L8a12

L8a11

L8a13

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a12 at Knotilus!
	L8a12 is $8_{2}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L8a12's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(2)^{2}t(1)^{3}-t(2)^{3}t(1)^{2}+t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(1)+t(2)t(1)-t(1)-t(2)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {2}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {2}{q^{17/2}}}+{\frac {1}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}z^{3}+3a^{9}z+a^{9}z^{-1}-a^{7}z^{5}-4a^{7}z^{3}-4a^{7}z-a^{7}z^{-1}-a^{5}z^{5}-4a^{5}z^{3}-3a^{5}z$ (db)
Kauffman polynomial	$-z^{3}a^{13}+2za^{13}-z^{4}a^{12}+z^{2}a^{12}-z^{5}a^{11}+z^{3}a^{11}-za^{11}-z^{6}a^{10}+2z^{4}a^{10}-3z^{2}a^{10}-z^{7}a^{9}+4z^{5}a^{9}-8z^{3}a^{9}+5za^{9}-a^{9}z^{-1}-2z^{6}a^{8}+6z^{4}a^{8}-5z^{2}a^{8}+a^{8}-z^{7}a^{7}+4z^{5}a^{7}-6z^{3}a^{7}+5za^{7}-a^{7}z^{-1}-z^{6}a^{6}+3z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+4z^{3}a^{5}-3za^{5}$ (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 -4                 1 1 -6               1 1 0 -8             1     1 -10           1 1     0 -12         2 1       1 -14       1 1         0 -16     1 2           -1 -18     1             1 -20 1 1               0 -22 1                 1

Integral Khovanov Homology (db, data source)

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

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.

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