L10a46

L10a45

L10a47

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Link Presentations

[edit Notes on L10a46's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X18,8,19,7 X20,10,5,9 X8,20,9,19 X12,18,13,17 X16,12,17,11 X10,16,11,15 X2536 X4,13,1,14
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -5, 4, -8, 7, -6, 10, -2, 8, -7, 6, -3, 5, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {\left(t(2)^{2}-t(2)+1\right)\left(t(1)t(2)^{3}-2t(2)^{3}-t(1)t(2)^{2}+t(2)^{2}+t(1)t(2)-t(2)-2t(1)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$9q^{9/2}-9q^{7/2}+10q^{5/2}-{\frac {1}{q^{5/2}}}-9q^{3/2}+{\frac {1}{q^{3/2}}}-q^{15/2}+3q^{13/2}-6q^{11/2}+6{\sqrt {q}}-{\frac {5}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-5}-3z^{3}a^{-5}-3za^{-5}-a^{-5}z^{-1}+z^{7}a^{-3}+5z^{5}a^{-3}+10z^{3}a^{-3}+11za^{-3}+5a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-9z^{3}a^{-1}+4az-14za^{-1}+4az^{-1}-8a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-9}+3z^{4}a^{-8}+6z^{5}a^{-7}-4z^{3}a^{-7}+za^{-7}+9z^{6}a^{-6}-15z^{4}a^{-6}+9z^{2}a^{-6}-2a^{-6}+8z^{7}a^{-5}-14z^{5}a^{-5}+4z^{3}a^{-5}+a^{-5}z^{-1}+4z^{8}a^{-4}-22z^{4}a^{-4}+23z^{2}a^{-4}-9a^{-4}+z^{9}a^{-3}+7z^{7}a^{-3}-26z^{5}a^{-3}+20z^{3}a^{-3}-9za^{-3}+5a^{-3}z^{-1}+5z^{8}a^{-2}-11z^{6}a^{-2}-9z^{4}a^{-2}+28z^{2}a^{-2}-14a^{-2}+z^{9}a^{-1}+az^{7}-6az^{5}-12z^{5}a^{-1}+13az^{3}+24z^{3}a^{-1}-12az-20za^{-1}+4az^{-1}+8a^{-1}z^{-1}+z^{8}-2z^{6}-5z^{4}+14z^{2}-8$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 16                     1 1 14                   2   -2 12                 4 1   3 10               5 2     -3 8             4 4       0 6           6 5         -1 4         3 4           -1 2       4 7             3 0     1 2               -1 -2     4                 4 -4 1 1                   0 -6 1                     1

Integral Khovanov Homology (db, data source)

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

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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