L10n89

L10n88

L10n90

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Link Presentations

[edit Notes on L10n89's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X20,15,17,16 X18,11,19,12 X12,17,13,18 X16,19,5,20 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {6, -5, 7, -4}, {9, -1, -3, 8, 10, -2, 5, -6, -8, 3, 4, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(3)-1)\left(t(1)t(3)^{2}+t(2)t(3)^{2}-t(3)^{2}-t(1)t(3)+2t(1)t(2)t(3)-t(2)t(3)+2t(3)+t(1)-t(1)t(2)+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$2q^{-10}-4q^{-9}+7q^{-8}-8q^{-7}+9q^{-6}-7q^{-5}+7q^{-4}-3q^{-3}+q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}+2a^{10}-5z^{2}a^{8}-2a^{8}z^{-2}-8a^{8}+3z^{4}a^{6}+8z^{2}a^{6}+a^{6}z^{-2}+6a^{6}+z^{4}a^{4}+z^{2}a^{4}$ (db)
Kauffman polynomial	$3a^{12}z^{4}-5a^{12}z^{2}+2a^{12}+a^{11}z^{7}+a^{11}z^{5}-2a^{11}z^{3}+a^{10}z^{8}+2a^{10}z^{6}-3a^{10}z^{4}+a^{10}z^{-2}-2a^{10}+5a^{9}z^{7}-3a^{9}z^{5}-8a^{9}z^{3}+8a^{9}z-2a^{9}z^{-1}+a^{8}z^{8}+8a^{8}z^{6}-18a^{8}z^{4}+16a^{8}z^{2}+2a^{8}z^{-2}-9a^{8}+4a^{7}z^{7}-a^{7}z^{5}-8a^{7}z^{3}+8a^{7}z-2a^{7}z^{-1}+6a^{6}z^{6}-11a^{6}z^{4}+10a^{6}z^{2}+a^{6}z^{-2}-6a^{6}+3a^{5}z^{5}-2a^{5}z^{3}+a^{4}z^{4}-a^{4}z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                 1 1 -5               3 1 -2 -7             4     4 -9           3 3     0 -11         6 4       2 -13       4 5         1 -15     3 4           -1 -17   1 4             3 -19 1 3               -2 -21 2                 2

Integral Khovanov Homology (db, data source)

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

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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