L11n269

L11n268

L11n270

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Link Presentations

[edit Notes on L11n269's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,17,8,16 X15,5,16,8 X18,11,19,12 X22,17,9,18 X20,13,21,14 X12,19,13,20 X14,21,15,22 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -8, 7, -9, -4, 3, 6, -5, 8, -7, 9, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(3)^{5}-t(1)t(3)^{4}-t(2)t(3)^{4}+t(1)t(3)^{3}+t(2)t(3)^{3}-t(1)t(3)^{2}-t(2)t(3)^{2}+t(1)t(3)+t(2)t(3)-t(1)t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$-q^{-8}+q^{-7}-2q^{-6}+2q^{-5}-3q^{-4}+3q^{-3}+q^{2}+2q^{-1}+1$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{4}\right)-4a^{6}z^{2}-2a^{6}z^{-2}-5a^{6}+a^{4}z^{6}+7a^{4}z^{4}+18a^{4}z^{2}+7a^{4}z^{-2}+18a^{4}-a^{2}z^{6}-8a^{2}z^{4}-20a^{2}z^{2}-8a^{2}z^{-2}-20a^{2}+z^{4}+5z^{2}+3z^{-2}+7$ (db)
Kauffman polynomial	$a^{9}z^{5}-4a^{9}z^{3}+3a^{9}z-a^{9}z^{-1}+a^{8}z^{6}-3a^{8}z^{4}+a^{8}+a^{7}z^{7}-3a^{7}z^{5}+3a^{7}z-a^{7}z^{-1}+a^{6}z^{8}-5a^{6}z^{6}+9a^{6}z^{4}-9a^{6}z^{2}-2a^{6}z^{-2}+7a^{6}+2a^{5}z^{7}-12a^{5}z^{5}+26a^{5}z^{3}-21a^{5}z+7a^{5}z^{-1}+a^{4}z^{8}-8a^{4}z^{6}+24a^{4}z^{4}-32a^{4}z^{2}-7a^{4}z^{-2}+22a^{4}+2a^{3}z^{7}-17a^{3}z^{5}+46a^{3}z^{3}-45a^{3}z+15a^{3}z^{-1}+a^{2}z^{8}-10a^{2}z^{6}+33a^{2}z^{4}-46a^{2}z^{2}-8a^{2}z^{-2}+28a^{2}+az^{7}-9az^{5}+24az^{3}-24az+8az^{-1}+z^{8}-8z^{6}+21z^{4}-23z^{2}-3z^{-2}+13$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 5                       1 1 3                       1 1 1                   1     1 -1               3         3 -3             1 4 1       2 -5           3             3 -7         1 2 1           0 -9       2 3               -1 -11       1 1               0 -13   1 2                   -1 -15                         0 -17 1                       -1

Integral Khovanov Homology (db, data source)

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

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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