L10a44

L10a43

L10a45

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Link Presentations

[edit Notes on L10a44's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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