L9n25

L9n24

L9n26

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n25 at Knotilus!
	L9n25 is $9_{18}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n25's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X11,16,12,17 X9,11,10,18 X17,5,18,10 X4,15,1,16
Gauss code	{1, -4, 3, -9}, {-2, -1, 5, -3, -7, 8}, {-6, 2, 4, -5, 9, 6, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L9n24 L9n26