L11a188

L11a187

L11a189

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a188 at Knotilus!

Link Presentations

[edit Notes on L11a188's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{5}-u^{2}v^{4}+u^{2}v^{3}-u^{2}v^{2}+u^{2}v-u^{2}+uv^{6}-3uv^{5}+3uv^{4}-3uv^{3}+3uv^{2}-3uv+u-v^{6}+v^{5}-v^{4}+v^{3}-v^{2}+v}{uv^{3}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {4}{q^{5/2}}}+{\frac {2}{q^{3/2}}}+{\frac {1}{q^{23/2}}}-{\frac {2}{q^{21/2}}}+{\frac {4}{q^{19/2}}}-{\frac {6}{q^{17/2}}}+{\frac {8}{q^{15/2}}}-{\frac {9}{q^{13/2}}}+{\frac {8}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{5}\right)-4a^{9}z^{3}-4a^{9}z-2a^{9}z^{-1}+a^{7}z^{7}+5a^{7}z^{5}+9a^{7}z^{3}+10a^{7}z+5a^{7}z^{-1}+a^{5}z^{7}+4a^{5}z^{5}+2a^{5}z^{3}-4a^{5}z-3a^{5}z^{-1}-a^{3}z^{5}-4a^{3}z^{3}-3a^{3}z$ (db)
Kauffman polynomial	$a^{14}z^{4}-2a^{14}z^{2}+2a^{13}z^{5}-3a^{13}z^{3}+3a^{12}z^{6}-6a^{12}z^{4}+5a^{12}z^{2}-a^{12}+3a^{11}z^{7}-6a^{11}z^{5}+6a^{11}z^{3}+3a^{10}z^{8}-8a^{10}z^{6}+10a^{10}z^{4}-2a^{10}z^{2}+3a^{9}z^{9}-13a^{9}z^{7}+28a^{9}z^{5}-28a^{9}z^{3}+12a^{9}z-2a^{9}z^{-1}+a^{8}z^{10}-11a^{8}z^{6}+25a^{8}z^{4}-22a^{8}z^{2}+5a^{8}+5a^{7}z^{9}-25a^{7}z^{7}+50a^{7}z^{5}-53a^{7}z^{3}+25a^{7}z-5a^{7}z^{-1}+a^{6}z^{10}-a^{6}z^{8}-9a^{6}z^{6}+18a^{6}z^{4}-15a^{6}z^{2}+5a^{6}+2a^{5}z^{9}-8a^{5}z^{7}+9a^{5}z^{5}-9a^{5}z^{3}+10a^{5}z-3a^{5}z^{-1}+2a^{4}z^{8}-9a^{4}z^{6}+10a^{4}z^{4}-2a^{4}z^{2}+a^{3}z^{7}-5a^{3}z^{5}+7a^{3}z^{3}-3a^{3}z$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                       1 1 -2                     1   -1 -4                   3 1   2 -6                 3 2     -1 -8               5 2       3 -10             4 4         0 -12           5 4           1 -14         3 4             1 -16       3 5               -2 -18     1 3                 2 -20   1 3                   -2 -22   1                     1 -24 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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