L10a100

L10a99

L10a101

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Link Presentations

[edit Notes on L10a100's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{3}v^{2}-u^{3}v+2u^{2}v^{3}-3u^{2}v^{2}+3u^{2}v-u^{2}-uv^{3}+3uv^{2}-3uv+2u-v^{2}+2v}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{25/2}}}+{\frac {2}{q^{23/2}}}-{\frac {4}{q^{21/2}}}+{\frac {6}{q^{19/2}}}-{\frac {7}{q^{17/2}}}+{\frac {8}{q^{15/2}}}-{\frac {8}{q^{13/2}}}+{\frac {5}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{11}+2za^{11}-z^{5}a^{9}-2z^{3}a^{9}+za^{9}+a^{9}z^{-1}-2z^{5}a^{7}-7z^{3}a^{7}-6za^{7}-a^{7}z^{-1}-z^{5}a^{5}-3z^{3}a^{5}-za^{5}$ (db)
Kauffman polynomial	$-z^{5}a^{15}+3z^{3}a^{15}-2za^{15}-2z^{6}a^{14}+5z^{4}a^{14}-2z^{2}a^{14}-2z^{7}a^{13}+3z^{5}a^{13}+z^{3}a^{13}-2z^{8}a^{12}+5z^{6}a^{12}-8z^{4}a^{12}+7z^{2}a^{12}-z^{9}a^{11}+z^{7}a^{11}-z^{5}a^{11}-za^{11}-4z^{8}a^{10}+13z^{6}a^{10}-19z^{4}a^{10}+7z^{2}a^{10}-z^{9}a^{9}+6z^{5}a^{9}-13z^{3}a^{9}+5za^{9}-a^{9}z^{-1}-2z^{8}a^{8}+4z^{6}a^{8}-z^{4}a^{8}-3z^{2}a^{8}+a^{8}-3z^{7}a^{7}+10z^{5}a^{7}-12z^{3}a^{7}+7za^{7}-a^{7}z^{-1}-2z^{6}a^{6}+5z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+3z^{3}a^{5}-za^{5}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                     1 1 -6                   2 1 -1 -8                 2     2 -10               3 2     -1 -12             5 2       3 -14           3 3         0 -16         4 5           -1 -18       2 3             1 -20     2 4               -2 -22   1 3                 2 -24   1                   -1 -26 1                     1

Integral Khovanov Homology (db, data source)

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

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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