L9a29

L9a28

L9a30

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a29 at Knotilus!
	L9a29 is $9_{19}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a29's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X16,11,17,12 X14,6,15,5 X4,16,5,15 X18,13,7,14 X12,17,13,18
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -9, 8, -6, 7, -5, 9, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{4}-u^{2}v^{3}+u^{2}v^{2}-u^{2}v-uv^{4}+uv^{3}-uv^{2}+uv-u-v^{3}+v^{2}-v+1}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {3}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {2}{q^{17/2}}}+{\frac {1}{q^{19/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{5}\right)-4a^{7}z^{3}-4a^{7}z-a^{7}z^{-1}+a^{5}z^{7}+6a^{5}z^{5}+12a^{5}z^{3}+10a^{5}z+3a^{5}z^{-1}-a^{3}z^{5}-5a^{3}z^{3}-7a^{3}z-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{2}+2a^{11}z^{3}+3a^{10}z^{4}-2a^{10}z^{2}+4a^{9}z^{5}-7a^{9}z^{3}+2a^{9}z+4a^{8}z^{6}-10a^{8}z^{4}+5a^{8}z^{2}-a^{8}+3a^{7}z^{7}-9a^{7}z^{5}+6a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+a^{6}z^{8}-11a^{6}z^{4}+13a^{6}z^{2}-3a^{6}+4a^{5}z^{7}-19a^{5}z^{5}+27a^{5}z^{3}-14a^{5}z+3a^{5}z^{-1}+a^{4}z^{8}-4a^{4}z^{6}+2a^{4}z^{4}+5a^{4}z^{2}-3a^{4}+a^{3}z^{7}-6a^{3}z^{5}+12a^{3}z^{3}-9a^{3}z+2a^{3}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                   1 1 -2                     0 -4               3 1   2 -6             1 1     0 -8           3 2       1 -10         2 2         0 -12       2 2           0 -14     1 2             1 -16   1 2               -1 -18   1                 1 -20 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

L9a28 L9a30