L11a254

L11a253

L11a255

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a254 at Knotilus!

Link Presentations

[edit Notes on L11a254's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X4,13,5,14 X20,16,21,15 X16,8,17,7 X6,21,7,22 X18,9,19,10 X22,17,9,18 X8,20,1,19
Gauss code	{1, -2, 3, -5, 4, -8, 7, -11}, {9, -1, 2, -3, 5, -4, 6, -7, 10, -9, 11, -6, 8, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{3}v^{5}-2u^{3}v^{4}+2u^{3}v^{3}-u^{3}v^{2}-u^{2}v^{5}+3u^{2}v^{4}-3u^{2}v^{3}+3u^{2}v^{2}-u^{2}v-uv^{4}+3uv^{3}-3uv^{2}+3uv-u-v^{3}+2v^{2}-2v+1}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {10}{q^{9/2}}}+{\frac {8}{q^{7/2}}}-{\frac {7}{q^{5/2}}}+{\frac {4}{q^{3/2}}}-{\frac {1}{q^{21/2}}}+{\frac {3}{q^{19/2}}}-{\frac {5}{q^{17/2}}}+{\frac {7}{q^{15/2}}}-{\frac {9}{q^{13/2}}}+{\frac {10}{q^{11/2}}}+{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{5}z^{9}+a^{7}z^{7}-7a^{5}z^{7}+a^{3}z^{7}+5a^{7}z^{5}-17a^{5}z^{5}+5a^{3}z^{5}+7a^{7}z^{3}-17a^{5}z^{3}+6a^{3}z^{3}+3a^{7}z-5a^{5}z+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{13}z^{3}+3a^{12}z^{4}-a^{12}z^{2}+5a^{11}z^{5}-4a^{11}z^{3}+a^{11}z+6a^{10}z^{6}-7a^{10}z^{4}+a^{10}z^{2}+6a^{9}z^{7}-9a^{9}z^{5}-a^{9}z^{3}+a^{9}z+6a^{8}z^{8}-15a^{8}z^{6}+8a^{8}z^{4}-3a^{8}z^{2}+5a^{7}z^{9}-17a^{7}z^{7}+17a^{7}z^{5}-11a^{7}z^{3}+4a^{7}z+2a^{6}z^{10}-2a^{6}z^{8}-17a^{6}z^{6}+27a^{6}z^{4}-9a^{6}z^{2}+8a^{5}z^{9}-40a^{5}z^{7}+61a^{5}z^{5}-33a^{5}z^{3}+5a^{5}z+a^{5}z^{-1}+2a^{4}z^{10}-7a^{4}z^{8}-a^{4}z^{6}+16a^{4}z^{4}-7a^{4}z^{2}-a^{4}+3a^{3}z^{9}-17a^{3}z^{7}+30a^{3}z^{5}-18a^{3}z^{3}+a^{3}z+a^{3}z^{-1}+a^{2}z^{8}-5a^{2}z^{6}+7a^{2}z^{4}-3a^{2}z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 2                       1 -1 0                     2   2 -2                   2 1   -1 -4                 5 2     3 -6               4 3       -1 -8             6 4         2 -10           4 4           0 -12         5 6             -1 -14       3 5               2 -16     2 4                 -2 -18   1 3                   2 -20   2                     -2 -22 1                       1

Integral Khovanov Homology (db, data source)

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

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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