L9a50

L9a49

L9a51

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a50 at Knotilus!
	L9a50 is $9_{1}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a50's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X18,16,11,15 X14,8,15,7 X10,12,5,11 X8,17,9,18 X16,9,17,10 X2536 X4,14,1,13
Gauss code	{1, -8, 2, -9}, {8, -1, 4, -6, 7, -5}, {5, -2, 9, -4, 3, -7, 6, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{2}t(2)^{2}-t(3)^{2}t(2)^{2}-t(1)t(3)t(2)^{2}+2t(3)t(2)^{2}-t(2)^{2}-2t(1)t(3)^{2}t(2)+t(3)^{2}t(2)-t(1)t(2)+2t(1)t(3)t(2)-2t(3)t(2)+2t(2)+t(1)t(3)^{2}+t(1)-2t(1)t(3)+t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{6}+3q^{5}-5q^{4}+7q^{3}+q^{-3}-7q^{2}-2q^{-2}+8q+5q^{-1}-5$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-4}-2z^{2}a^{-4}-a^{-4}+z^{6}a^{-2}+4z^{4}a^{-2}+a^{2}z^{2}+6z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+2a^{2}+4a^{-2}-2z^{4}-6z^{2}-2z^{-2}-5$ (db)
Kauffman polynomial	$z^{8}a^{-2}+z^{8}+2az^{7}+6z^{7}a^{-1}+4z^{7}a^{-3}+a^{2}z^{6}+8z^{6}a^{-2}+6z^{6}a^{-4}+3z^{6}-6az^{5}-14z^{5}a^{-1}-3z^{5}a^{-3}+5z^{5}a^{-5}-4a^{2}z^{4}-27z^{4}a^{-2}-9z^{4}a^{-4}+3z^{4}a^{-6}-19z^{4}+3az^{3}+3z^{3}a^{-1}-5z^{3}a^{-3}-4z^{3}a^{-5}+z^{3}a^{-7}+6a^{2}z^{2}+23z^{2}a^{-2}+6z^{2}a^{-4}-z^{2}a^{-6}+22z^{2}+3az+5za^{-1}+3za^{-3}+za^{-5}-4a^{2}-8a^{-2}-2a^{-4}-9-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 13                   1 -1 11                 2   2 9               3 1   -2 7             4 2     2 5           4 4       0 3         4 3         1 1       3 6           3 -1     2 2             0 -3   1 4               3 -5   1                 -1 -7 1                   1

Integral Khovanov Homology (db, data source)

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

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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