L9a45

L9a44

L9a46

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a45 at Knotilus!
	L9a45 is $9_{7}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a45's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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