L9n23

L9n22

L9n24

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n23 at Knotilus!
	L9n23 is $9_{13}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n23's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X13,2,14,3 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 $u$, $v$, $w$, ...)	${\frac {-t(1)t(3)^{2}t(2)^{2}+t(1)t(3)t(2)^{2}+t(1)t(3)^{2}t(2)-t(2)-t(3)+1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$1-q^{-1}+2q^{-2}-q^{-3}+3q^{-4}-q^{-5}+2q^{-6}-q^{-7}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{8}+z^{4}a^{6}+4z^{2}a^{6}+a^{6}z^{-2}+3a^{6}-z^{6}a^{4}-5z^{4}a^{4}-7z^{2}a^{4}-2a^{4}z^{-2}-5a^{4}+z^{4}a^{2}+4z^{2}a^{2}+a^{2}z^{-2}+3a^{2}$ (db)
Kauffman polynomial	$za^{9}+2z^{2}a^{8}-2a^{8}+z^{5}a^{7}-3z^{3}a^{7}+3za^{7}+2z^{6}a^{6}-9z^{4}a^{6}+13z^{2}a^{6}+a^{6}z^{-2}-8a^{6}+z^{7}a^{5}-3z^{5}a^{5}-z^{3}a^{5}+5za^{5}-2a^{5}z^{-1}+3z^{6}a^{4}-14z^{4}a^{4}+18z^{2}a^{4}+2a^{4}z^{-2}-9a^{4}+z^{7}a^{3}-4z^{5}a^{3}+2z^{3}a^{3}+3za^{3}-2a^{3}z^{-1}+z^{6}a^{2}-5z^{4}a^{2}+7z^{2}a^{2}+a^{2}z^{-2}-4a^{2}$ (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 1               1 1 -1                 0 -3           2 1   1 -5       1 1 1     1 -7       3 1       2 -9   1 1 2         2 -11   2 1           1 -13   1             1 -15 1               -1

Integral Khovanov Homology (db, data source)

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

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.

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