L11n270

L11n269

L11n271

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Link Presentations

[edit Notes on L11n270's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X11,19,12,18 X17,9,18,22 X13,21,14,20 X19,13,20,12 X21,15,22,14 X2536 X4,9,1,10
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 $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{5}+t(2)t(3)^{5}-t(3)^{5}-t(1)t(3)^{4}-t(2)t(3)^{4}+t(1)t(3)^{3}+t(2)t(3)^{3}-t(1)t(3)^{2}-t(2)t(3)^{2}+t(1)t(3)+t(2)t(3)-t(1)+t(1)t(2)-t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{-5}-2q^{4}-q^{-4}+2q^{3}+4q^{-3}-3q^{2}-2q^{-2}+4q+5q^{-1}-4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$2z^{6}-3a^{2}z^{4}-2z^{4}a^{-2}+12z^{4}+a^{4}z^{2}-14a^{2}z^{2}-8z^{2}a^{-2}+24z^{2}+4a^{4}-18a^{2}-7a^{-2}+21+3a^{4}z^{-2}-8a^{2}z^{-2}-2a^{-2}z^{-2}+7z^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+a^{4}z^{8}+5a^{2}z^{8}+4z^{8}-4a^{3}z^{7}+4z^{7}a^{-1}-7a^{4}z^{6}-30a^{2}z^{6}+2z^{6}a^{-2}-21z^{6}-2a^{3}z^{5}-21az^{5}-17z^{5}a^{-1}+2z^{5}a^{-3}+18a^{4}z^{4}+59a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}+38z^{4}+22a^{3}z^{3}+50az^{3}+24z^{3}a^{-1}-4z^{3}a^{-3}-22a^{4}z^{2}-51a^{2}z^{2}-8z^{2}a^{-2}-37z^{2}-24a^{3}z-45az-21za^{-1}+3za^{-3}+3za^{-5}+13a^{4}+28a^{2}+7a^{-2}+a^{-4}+22+8a^{3}z^{-1}+15az^{-1}+7a^{-1}z^{-1}-a^{-3}z^{-1}-a^{-5}z^{-1}-3a^{4}z^{-2}-8a^{2}z^{-2}-2a^{-2}z^{-2}-7z^{-2}$ (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 χ 9                   2 -2 7                 1 1 0 5               3 2   -1 3             1 1 1   1 1           4 4       0 -1         1 1 1       1 -3       1 4           3 -5     3 1             2 -7   1 4               3 -9                     0 -11 1                   1

Integral Khovanov Homology (db, data source)

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

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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