L9n10

L9n9

L9n11

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n10 at Knotilus!
	L9n10 is $9_{58}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n10's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X18,16,5,15 X16,12,17,11 X12,18,13,17 X13,8,14,9 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 $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{3}-t(2)^{3}-2t(1)t(2)^{2}+3t(2)^{2}+3t(1)t(2)-2t(2)-t(1)+1}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{9/2}}}+{\frac {3}{q^{7/2}}}+q^{5/2}-{\frac {5}{q^{5/2}}}-3q^{3/2}+{\frac {5}{q^{3/2}}}+4{\sqrt {q}}-{\frac {5}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}+z^{3}a^{3}-a^{3}z^{-1}-z^{5}a-3z^{3}a-3za+z^{3}a^{-1}+za^{-1}$ (db)
Kauffman polynomial	$3a^{5}z^{3}-4a^{5}z+a^{5}z^{-1}+a^{4}z^{6}+a^{4}z^{2}-a^{4}+a^{3}z^{7}+2a^{3}z^{3}-3a^{3}z+a^{3}z^{-1}+4a^{2}z^{6}-5a^{2}z^{4}+z^{4}a^{-2}+2a^{2}z^{2}-z^{2}a^{-2}+az^{7}+3az^{5}+3z^{5}a^{-1}-6az^{3}-5z^{3}a^{-1}+2az+za^{-1}+3z^{6}-4z^{4}$ (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 χ 6               1 -1 4             2   2 2           2 1   -1 0         3 2     1 -2       3 3       0 -4     2 2         0 -6   1 3           2 -8 1 2             -1 -10 2               2

Integral Khovanov Homology (db, data source)

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

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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