L10a13

L10a12

L10a14

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Link Presentations

[edit Notes on L10a13's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,13,17,14 X14,7,15,8 X8,15,9,16 X20,18,5,17 X18,12,19,11 X12,20,13,19 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 4, -5, 10, -2, 7, -8, 3, -4, 5, -3, 6, -7, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{5}+2t(1)t(2)^{4}-4t(2)^{4}-5t(1)t(2)^{3}+7t(2)^{3}+7t(1)t(2)^{2}-5t(2)^{2}-4t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{5/2}-3q^{3/2}+6{\sqrt {q}}-{\frac {10}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {13}{q^{5/2}}}+{\frac {11}{q^{7/2}}}-{\frac {10}{q^{9/2}}}+{\frac {6}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+3a^{5}z^{3}+6a^{5}z+4a^{5}z^{-1}-2a^{3}z^{5}-6a^{3}z^{3}-8a^{3}z-4a^{3}z^{-1}-az^{5}-az^{3}+z^{3}a^{-1}+az^{-1}+za^{-1}$ (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-3a^{6}z^{8}-7a^{4}z^{8}-4a^{2}z^{8}-3a^{7}z^{7}-9a^{5}z^{7}-12a^{3}z^{7}-6az^{7}-a^{8}z^{6}+3a^{6}z^{6}+7a^{4}z^{6}-2a^{2}z^{6}-5z^{6}+9a^{7}z^{5}+32a^{5}z^{5}+33a^{3}z^{5}+7az^{5}-3z^{5}a^{-1}+3a^{8}z^{4}+10a^{6}z^{4}+16a^{4}z^{4}+15a^{2}z^{4}-z^{4}a^{-2}+5z^{4}-9a^{7}z^{3}-32a^{5}z^{3}-31a^{3}z^{3}-5az^{3}+3z^{3}a^{-1}-3a^{8}z^{2}-13a^{6}z^{2}-21a^{4}z^{2}-15a^{2}z^{2}+z^{2}a^{-2}-3z^{2}+4a^{7}z+17a^{5}z+16a^{3}z+2az-za^{-1}+a^{8}+4a^{6}+7a^{4}+4a^{2}+1-a^{7}z^{-1}-4a^{5}z^{-1}-4a^{3}z^{-1}-az^{-1}$ (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 χ 6                     1 -1 4                   2   2 2                 4 1   -3 0               6 2     4 -2             7 5       -2 -4           6 5         1 -6         5 7           2 -8       5 6             -1 -10     2 6               4 -12   1 4                 -3 -14   2                   2 -16 1                     -1

Integral Khovanov Homology (db, data source)

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

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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