L8a15

L8a14

L8a16

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a15 at Knotilus!
	L8a15 is $8_{3}^{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a15's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uvw-2uv-2uw+2u-2vw+2v+2w-1}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}}}$ (db)
Jones polynomial	$q^{-7}-q^{-6}+4q^{-5}-4q^{-4}+6q^{-3}-4q^{-2}+q+4q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}-2a^{6}z^{-2}-3a^{6}+3a^{4}z^{2}+a^{4}z^{-2}+3a^{4}-a^{2}z^{4}-a^{2}z^{2}+z^{2}$ (db)
Kauffman polynomial	$z^{4}a^{8}-3z^{2}a^{8}-a^{8}z^{-2}+3a^{8}+z^{5}a^{7}-3za^{7}+2a^{7}z^{-1}+z^{6}a^{6}+2z^{4}a^{6}-6z^{2}a^{6}-2a^{6}z^{-2}+5a^{6}+z^{7}a^{5}+2z^{3}a^{5}-3za^{5}+2a^{5}z^{-1}+4z^{6}a^{4}-5z^{4}a^{4}-a^{4}z^{-2}+3a^{4}+z^{7}a^{3}+2z^{5}a^{3}-3z^{3}a^{3}+3z^{6}a^{2}-5z^{4}a^{2}+2z^{2}a^{2}+3z^{5}a-5z^{3}a+z^{4}-z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 3                 1 1 1               2   -2 -1             2 1   1 -3           3 3     0 -5         3 1       2 -7       1 3         2 -9     3 3           0 -11   1 4             3 -13                   0 -15 1                 1

Integral Khovanov Homology (db, data source)

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

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.

L8a14 L8a16