L10a27

L10a26

L10a28

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Link Presentations

[edit Notes on L10a27's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-2t(2)^{3}+t(2)^{2}-2t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{15/2}+3q^{13/2}-5q^{11/2}+7q^{9/2}-9q^{7/2}+9q^{5/2}-8q^{3/2}+6{\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-5}-3z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+z^{7}a^{-3}+5z^{5}a^{-3}+9z^{3}a^{-3}+8za^{-3}+3a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-8z^{3}a^{-1}+3az-9za^{-1}+2az^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-9}+3z^{4}a^{-8}-z^{2}a^{-8}+5z^{5}a^{-7}-4z^{3}a^{-7}+za^{-7}+6z^{6}a^{-6}-7z^{4}a^{-6}+2z^{2}a^{-6}-a^{-6}+6z^{7}a^{-5}-10z^{5}a^{-5}+3z^{3}a^{-5}-2za^{-5}+a^{-5}z^{-1}+4z^{8}a^{-4}-5z^{6}a^{-4}-8z^{4}a^{-4}+10z^{2}a^{-4}-3a^{-4}+z^{9}a^{-3}+7z^{7}a^{-3}-32z^{5}a^{-3}+34z^{3}a^{-3}-15za^{-3}+3a^{-3}z^{-1}+6z^{8}a^{-2}-19z^{6}a^{-2}+10z^{4}a^{-2}+7z^{2}a^{-2}-3a^{-2}+z^{9}a^{-1}+az^{7}+2z^{7}a^{-1}-5az^{5}-22z^{5}a^{-1}+9az^{3}+35z^{3}a^{-1}-7az-19za^{-1}+2az^{-1}+4a^{-1}z^{-1}+2z^{8}-8z^{6}+8z^{4}-2$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 16                     1 1 14                   2   -2 12                 3 1   2 10               4 2     -2 8             5 3       2 6           4 4         0 4         4 5           -1 2       4 6             2 0     1 2               -1 -2   1 4                 3 -4   1                   -1 -6 1                     1

Integral Khovanov Homology (db, data source)

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

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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