L11a208

L11a207

L11a209

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a208 at Knotilus!

Link Presentations

[edit Notes on L11a208's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X20,13,21,14 X10,4,11,3 X16,6,17,5 X4,12,5,11 X22,15,7,16 X12,19,13,20 X14,21,15,22 X2,18,3,17
Gauss code	{1, -11, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -9, 4, -10, 8, -6, 11, -2, 9, -4, 10, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {\left(v^{4}-v^{3}+v^{2}-v+1\right)(uv-u+1)(uv-v+1)}{uv^{3}}}$ (db)
Jones polynomial	${\frac {13}{q^{9/2}}}-{\frac {14}{q^{7/2}}}-q^{5/2}+{\frac {13}{q^{5/2}}}+3q^{3/2}-{\frac {13}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {9}{q^{11/2}}}-6{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{5}z^{7}-5a^{5}z^{5}-8a^{5}z^{3}-5a^{5}z-2a^{5}z^{-1}+a^{3}z^{9}+7a^{3}z^{7}+18a^{3}z^{5}+21a^{3}z^{3}+12a^{3}z+5a^{3}z^{-1}-az^{7}-5az^{5}-8az^{3}-6az-3az^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}-a^{10}z^{2}+3a^{9}z^{5}-3a^{9}z^{3}+5a^{8}z^{6}-6a^{8}z^{4}+3a^{8}z^{2}-a^{8}+6a^{7}z^{7}-8a^{7}z^{5}+4a^{7}z^{3}+6a^{6}z^{8}-10a^{6}z^{6}+5a^{6}z^{4}+2a^{6}z^{2}+5a^{5}z^{9}-12a^{5}z^{7}+15a^{5}z^{5}-13a^{5}z^{3}+6a^{5}z-2a^{5}z^{-1}+2a^{4}z^{10}+a^{4}z^{8}-15a^{4}z^{6}+19a^{4}z^{4}-12a^{4}z^{2}+5a^{4}+9a^{3}z^{9}-35a^{3}z^{7}+50a^{3}z^{5}-38a^{3}z^{3}+15a^{3}z-5a^{3}z^{-1}+2a^{2}z^{10}-2a^{2}z^{8}-12a^{2}z^{6}+19a^{2}z^{4}-12a^{2}z^{2}+5a^{2}+4az^{9}-16az^{7}+z^{7}a^{-1}+20az^{5}-4z^{5}a^{-1}-14az^{3}+4z^{3}a^{-1}+8az-3az^{-1}-za^{-1}+3z^{8}-12z^{6}+12z^{4}-2z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     2   -2 2                   4 1   3 0                 5 3     -2 -2               8 3       5 -4             6 6         0 -6           8 7           1 -8         5 6             1 -10       4 8               -4 -12     2 5                 3 -14   1 4                   -3 -16   2                     2 -18 1                       -1

Integral Khovanov Homology (db, data source)

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

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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