L10a148

L10a147

L10a149

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Link Presentations

[edit Notes on L10a148's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X16,5,17,6 X12,15,5,16 X8,20,9,19 X18,8,19,7 X20,10,13,9 X10,14,11,13 X2,11,3,12 X4,18,1,17
Gauss code	{1, -9, 2, -10}, {3, -1, 6, -5, 7, -8, 9, -4}, {8, -2, 4, -3, 10, -6, 5, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{3}w^{3}-uv^{3}w^{2}-uv^{2}w^{3}+3uv^{2}w^{2}-uv^{2}w-2uvw^{2}+3uvw-uv-uw+u-v^{3}w^{3}+v^{3}w^{2}+v^{2}w^{3}-3v^{2}w^{2}+2v^{2}w+vw^{2}-3vw+v+w-1}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+5q^{5}-7q^{4}+10q^{3}+q^{-3}-9q^{2}-3q^{-2}+10q+5q^{-1}-6$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}+4z^{4}a^{-4}+4z^{2}a^{-4}+a^{-4}z^{-2}+a^{-4}-z^{8}a^{-2}-6z^{6}a^{-2}-12z^{4}a^{-2}-9z^{2}a^{-2}-2a^{-2}z^{-2}-3a^{-2}+z^{6}+4z^{4}+4z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$2z^{9}a^{-1}+2z^{9}a^{-3}+8z^{8}a^{-2}+4z^{8}a^{-4}+4z^{8}+3az^{7}-3z^{7}a^{-1}-2z^{7}a^{-3}+4z^{7}a^{-5}+a^{2}z^{6}-30z^{6}a^{-2}-10z^{6}a^{-4}+4z^{6}a^{-6}-15z^{6}-10az^{5}-5z^{5}a^{-1}-3z^{5}a^{-3}-5z^{5}a^{-5}+3z^{5}a^{-7}-3a^{2}z^{4}+41z^{4}a^{-2}+15z^{4}a^{-4}-5z^{4}a^{-6}+z^{4}a^{-8}+17z^{4}+6az^{3}+7z^{3}a^{-1}+7z^{3}a^{-3}+2z^{3}a^{-5}-4z^{3}a^{-7}+a^{2}z^{2}-22z^{2}a^{-2}-10z^{2}a^{-4}+z^{2}a^{-6}-z^{2}a^{-8}-9z^{2}-3za^{-1}-3za^{-3}+5a^{-2}+3a^{-4}+3+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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 15                     1 1 13                   2   -2 11                 3 1   2 9               5 3     -2 7             5 2       3 5           4 5         1 3         6 5           1 1       3 7             4 -1     2 3               -1 -3   1 3                 2 -5   2                   -2 -7 1                     1

Integral Khovanov Homology (db, data source)

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

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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