L11a128

L11a127

L11a129

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a128 at Knotilus!

Link Presentations

[edit Notes on L11a128's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X16,10,17,9 X18,12,19,11 X10,18,11,17 X12,16,13,15 X22,19,5,20 X20,7,21,8 X8,21,9,22 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 8, -9, 3, -5, 4, -6, 11, -2, 6, -3, 5, -4, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {\left(v^{2}-v+1\right)^{2}(uv-2u-2v+1)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-8q^{3/2}+13{\sqrt {q}}-{\frac {16}{\sqrt {q}}}+{\frac {17}{q^{3/2}}}-{\frac {18}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {10}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {2}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-2a^{7}z^{-1}+3z^{3}a^{5}+9za^{5}+7a^{5}z^{-1}-3z^{5}a^{3}-11z^{3}a^{3}-15za^{3}-7a^{3}z^{-1}+z^{7}a+4z^{5}a+7z^{3}a+6za+2az^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}-za^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-2a^{5}z^{9}-6a^{3}z^{9}-4az^{9}-2a^{6}z^{8}-4a^{4}z^{8}-9a^{2}z^{8}-7z^{8}-2a^{7}z^{7}-a^{5}z^{7}+6a^{3}z^{7}-2az^{7}-7z^{7}a^{-1}-a^{8}z^{6}+5a^{4}z^{6}+16a^{2}z^{6}-4z^{6}a^{-2}+8z^{6}+6a^{7}z^{5}+11a^{5}z^{5}+5a^{3}z^{5}+12az^{5}+11z^{5}a^{-1}-z^{5}a^{-3}+4a^{8}z^{4}+13a^{6}z^{4}+11a^{4}z^{4}-2a^{2}z^{4}+6z^{4}a^{-2}+2z^{4}-6a^{7}z^{3}-17a^{5}z^{3}-15a^{3}z^{3}-9az^{3}-4z^{3}a^{-1}+z^{3}a^{-3}-5a^{8}z^{2}-18a^{6}z^{2}-24a^{4}z^{2}-13a^{2}z^{2}-2z^{2}a^{-2}-4z^{2}+4a^{7}z+16a^{5}z+15a^{3}z+4az+za^{-1}+2a^{8}+8a^{6}+13a^{4}+8a^{2}+2-2a^{7}z^{-1}-7a^{5}z^{-1}-7a^{3}z^{-1}-2az^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   5 1   4 2                 8 3     -5 0               8 5       3 -2             10 9         -1 -4           8 7           1 -6         5 10             5 -8       5 8               -3 -10     1 6                 5 -12   1 4                   -3 -14   1                     1 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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