L10n80

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Link Presentations

[edit Notes on L10n80's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,18,12,19 X7,14,8,15 X13,8,14,9 X15,13,16,20 X19,17,20,16 X17,12,18,5 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, -4, 5, 10, -2, -3, 8}, {-5, 4, -6, 7, -8, 3, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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