L10a99

L10a98

L10a100

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Link Presentations

[edit Notes on L10a99's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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