L10n85

L10n84

L10n86

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Link Presentations

[edit Notes on L10n85's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X20,14,15,13 X7,16,8,17 X15,8,16,9 X18,12,19,11 X12,20,13,19 X14,18,5,17 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {-5, 4, 8, -6, 7, -3}, {9, -1, -4, 5, 10, -2, 6, -7, 3, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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