L10a119

L10a118

L10a120

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a119 at Knotilus!

Link Presentations

[edit Notes on L10a119's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-10$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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