L11n195

L11n194

L11n196

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Link Presentations

[edit Notes on L11n195's Link Presentations]

Planar diagram presentation	X8192 X2,11,3,12 X12,3,13,4 X16,5,17,6 X6718 X4,15,5,16 X20,14,21,13 X9,18,10,19 X19,10,20,11 X22,18,7,17 X14,22,15,21
Gauss code	{1, -2, 3, -6, 4, -5}, {5, -1, -8, 9, 2, -3, 7, -11, 6, -4, 10, 8, -9, -7, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-2t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}-t(2)^{3}+3t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+3t(2)^{2}-t(1)^{2}t(2)+4t(1)t(2)-2t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {1}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {9}{q^{7/2}}}+{\frac {10}{q^{9/2}}}-{\frac {10}{q^{11/2}}}+{\frac {8}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+3z^{3}a^{7}+3za^{7}+a^{7}z^{-1}-z^{7}a^{5}-5z^{5}a^{5}-9z^{3}a^{5}-6za^{5}-a^{5}z^{-1}+z^{5}a^{3}+2z^{3}a^{3}$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-6a^{10}z^{4}+2a^{10}z^{2}+4a^{9}z^{7}-7a^{9}z^{5}+2a^{9}z^{3}-a^{9}z+3a^{8}z^{8}-3a^{8}z^{6}-a^{8}z^{4}+a^{7}z^{9}+3a^{7}z^{7}-5a^{7}z^{5}-a^{7}z^{3}+4a^{7}z-a^{7}z^{-1}+4a^{6}z^{8}-6a^{6}z^{6}+6a^{6}z^{4}-3a^{6}z^{2}+a^{6}+a^{5}z^{9}-a^{5}z^{7}+7a^{5}z^{5}-10a^{5}z^{3}+6a^{5}z-a^{5}z^{-1}+a^{4}z^{8}+2a^{4}z^{4}-2a^{4}z^{2}+4a^{3}z^{5}-5a^{3}z^{3}+a^{2}z^{4}-a^{2}z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 0                   1 -1 -2                 3   3 -4               4 2   -2 -6             5 2     3 -8           5 4       -1 -10         5 5         0 -12       3 5           2 -14     3 5             -2 -16   1 4               3 -18   2                 -2 -20 1                   1

Integral Khovanov Homology (db, data source)

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

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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