L11n291

L11n290

L11n292

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Link Presentations

[edit Notes on L11n291's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(2)^{2}t(3)^{4}+t(1)t(2)t(3)^{4}+2t(1)t(2)^{2}t(3)^{3}-2t(1)t(2)t(3)^{3}+t(2)t(3)^{3}-t(3)^{3}-2t(1)t(2)^{2}t(3)^{2}+3t(1)t(2)t(3)^{2}-3t(2)t(3)^{2}+2t(3)^{2}+t(1)t(2)^{2}t(3)-t(1)t(2)t(3)+2t(2)t(3)-2t(3)-t(2)+1}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{3}-3q^{2}+5q-7+9q^{-1}-8q^{-2}+9q^{-3}-5q^{-4}+4q^{-5}-q^{-6}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{2}z^{8}+a^{4}z^{6}-6a^{2}z^{6}+z^{6}+5a^{4}z^{4}-12a^{2}z^{4}+4z^{4}-a^{6}z^{2}+7a^{4}z^{2}-9a^{2}z^{2}+4z^{2}-a^{6}+a^{4}-a^{2}+1+a^{6}z^{-2}-2a^{4}z^{-2}+a^{2}z^{-2}$ (db)
Kauffman polynomial	$2a^{3}z^{9}+2az^{9}+4a^{4}z^{8}+8a^{2}z^{8}+4z^{8}+2a^{5}z^{7}-3a^{3}z^{7}-2az^{7}+3z^{7}a^{-1}-17a^{4}z^{6}-32a^{2}z^{6}+z^{6}a^{-2}-14z^{6}-5a^{5}z^{5}-7a^{3}z^{5}-12az^{5}-10z^{5}a^{-1}+4a^{6}z^{4}+32a^{4}z^{4}+44a^{2}z^{4}-3z^{4}a^{-2}+13z^{4}+a^{7}z^{3}+9a^{5}z^{3}+19a^{3}z^{3}+18az^{3}+7z^{3}a^{-1}-5a^{6}z^{2}-21a^{4}z^{2}-24a^{2}z^{2}+z^{2}a^{-2}-7z^{2}-a^{7}z-3a^{5}z-7a^{3}z-7az-2za^{-1}+3a^{4}+4a^{2}+2-2a^{5}z^{-1}-2a^{3}z^{-1}+a^{6}z^{-2}+2a^{4}z^{-2}+a^{2}z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                 2   -2 3               3 1   2 1             4 2     -2 -1           5 3       2 -3         4 5         1 -5       5 4           1 -7     2 6             4 -9   2 3               -1 -11   3                 3 -13 1                   -1

Integral Khovanov Homology (db, data source)

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

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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