L10n83

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Link Presentations

[edit Notes on L10n83's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X5,16,6,17 X8493 X17,20,18,15 X19,10,20,11 X9,18,10,19 X15,14,16,5 X2,12,3,11
Gauss code	{1, -10, 5, -3}, {-9, 4, -6, 8, -7, 6}, {-4, -1, 2, -5, -8, 7, 10, -2, 3, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(w-1)^{2}(vw+1)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$1-2q^{-1}+5q^{-2}-4q^{-3}+6q^{-4}-5q^{-5}+5q^{-6}-3q^{-7}+q^{-8}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$z^{4}a^{6}+2z^{2}a^{6}+a^{6}z^{-2}+a^{6}-z^{6}a^{4}-4z^{4}a^{4}-5z^{2}a^{4}-2a^{4}z^{-2}-4a^{4}+z^{4}a^{2}+3z^{2}a^{2}+a^{2}z^{-2}+3a^{2}$ (db)
Kauffman polynomial	$z^{2}a^{10}-a^{10}+3z^{3}a^{9}-za^{9}+2z^{6}a^{8}-5z^{4}a^{8}+9z^{2}a^{8}-3a^{8}+3z^{7}a^{7}-9z^{5}a^{7}+10z^{3}a^{7}-3za^{7}+z^{8}a^{6}+3z^{6}a^{6}-18z^{4}a^{6}+20z^{2}a^{6}+a^{6}z^{-2}-7a^{6}+5z^{7}a^{5}-15z^{5}a^{5}+9z^{3}a^{5}+za^{5}-2a^{5}z^{-1}+z^{8}a^{4}+2z^{6}a^{4}-17z^{4}a^{4}+18z^{2}a^{4}+2a^{4}z^{-2}-8a^{4}+2z^{7}a^{3}-6z^{5}a^{3}+2z^{3}a^{3}+3za^{3}-2a^{3}z^{-1}+z^{6}a^{2}-4z^{4}a^{2}+6z^{2}a^{2}+a^{2}z^{-2}-4a^{2}$ (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 χ 1                 1 1 -1               1   -1 -3             4 1   3 -5           2 3     1 -7         4 2       2 -9     1 2 2         1 -11     4 4           0 -13   1 3             2 -15   2               -2 -17 1                 1

Integral Khovanov Homology (db, data source)

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

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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