L9a46

L9a45

L9a47

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a46 at Knotilus!
	L9a46 is $9_{10}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a46's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X16,12,17,11 X18,9,11,10 X10,17,5,18 X4,15,1,16
Gauss code	{1, -4, 3, -9}, {2, -1, 5, -3, 7, -8}, {6, -2, 4, -5, 9, -6, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)^{2}(w-1)^{2}}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{-5}+q^{4}+4q^{-4}-4q^{3}-6q^{-3}+7q^{2}+10q^{-2}-9q-10q^{-1}+12$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{4}z^{2}+a^{4}z^{-2}+2a^{2}z^{4}+z^{4}a^{-2}+3a^{2}z^{2}+z^{2}a^{-2}-2a^{2}z^{-2}-a^{2}-z^{6}-3z^{4}-3z^{2}+z^{-2}+1$ (db)
Kauffman polynomial	$a^{5}z^{5}-a^{5}z^{3}+4a^{4}z^{6}-8a^{4}z^{4}+z^{4}a^{-4}+4a^{4}z^{2}-a^{4}z^{-2}+a^{4}+5a^{3}z^{7}-8a^{3}z^{5}+4z^{5}a^{-3}+2a^{3}z^{3}-3z^{3}a^{-3}-a^{3}z+2a^{3}z^{-1}+2a^{2}z^{8}+8a^{2}z^{6}+7z^{6}a^{-2}-23a^{2}z^{4}-9z^{4}a^{-2}+12a^{2}z^{2}+4z^{2}a^{-2}-2a^{2}z^{-2}+a^{2}+11az^{7}+6z^{7}a^{-1}-17az^{5}-4z^{5}a^{-1}+4az^{3}-2z^{3}a^{-1}-az+2az^{-1}+2z^{8}+11z^{6}-25z^{4}+12z^{2}-z^{-2}+1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   1 1 7                 3   -3 5               4 1   3 3             5 3     -2 1           7 4       3 -1         5 7         2 -3       5 5           0 -5     3 7             4 -7   1 3               -2 -9   3                 3 -11 1                   -1

Integral Khovanov Homology (db, data source)

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

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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