L9n19

L9n18

L9n20

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n19 at Knotilus!
	L9n19 is $9_{61}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n19's Link Presentations]

Planar diagram presentation	X10,1,11,2 X3,12,4,13 X18,5,9,6 X6,9,7,10 X16,12,17,11 X7,14,8,15 X13,4,14,5 X15,8,16,1 X2,17,3,18
Gauss code	{1, -9, -2, 7, 3, -4, -6, 8}, {4, -1, 5, 2, -7, 6, -8, -5, 9, -3}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {\left(t(1)^{2}+t(2)\right)\left(t(1)t(2)^{2}+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}-a^{7}z^{-1}-z^{5}a^{5}-5z^{3}a^{5}-5za^{5}$ (db)
Kauffman polynomial	$a^{11}z^{5}-5a^{11}z^{3}+5a^{11}z-a^{9}z+a^{9}z^{-1}-a^{8}-a^{7}z+a^{7}z^{-1}+a^{5}z^{5}-5a^{5}z^{3}+5a^{5}z$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                 1 1 -6                 1 1 -8           1 1     0 -10                   0 -12       1 2 1       0 -14                   0 -16     1 1           0 -18 1                 1 -20 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-7$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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