L8a13

L8a12

L8a14

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a13 at Knotilus!
	L8a13 is $8_{4}^{2}$ in the Rolfsen table of links. Contains two L4a1 configurations.

Symmetric form

(alternate)

National swiss museum

Povray depiction

Link Presentations

[edit Notes on L8a13's Link Presentations]

Planar diagram presentation	X10,1,11,2 X16,7,9,8 X12,3,13,4 X6,13,7,14 X14,5,15,6 X4,15,5,16 X2,9,3,10 X8,11,1,12
Gauss code	{1, -7, 3, -6, 5, -4, 2, -8}, {7, -1, 8, -3, 4, -5, 6, -2}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

L8a12 L8a14