}-5a^{6}+a^{5}z^{7}-3a^{5}z^{5}-a^{5}z^{3}+5a^{5}z-2a^{5}z^{-1}+4a^{4}z^{6}-17a^{4}z^{4}+20a^{4}z^{2}+2a^{4}z^{-2}-10a^{4}+a^{3}z^{7}-a^{3}z^{5}-6a^{3}z^{3}+7a^{3}z-2a^{3}z^{-1}+3a^{2}z^{6}-12a^{2}z^{4}+15a^{2}z^{2}+a^{2}z^{-2}-7a^{2}+2az^{5}-5az^{3}+3az+za^{-1}+3z^{2}-1$ (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 χ 3               1 -1 1             2   2 -1           2 3   1 -3         1 1 1   1 -5       1 2       1 -7     2 1         1 -9   1 3           2 -11                 0 -13 1               1

Integral Khovanov Homology (db, data source)

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

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.

L9n25 L9n27