L9a48

L9a47

L9a49

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a48 at Knotilus!
	L9a48 is $9_{5}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a48's Link Presentations]

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

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

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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