L9a4

L9a3

L9a5

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a4 at Knotilus!
	L9a4 is $9_{18}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a4's Link Presentations]

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

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

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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