L11a186

L11a185

L11a187

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a186 at Knotilus!

Link Presentations

[edit Notes on L11a186's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X22,18,7,17 X16,5,17,6 X4,15,5,16 X14,22,15,21 X18,12,19,11 X20,14,21,13 X12,20,13,19 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 7, -9, 8, -6, 5, -4, 3, -7, 9, -8, 6, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{5}-2u^{2}v^{4}+2u^{2}v^{3}-2u^{2}v^{2}+2u^{2}v+uv^{6}-3uv^{5}+5uv^{4}-5uv^{3}+5uv^{2}-3uv+u+2v^{5}-2v^{4}+2v^{3}-2v^{2}+v}{uv^{3}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {2}{q^{9/2}}}+6q^{7/2}-{\frac {5}{q^{7/2}}}-9q^{5/2}+{\frac {7}{q^{5/2}}}+12q^{3/2}-{\frac {11}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-13{\sqrt {q}}+{\frac {12}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{3}z^{5}+z^{5}a^{-3}+4a^{3}z^{3}+3z^{3}a^{-3}+5a^{3}z+2za^{-3}+3a^{3}z^{-1}-az^{7}-z^{7}a^{-1}-5az^{5}-4z^{5}a^{-1}-10az^{3}-4z^{3}a^{-1}-11az+za^{-1}-5az^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-z^{2}a^{-6}+a^{5}z^{7}-5a^{5}z^{5}+3z^{5}a^{-5}+7a^{5}z^{3}-3z^{3}a^{-5}-2a^{5}z+2a^{4}z^{8}-8a^{4}z^{6}+5z^{6}a^{-4}+8a^{4}z^{4}-6z^{4}a^{-4}-a^{4}z^{2}+3z^{2}a^{-4}-a^{-4}+3a^{3}z^{9}-14a^{3}z^{7}+6z^{7}a^{-3}+26a^{3}z^{5}-9z^{5}a^{-3}-28a^{3}z^{3}+7z^{3}a^{-3}+15a^{3}z-2za^{-3}-3a^{3}z^{-1}+a^{2}z^{10}+2a^{2}z^{8}+5z^{8}a^{-2}-18a^{2}z^{6}-6z^{6}a^{-2}+28a^{2}z^{4}-20a^{2}z^{2}+4z^{2}a^{-2}+5a^{2}+6az^{9}+3z^{9}a^{-1}-22az^{7}-z^{7}a^{-1}+36az^{5}-7z^{5}a^{-1}-40az^{3}+5z^{3}a^{-1}+22az+3za^{-1}-5az^{-1}-2a^{-1}z^{-1}+z^{10}+5z^{8}-21z^{6}+27z^{4}-19z^{2}+5$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     2   2 8                   4 1   -3 6                 5 2     3 4               7 4       -3 2             6 5         1 0           7 8           1 -2         4 5             -1 -4       3 7               4 -6     2 4                 -2 -8     3                   3 -10 1 2                     -1 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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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