L11n168

L11n167

L11n169

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Link Presentations

[edit Notes on L11n168's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,11,19,12 X19,7,20,22 X15,21,16,20 X21,17,22,16 X12,17,13,18 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 5, -9, 11, -4, -7, 8, 9, -5, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-t(1)^{2}t(2)^{3}+3t(1)t(2)^{3}-t(2)^{3}+2t(1)^{2}t(2)^{2}-3t(1)t(2)^{2}+2t(2)^{2}-t(1)^{2}t(2)+3t(1)t(2)-t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {7}{q^{9/2}}}-{\frac {7}{q^{7/2}}}+{\frac {4}{q^{5/2}}}-{\frac {3}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {2}{q^{17/2}}}-{\frac {4}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {7}{q^{11/2}}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+4z^{3}a^{7}+5za^{7}+a^{7}z^{-1}-z^{7}a^{5}-6z^{5}a^{5}-13z^{3}a^{5}-10za^{5}-a^{5}z^{-1}+z^{5}a^{3}+3z^{3}a^{3}+2za^{3}$ (db)
Kauffman polynomial	$a^{11}z^{5}-3a^{11}z^{3}+2a^{11}z+2a^{10}z^{6}-5a^{10}z^{4}+2a^{10}z^{2}+2a^{9}z^{7}-3a^{9}z^{5}-a^{9}z^{3}+2a^{8}z^{8}-5a^{8}z^{6}+7a^{8}z^{4}-5a^{8}z^{2}+a^{7}z^{9}-2a^{7}z^{7}+5a^{7}z^{5}-6a^{7}z^{3}+5a^{7}z-a^{7}z^{-1}+3a^{6}z^{8}-10a^{6}z^{6}+18a^{6}z^{4}-10a^{6}z^{2}+a^{6}+a^{5}z^{9}-4a^{5}z^{7}+12a^{5}z^{5}-14a^{5}z^{3}+9a^{5}z-a^{5}z^{-1}+a^{4}z^{8}-3a^{4}z^{6}+7a^{4}z^{4}-5a^{4}z^{2}+3a^{3}z^{5}-6a^{3}z^{3}+2a^{3}z+a^{2}z^{4}-2a^{2}z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 0                   1 -1 -2                 2   2 -4               3 2   -1 -6             4 1     3 -8           3 3       0 -10         4 4         0 -12       2 3           1 -14     2 4             -2 -16   1 3               2 -18   1                 -1 -20 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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