L10a6

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Link Presentations

[edit Notes on L10a6's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X12,6,13,5 X8493 X16,13,17,14 X14,9,15,10 X10,15,11,16 X20,12,5,11 X2,18,3,17
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -5, 7, -8, 9, -4, 6, -7, 8, -6, 10, -2, 3, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{4}-3v^{3}+5v^{2}-3v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}-q^{7/2}+{\frac {13}{q^{7/2}}}+4q^{5/2}-{\frac {17}{q^{5/2}}}-9q^{3/2}+{\frac {17}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+13{\sqrt {q}}-{\frac {17}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z-2a^{3}z^{5}-5a^{3}z^{3}-4a^{3}z+az^{7}+4az^{5}-z^{5}a^{-1}+7az^{3}-2z^{3}a^{-1}+5az-2za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-a^{7}z^{3}+4a^{6}z^{6}-6a^{6}z^{4}+3a^{6}z^{2}+7a^{5}z^{7}-11a^{5}z^{5}+7a^{5}z^{3}-2a^{5}z+6a^{4}z^{8}-a^{4}z^{6}-12a^{4}z^{4}+8a^{4}z^{2}+2a^{3}z^{9}+16a^{3}z^{7}-39a^{3}z^{5}+z^{5}a^{-3}+28a^{3}z^{3}-z^{3}a^{-3}-8a^{3}z+13a^{2}z^{8}-14a^{2}z^{6}+4z^{6}a^{-2}-7a^{2}z^{4}-5z^{4}a^{-2}+7a^{2}z^{2}+2z^{2}a^{-2}+2az^{9}+17az^{7}+8z^{7}a^{-1}-42az^{5}-14z^{5}a^{-1}+31az^{3}+10z^{3}a^{-1}-10az-4za^{-1}+az^{-1}+a^{-1}z^{-1}+7z^{8}-5z^{6}-6z^{4}+4z^{2}-1$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   3   -3 4                 6 1   5 2               7 3     -4 0             10 6       4 -2           9 9         0 -4         8 8           0 -6       5 9             4 -8     3 8               -5 -10   1 5                 4 -12   3                   -3 -14 1                     1

Integral Khovanov Homology (db, data source)

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

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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