L10n72

L10n71

L10n73

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Link Presentations

[edit Notes on L10n72's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,17,8,16 X9,11,10,20 X11,18,12,19 X15,9,16,8 X19,5,20,10 X17,14,18,15 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, -3, 6, -4, 7}, {-5, -2, 10, 8, -6, 3, -8, 5, -7, 4}

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

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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