L10n110

L10n109

L10n111

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Link Presentations

[edit Notes on L10n110's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X13,17,14,20 X19,11,20,16 X7,19,8,18 X15,8,16,9 X9,14,10,15 X17,5,18,10 X2536 X11,1,12,4
Gauss code	{1, -9, -2, 10}, {-8, 5, -4, 3}, {9, -1, -5, 6, -7, 8}, {-10, 2, -3, 7, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{2}-t(1)t(2)t(3)^{2}-t(1)t(4)t(3)^{2}+t(1)t(2)t(4)t(3)^{2}+t(4)t(3)^{2}-t(1)t(4)^{2}t(3)-t(2)t(4)^{2}t(3)+t(4)^{2}t(3)-t(1)t(3)+t(1)t(2)t(3)-t(2)t(3)+2t(1)t(4)t(3)-2t(1)t(2)t(4)t(3)+2t(2)t(4)t(3)-2t(4)t(3)+t(2)t(4)^{2}-t(4)^{2}+t(1)t(2)t(4)-t(2)t(4)+t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$-6q^{9/2}+6q^{7/2}-10q^{5/2}+7q^{3/2}-{\frac {3}{q^{3/2}}}-q^{13/2}+3q^{11/2}-8{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-a^{-5}z^{-3}-za^{-5}-2a^{-5}z^{-1}+z^{5}a^{-3}+3z^{3}a^{-3}+3a^{-3}z^{-3}+7za^{-3}+7a^{-3}z^{-1}-4z^{3}a^{-1}+az^{-3}-3a^{-1}z^{-3}+3az-9za^{-1}+3az^{-1}-8a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{8}a^{-2}-2z^{8}a^{-4}-5z^{7}a^{-1}-9z^{7}a^{-3}-4z^{7}a^{-5}-z^{6}a^{-2}-z^{6}a^{-4}-3z^{6}a^{-6}-3z^{6}+17z^{5}a^{-1}+27z^{5}a^{-3}+9z^{5}a^{-5}-z^{5}a^{-7}+11z^{4}a^{-2}+11z^{4}a^{-4}+6z^{4}a^{-6}+6z^{4}-6az^{3}-33z^{3}a^{-1}-36z^{3}a^{-3}-7z^{3}a^{-5}+2z^{3}a^{-7}-23z^{2}a^{-2}-13z^{2}a^{-4}-z^{2}a^{-6}-11z^{2}+11az+27za^{-1}+25za^{-3}+8za^{-5}-za^{-7}+19a^{-2}+10a^{-4}+10-5az^{-1}-12a^{-1}z^{-1}-12a^{-3}z^{-1}-5a^{-5}z^{-1}-6a^{-2}z^{-2}-3a^{-4}z^{-2}-3z^{-2}+az^{-3}+3a^{-1}z^{-3}+3a^{-3}z^{-3}+a^{-5}z^{-3}$ (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   \ -2 -1 0 1 2 3 4 5 6 χ 14                 1 1 12               2   -2 10             4 1   3 8           4 4     0 6         6 2       4 4       3 6         3 2     5 4           1 0   1 5             4 -2 2 3               -1 -4 3                 3

Integral Khovanov Homology (db, data source)

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

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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