L9n11

L9n10

L9n12

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n11 at Knotilus!
	L9n11 is $9_{57}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n11's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X15,18,16,5 X11,16,12,17 X17,12,18,13 X8,14,9,13 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 $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{3}-t(2)^{3}-2t(1)t(2)^{2}+t(2)^{2}+t(1)t(2)-2t(2)-t(1)+1}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {2}{\sqrt {q}}}-{\frac {3}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}\left(-z^{3}\right)-a^{5}z+a^{5}z^{-1}+a^{3}z^{5}+3a^{3}z^{3}+a^{3}z-a^{3}z^{-1}-az^{3}-2az$ (db)
Kauffman polynomial	$-z^{2}a^{8}-3z^{3}a^{7}+za^{7}-z^{6}a^{6}+z^{4}a^{6}-2z^{2}a^{6}-z^{7}a^{5}+2z^{5}a^{5}-4z^{3}a^{5}+4za^{5}-a^{5}z^{-1}-3z^{6}a^{4}+7z^{4}a^{4}-4z^{2}a^{4}+a^{4}-z^{7}a^{3}+z^{5}a^{3}+2z^{3}a^{3}+za^{3}-a^{3}z^{-1}-2z^{6}a^{2}+6z^{4}a^{2}-3z^{2}a^{2}-z^{5}a+3z^{3}a-2za$ (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 2               1 1 0             1   -1 -2           2 1   1 -4         2 2     0 -6       2 1       1 -8     1 2         1 -10   2 2           0 -12   2             2 -14 1               -1

Integral Khovanov Homology (db, data source)

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

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.

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