L11n350

L11n349

L11n351

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Link Presentations

[edit Notes on L11n350's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X11,21,12,20 X7,18,8,19 X9,16,10,17 X17,8,18,9 X19,13,20,22 X13,10,14,11 X21,5,22,12 X2536 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 6, -5, 8, -3, 9}, {-8, -2, 11, 5, -6, 4, -7, 3, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {-t(3)t(2)^{3}+t(2)^{3}-t(1)t(3)^{2}t(2)^{2}+t(3)^{2}t(2)^{2}+t(1)t(3)t(2)^{2}-t(3)t(2)^{2}-t(2)^{2}+t(1)t(3)^{3}t(2)+t(1)t(3)^{2}t(2)-t(3)^{2}t(2)-t(1)t(3)t(2)+t(3)t(2)-t(1)t(3)^{3}+t(1)t(3)^{2}}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+2q^{4}-2q^{3}+2q^{2}-q+2+q^{-1}-q^{-2}+2q^{-3}-q^{-4}+q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{2}a^{4}+a^{4}z^{-2}+2a^{4}-z^{4}a^{2}-4z^{2}a^{2}-2a^{2}z^{-2}-5a^{2}+z^{-2}+2+z^{4}a^{-2}+3z^{2}a^{-2}+2a^{-2}-z^{2}a^{-4}-a^{-4}$ (db)
Kauffman polynomial	$z^{5}a^{-5}-3z^{3}a^{-5}+za^{-5}+a^{4}z^{8}-7a^{4}z^{6}+2z^{6}a^{-4}+16a^{4}z^{4}-7z^{4}a^{-4}-15a^{4}z^{2}+4z^{2}a^{-4}-a^{4}z^{-2}+6a^{4}-a^{-4}+a^{3}z^{9}-6a^{3}z^{7}+z^{7}a^{-3}+9a^{3}z^{5}-3z^{5}a^{-3}-a^{3}z^{3}-5a^{3}z+za^{-3}+2a^{3}z^{-1}+3a^{2}z^{8}-21a^{2}z^{6}+z^{6}a^{-2}+44a^{2}z^{4}-4z^{4}a^{-2}-36a^{2}z^{2}+2z^{2}a^{-2}-2a^{2}z^{-2}+13a^{2}+az^{9}-6az^{7}+z^{7}a^{-1}+6az^{5}-7z^{5}a^{-1}+6az^{3}+10z^{3}a^{-1}-8az-3za^{-1}+2az^{-1}+2z^{8}-15z^{6}+31z^{4}-23z^{2}-z^{-2}+9$ (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                     1   1 7                   1 1   0 5               1 2 1     0 3             1 1 1       1 1           1 4 2         1 -1         1 2 4           3 -3       1 1               0 -5     1 1 1               1 -7   1 2                   1 -9                         0 -11 1                       1

Integral Khovanov Homology (db, data source)

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

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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