L10a11

L10a10

L10a12

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Link Presentations

[edit Notes on L10a11's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,11,17,12 X14,7,15,8 X8,15,9,16 X20,17,5,18 X18,14,19,13 X12,20,13,19 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 4, -5, 10, -2, 3, -8, 7, -4, 5, -3, 6, -7, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uv^{4}-7uv^{3}+7uv^{2}-4uv+u+v^{5}-4v^{4}+7v^{3}-7v^{2}+2v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	${\frac {14}{q^{9/2}}}-{\frac {14}{q^{7/2}}}+{\frac {10}{q^{5/2}}}-{\frac {7}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {7}{q^{15/2}}}+{\frac {11}{q^{13/2}}}-{\frac {13}{q^{11/2}}}-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}-3z^{3}a^{7}-5za^{7}-3a^{7}z^{-1}+2z^{5}a^{5}+5z^{3}a^{5}+6za^{5}+4a^{5}z^{-1}+z^{5}a^{3}-3za^{3}-2a^{3}z^{-1}-z^{3}a-za$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+5z^{4}a^{10}-3z^{2}a^{10}+a^{10}-5z^{7}a^{9}+8z^{5}a^{9}-5z^{3}a^{9}+3za^{9}-a^{9}z^{-1}-4z^{8}a^{8}+10z^{4}a^{8}-8z^{2}a^{8}+3a^{8}-z^{9}a^{7}-13z^{7}a^{7}+33z^{5}a^{7}-31z^{3}a^{7}+14za^{7}-3a^{7}z^{-1}-8z^{8}a^{6}+7z^{6}a^{6}+7z^{4}a^{6}-9z^{2}a^{6}+3a^{6}-z^{9}a^{5}-13z^{7}a^{5}+34z^{5}a^{5}-34z^{3}a^{5}+17za^{5}-4a^{5}z^{-1}-4z^{8}a^{4}+z^{6}a^{4}+7z^{4}a^{4}-6z^{2}a^{4}+2a^{4}-5z^{7}a^{3}+9z^{5}a^{3}-8z^{3}a^{3}+6za^{3}-2a^{3}z^{-1}-3z^{6}a^{2}+5z^{4}a^{2}-2z^{2}a^{2}-z^{5}a+2z^{3}a-za$ (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 χ 2                     1 1 0                   2   -2 -2                 5 1   4 -4               6 3     -3 -6             8 4       4 -8           6 6         0 -10         7 8           -1 -12       5 7             2 -14     2 6               -4 -16   1 5                 4 -18   2                   -2 -20 1                     1

Integral Khovanov Homology (db, data source)

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

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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