L11n267

L11n266

L11n268

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Link Presentations

[edit Notes on L11n267's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,19,12,18 X15,21,16,20 X17,9,18,22 X21,17,22,16 X19,13,20,12 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, -5, 9, 4, -3, -6, 8, -7, 5, -9, 6, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uvw^{3}-uvw^{2}+uv+uw^{5}-uw^{4}+uw-u+vw^{5}-vw^{4}+vw-v-w^{5}+w^{3}-w^{2}}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{5}+q^{4}-q^{3}-q^{2}+q-1+4q^{-1}-2q^{-2}+4q^{-3}-q^{-4}+q^{-5}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$2z^{6}-3a^{2}z^{4}-z^{4}a^{-2}+12z^{4}+a^{4}z^{2}-14a^{2}z^{2}-6z^{2}a^{-2}-z^{2}a^{-4}+23z^{2}+4a^{4}-18a^{2}-7a^{-2}+21+3a^{4}z^{-2}-8a^{2}z^{-2}-2a^{-2}z^{-2}+7z^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+a^{4}z^{8}+5a^{2}z^{8}+4z^{8}-4a^{3}z^{7}-az^{7}+3z^{7}a^{-1}-7a^{4}z^{6}-31a^{2}z^{6}-z^{6}a^{-2}+z^{6}a^{-4}-26z^{6}-2a^{3}z^{5}-21az^{5}-20z^{5}a^{-1}+z^{5}a^{-5}+18a^{4}z^{4}+61a^{2}z^{4}+3z^{4}a^{-2}-4z^{4}a^{-4}+50z^{4}+21a^{3}z^{3}+53az^{3}+33z^{3}a^{-1}-3z^{3}a^{-3}-4z^{3}a^{-5}-22a^{4}z^{2}-54a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}-42z^{2}-24a^{3}z-45az-21za^{-1}+3za^{-3}+3za^{-5}+13a^{4}+28a^{2}+7a^{-2}+a^{-4}+22+8a^{3}z^{-1}+15az^{-1}+7a^{-1}z^{-1}-a^{-3}z^{-1}-a^{-5}z^{-1}-3a^{4}z^{-2}-8a^{2}z^{-2}-2a^{-2}z^{-2}-7z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                         0 7                   1 1   0 5               3 1       -2 3             1 2 1       0 1           3 4 1         0 -1         1 1 3           3 -3       1 3               2 -5     3 1                 2 -7   1 4                   3 -9                         0 -11 1                       1

Integral Khovanov Homology (db, data source)

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

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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