L10n74

L10n73

L10n75

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Link Presentations

[edit Notes on L10n74's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X9,18,10,19 X4,17,1,18 X19,12,20,5 X11,20,12,13 X13,10,14,11
Gauss code	{1, 4, -3, -7}, {-2, -1, 5, 3, -6, 10, -9, 8}, {-10, 2, -4, -5, 7, 6, -8, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{3}w^{2}+uv^{2}w^{3}-uv^{2}w^{2}+vw-v-w}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{-3}-q^{-4}+2q^{-5}-q^{-6}+2q^{-7}-2q^{-8}+2q^{-9}+q^{-11}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{12}z^{-2}+a^{12}-z^{4}a^{10}-5z^{2}a^{10}-2a^{10}z^{-2}-5a^{10}+z^{6}a^{8}+5z^{4}a^{8}+6z^{2}a^{8}+a^{8}z^{-2}+2a^{8}+z^{6}a^{6}+5z^{4}a^{6}+6z^{2}a^{6}+2a^{6}$ (db)
Kauffman polynomial	$a^{14}z^{2}-2a^{14}+a^{13}z+a^{12}z^{4}-3a^{12}z^{2}-a^{12}z^{-2}+3a^{12}+a^{11}z^{7}-6a^{11}z^{5}+12a^{11}z^{3}-8a^{11}z+2a^{11}z^{-1}+a^{10}z^{8}-6a^{10}z^{6}+13a^{10}z^{4}-15a^{10}z^{2}-2a^{10}z^{-2}+9a^{10}+2a^{9}z^{7}-10a^{9}z^{5}+14a^{9}z^{3}-8a^{9}z+2a^{9}z^{-1}+a^{8}z^{8}-5a^{8}z^{6}+7a^{8}z^{4}-5a^{8}z^{2}-a^{8}z^{-2}+3a^{8}+a^{7}z^{7}-4a^{7}z^{5}+2a^{7}z^{3}+a^{7}z+a^{6}z^{6}-5a^{6}z^{4}+6a^{6}z^{2}-2a^{6}$ (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 -5                 1 1 -7               1 1 0 -9             1     1 -11         1 1 1     1 -13       1 3 1       1 -15       1 1         0 -17     2 2           0 -19 1   1             2 -21 2 1               1 -23 1                 1

Integral Khovanov Homology (db, data source)

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

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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