L11n28

L11n27

L11n29

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Link Presentations

[edit Notes on L11n28's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X8493 X13,22,14,5 X21,14,22,15 X11,18,12,19 X9,20,10,21 X19,10,20,11 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -9, 10, -8, 4, -6, 7, 11, -2, 3, 8, -10, 9, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {1}{q^{13/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}(-z)-a^{9}z^{-1}+a^{7}z^{3}+3a^{7}z+a^{7}z^{-1}-a^{5}z^{3}+2a^{5}z^{-1}+a^{3}z^{5}+2a^{3}z^{3}-a^{3}z-2a^{3}z^{-1}-az^{3}-az$ (db)
Kauffman polynomial	$a^{10}z^{8}-7a^{10}z^{6}+15a^{10}z^{4}-13a^{10}z^{2}+4a^{10}+a^{9}z^{9}-7a^{9}z^{7}+13a^{9}z^{5}-7a^{9}z^{3}-a^{9}z^{-1}+2a^{8}z^{8}-17a^{8}z^{6}+40a^{8}z^{4}-32a^{8}z^{2}+9a^{8}+a^{7}z^{9}-7a^{7}z^{7}+11a^{7}z^{5}+a^{7}z^{3}-a^{7}z-a^{7}z^{-1}+2a^{6}z^{8}-13a^{6}z^{6}+28a^{6}z^{4}-18a^{6}z^{2}+4a^{6}+3a^{5}z^{7}-10a^{5}z^{5}+12a^{5}z^{3}-5a^{5}z+2a^{5}z^{-1}+a^{4}z^{8}-4a^{4}z^{4}+2a^{4}z^{2}-2a^{4}+3a^{3}z^{7}-7a^{3}z^{5}+2a^{3}z^{3}-3a^{3}z+2a^{3}z^{-1}+3a^{2}z^{6}-7a^{2}z^{4}+a^{2}z^{2}+az^{5}-2az^{3}+az$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     2   -2 -2                   3 1   2 -4                 3 4     1 -6             1 4 1       2 -8             1 3         2 -10         1 4 4           -1 -12       1   1             2 -14       1 3               -2 -16   1 1                   0 -18                         0 -20 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $i=0$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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