L9a14

L9a13

L9a15

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a14 at Knotilus!
	L9a14 is $9_{13}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a14's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(v^{2}-v+1\right)\left(v^{2}+v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-4q^{9/2}+2q^{7/2}-3q^{5/2}+q^{3/2}+q^{19/2}-2q^{17/2}+3q^{15/2}-3q^{13/2}+4q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{5}a^{-7}+4z^{3}a^{-7}+4za^{-7}+2a^{-7}z^{-1}-z^{7}a^{-5}-6z^{5}a^{-5}-12z^{3}a^{-5}-11za^{-5}-5a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+7za^{-3}+3a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-4}-z^{8}a^{-6}-z^{7}a^{-3}-4z^{7}a^{-5}-3z^{7}a^{-7}+4z^{6}a^{-4}+z^{6}a^{-6}-3z^{6}a^{-8}+6z^{5}a^{-3}+20z^{5}a^{-5}+11z^{5}a^{-7}-3z^{5}a^{-9}-z^{4}a^{-4}+8z^{4}a^{-6}+6z^{4}a^{-8}-3z^{4}a^{-10}-12z^{3}a^{-3}-30z^{3}a^{-5}-12z^{3}a^{-7}+4z^{3}a^{-9}-2z^{3}a^{-11}-8z^{2}a^{-4}-12z^{2}a^{-6}+3z^{2}a^{-10}-z^{2}a^{-12}+10za^{-3}+17za^{-5}+7za^{-7}+5a^{-4}+5a^{-6}-a^{-10}-3a^{-3}z^{-1}-5a^{-5}z^{-1}-2a^{-7}z^{-1}$ (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 20                   1 -1 18                 1   1 16               2 1   -1 14             1 1     0 12           3 2       -1 10         1 1         0 8       1 3           2 6     2 1             1 4   1 3               2 2                     0 0 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L9a13 L9a15