L11n320

L11n319

L11n321

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Link Presentations

[edit Notes on L11n320's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X9,18,10,19 X4,17,1,18 X19,13,20,22 X13,10,14,11 X21,5,22,12 X11,21,12,20
Gauss code	{1, 4, -3, -7}, {-2, -1, 5, 3, -6, 9, -11, 10}, {-9, 2, -4, -5, 7, 6, -8, 11, -10, 8}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{3}t(2)^{3}-t(1)t(3)^{2}t(2)^{3}-t(1)t(3)^{3}t(2)^{2}+2t(1)t(3)^{2}t(2)^{2}-2t(3)t(2)+t(2)+t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q+2-2q^{-1}+3q^{-2}-2q^{-3}+4q^{-4}-2q^{-5}+2q^{-6}-q^{-7}+q^{-8}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+a^{8}z^{-2}+2a^{8}-a^{6}z^{6}-6a^{6}z^{4}-11a^{6}z^{2}-2a^{6}z^{-2}-9a^{6}+a^{4}z^{8}+7a^{4}z^{6}+17a^{4}z^{4}+19a^{4}z^{2}+a^{4}z^{-2}+9a^{4}-a^{2}z^{6}-5a^{2}z^{4}-6a^{2}z^{2}-2a^{2}$ (db)
Kauffman polynomial	$z^{2}a^{10}-a^{10}+z^{3}a^{9}-za^{9}+z^{4}a^{8}-2z^{2}a^{8}-a^{8}z^{-2}+3a^{8}+z^{7}a^{7}-5z^{5}a^{7}+9z^{3}a^{7}-8za^{7}+2a^{7}z^{-1}+2z^{8}a^{6}-12z^{6}a^{6}+25z^{4}a^{6}-25z^{2}a^{6}-2a^{6}z^{-2}+11a^{6}+z^{9}a^{5}-3z^{7}a^{5}-5z^{5}a^{5}+17z^{3}a^{5}-12za^{5}+2a^{5}z^{-1}+4z^{8}a^{4}-23z^{6}a^{4}+41z^{4}a^{4}-32z^{2}a^{4}-a^{4}z^{-2}+11a^{4}+z^{9}a^{3}-3z^{7}a^{3}-5z^{5}a^{3}+15z^{3}a^{3}-7za^{3}+2z^{8}a^{2}-11z^{6}a^{2}+17z^{4}a^{2}-10z^{2}a^{2}+3a^{2}+z^{7}a-5z^{5}a+6z^{3}a-2za$ (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 χ 3                   1 -1 1                 1   1 -1               1 1   0 -3           1 3 1     1 -5         1 2 2       1 -7         4 2         2 -9     2 2 2           2 -11   1 4 3             0 -13   1 2               1 -15 1 1                 0 -17 1                   1

Integral Khovanov Homology (db, data source)

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

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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