L9a43

L9a42

L9a44

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a43 at Knotilus!
	L9a43 is $9_{4}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a43's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{3}+t(2)t(3)^{3}-t(3)^{3}-3t(1)t(3)^{2}+2t(1)t(2)t(3)^{2}-3t(2)t(3)^{2}+2t(3)^{2}+3t(1)t(3)-2t(1)t(2)t(3)+3t(2)t(3)-2t(3)-t(1)+t(1)t(2)-t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-11}+2q^{-10}-5q^{-9}+8q^{-8}-8q^{-7}+10q^{-6}-7q^{-5}+7q^{-4}-3q^{-3}+q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{12}z^{-2}+4a^{10}z^{-2}+4a^{10}-6a^{8}z^{2}-5a^{8}z^{-2}-11a^{8}+3a^{6}z^{4}+8a^{6}z^{2}+2a^{6}z^{-2}+7a^{6}+a^{4}z^{4}+a^{4}z^{2}$ (db)
Kauffman polynomial	$a^{13}z^{5}-3a^{13}z^{3}+3a^{13}z-a^{13}z^{-1}+2a^{12}z^{6}-4a^{12}z^{4}+3a^{12}z^{2}+a^{12}z^{-2}-2a^{12}+2a^{11}z^{7}+a^{11}z^{5}-11a^{11}z^{3}+13a^{11}z-5a^{11}z^{-1}+a^{10}z^{8}+6a^{10}z^{6}-16a^{10}z^{4}+14a^{10}z^{2}+4a^{10}z^{-2}-10a^{10}+6a^{9}z^{7}-3a^{9}z^{5}-16a^{9}z^{3}+21a^{9}z-9a^{9}z^{-1}+a^{8}z^{8}+10a^{8}z^{6}-24a^{8}z^{4}+23a^{8}z^{2}+5a^{8}z^{-2}-14a^{8}+4a^{7}z^{7}-10a^{7}z^{3}+11a^{7}z-5a^{7}z^{-1}+6a^{6}z^{6}-11a^{6}z^{4}+11a^{6}z^{2}+2a^{6}z^{-2}-7a^{6}+3a^{5}z^{5}-2a^{5}z^{3}+a^{4}z^{4}-a^{4}z^{2}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                   1 1 -5                 3 1 -2 -7               4     4 -9             3 3     0 -11           7 4       3 -13         4 6         2 -15       4 4           0 -17     1 4             3 -19   1 4               -3 -21   1                 1 -23 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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