L11n421

L11n420

L11n422

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Link Presentations

[edit Notes on L11n421's Link Presentations]

Planar diagram presentation	X8192 X16,8,17,7 X3,10,4,11 X17,2,18,3 X9,19,10,18 X20,12,21,11 X14,6,15,5 X22,15,13,16 X6,14,1,13 X4,19,5,20 X12,22,7,21
Gauss code	{1, 4, -3, -10, 7, -9}, {2, -1, -5, 3, 6, -11}, {9, -7, 8, -2, -4, 5, 10, -6, 11, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{2}t(3)^{3}-t(2)^{2}t(3)^{3}-2t(1)t(2)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}+t(1)t(2)t(3)^{2}-t(1)^{2}t(3)+2t(1)t(3)-t(1)t(2)t(3)+t(1)^{2}-t(1)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{5}-q^{4}+3q^{3}-3q^{2}+4q-3+4q^{-1}-2q^{-2}+2q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}-a^{2}z^{4}-2z^{4}a^{-2}-3a^{2}z^{2}-8z^{2}a^{-2}-2a^{-2}z^{-2}-a^{2}-8a^{-2}+z^{6}+5z^{4}+8z^{2}+z^{-2}+6$ (db)
Kauffman polynomial	$z^{2}a^{-6}-a^{-6}+z^{3}a^{-5}+2z^{4}a^{-4}-4z^{2}a^{-4}-a^{-4}z^{-2}+4a^{-4}+a^{3}z^{7}+z^{7}a^{-3}-5a^{3}z^{5}-4z^{5}a^{-3}+6a^{3}z^{3}+7z^{3}a^{-3}-a^{3}z-6za^{-3}+2a^{-3}z^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}-11a^{2}z^{6}-11z^{6}a^{-2}+18a^{2}z^{4}+23z^{4}a^{-2}-11a^{2}z^{2}-25z^{2}a^{-2}-2a^{-2}z^{-2}+2a^{2}+12a^{-2}+az^{9}+z^{9}a^{-1}-3az^{7}-3z^{7}a^{-1}-3az^{5}-2z^{5}a^{-1}+9az^{3}+9z^{3}a^{-1}-3az-8za^{-1}+2a^{-1}z^{-1}+4z^{8}-22z^{6}+39z^{4}-31z^{2}-z^{-2}+10$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 11                   1 1 9                 1 1 0 7               2     2 5             2 2     0 3           2 1       1 1         2 3         1 -1       2 1           1 -3     1 3             2 -5   1 1               0 -7   1                 1 -9 1                   -1

Integral Khovanov Homology (db, data source)

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

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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