L11n125

L11n124

L11n126

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Link Presentations

[edit Notes on L11n125's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X9,22,10,5 X7,19,8,18 X17,9,18,8 X19,13,20,12 X11,21,12,20 X15,10,16,11 X21,16,22,17 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, -3, 8, -7, 6, 11, -2, -8, 9, -5, 4, -6, 7, -9, 3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{5}-t(1)t(2)^{4}-3t(2)^{4}+4t(2)^{3}+4t(1)t(2)^{2}-3t(1)t(2)-t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-q^{5/2}-{\frac {5}{q^{5/2}}}+q^{3/2}+{\frac {4}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {4}{q^{11/2}}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+2z^{3}a^{5}+3za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-8za^{3}-4a^{3}z^{-1}+z^{5}a+6z^{3}a+7za+3az^{-1}-z^{3}a^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-3a^{6}z^{8}-4a^{4}z^{8}-2a^{2}z^{8}-z^{8}-3a^{7}z^{7}-a^{5}z^{7}+2a^{3}z^{7}-az^{7}-z^{7}a^{-1}-a^{8}z^{6}+10a^{6}z^{6}+18a^{4}z^{6}+14a^{2}z^{6}+7z^{6}+11a^{7}z^{5}+18a^{5}z^{5}+12a^{3}z^{5}+11az^{5}+6z^{5}a^{-1}+3a^{8}z^{4}-5a^{6}z^{4}-23a^{4}z^{4}-27a^{2}z^{4}-12z^{4}-9a^{7}z^{3}-27a^{5}z^{3}-34a^{3}z^{3}-25az^{3}-9z^{3}a^{-1}-a^{8}z^{2}+a^{6}z^{2}+10a^{4}z^{2}+15a^{2}z^{2}+7z^{2}+3a^{7}z+15a^{5}z+23a^{3}z+16az+5za^{-1}-a^{6}-2a^{4}-3a^{2}-1-2a^{5}z^{-1}-4a^{3}z^{-1}-3az^{-1}-a^{-1}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                         0 2                 1 1 1   -1 0               3 1       2 -2             3 2 1       -2 -4           3 3 1         1 -6         3 3             0 -8       3 3 1             1 -10     2 4                 2 -12   1 2                   -1 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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