L9a38

L9a37

L9a39

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a38 at Knotilus!
	L9a38 is $9_{5}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a38's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X14,6,15,5 X18,14,9,13 X16,8,17,7 X6,16,7,15 X8,18,1,17
Gauss code	{1, -2, 3, -4, 5, -8, 7, -9}, {4, -1, 2, -3, 6, -5, 8, -7, 9, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(1)t(2)+1)^{2}}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-2q^{9/2}+3q^{7/2}-{\frac {1}{q^{7/2}}}-5q^{5/2}+{\frac {2}{q^{5/2}}}+5q^{3/2}-{\frac {3}{q^{3/2}}}+q^{11/2}-6{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-6z^{5}a^{-1}+z^{5}a^{-3}+4az^{3}-12z^{3}a^{-1}+4z^{3}a^{-3}+4az-8za^{-1}+4za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-2z^{2}a^{-6}+2z^{5}a^{-5}-4z^{3}a^{-5}+za^{-5}+2z^{6}a^{-4}-3z^{4}a^{-4}+z^{2}a^{-4}+2z^{7}a^{-3}+a^{3}z^{5}-5z^{5}a^{-3}-3a^{3}z^{3}+8z^{3}a^{-3}+a^{3}z-4za^{-3}+z^{8}a^{-2}+2a^{2}z^{6}-z^{6}a^{-2}-6a^{2}z^{4}+3a^{2}z^{2}+3z^{2}a^{-2}+2az^{7}+4z^{7}a^{-1}-6az^{5}-14z^{5}a^{-1}+6az^{3}+21z^{3}a^{-1}-4az-10za^{-1}+az^{-1}+a^{-1}z^{-1}+z^{8}-z^{6}-2z^{4}+3z^{2}-1$ (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 12                   1 -1 10                 1   1 8               2 1   -1 6             3 1     2 4           2 2       0 2         4 3         1 0       2 4           2 -2     1 2             -1 -4   1 2               1 -6   1                 -1 -8 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L9a37 L9a39