L10a35

L10a34

L10a36

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Link Presentations

[edit Notes on L10a35's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-2t(2)^{3}+t(2)^{2}-2t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {10}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-3za^{7}-2a^{7}z^{-1}+2z^{5}a^{5}+8z^{3}a^{5}+10za^{5}+5a^{5}z^{-1}-z^{7}a^{3}-5z^{5}a^{3}-9z^{3}a^{3}-8za^{3}-3a^{3}z^{-1}+z^{5}a+3z^{3}a+za$ (db)
Kauffman polynomial	$a^{10}z^{4}-2a^{10}z^{2}+a^{10}+2a^{9}z^{5}-2a^{9}z^{3}+3a^{8}z^{6}-3a^{8}z^{4}+a^{8}z^{2}+3a^{7}z^{7}-3a^{7}z^{5}+4a^{7}z^{3}-5a^{7}z+2a^{7}z^{-1}+2a^{6}z^{8}+a^{6}z^{6}-7a^{6}z^{4}+10a^{6}z^{2}-5a^{6}+a^{5}z^{9}+3a^{5}z^{7}-12a^{5}z^{5}+18a^{5}z^{3}-15a^{5}z+5a^{5}z^{-1}+5a^{4}z^{8}-12a^{4}z^{6}+5a^{4}z^{4}+7a^{4}z^{2}-5a^{4}+a^{3}z^{9}+3a^{3}z^{7}-18a^{3}z^{5}+22a^{3}z^{3}-12a^{3}z+3a^{3}z^{-1}+3a^{2}z^{8}-9a^{2}z^{6}+5a^{2}z^{4}+a^{2}z^{2}+3az^{7}-11az^{5}+10az^{3}-2az+z^{6}-3z^{4}+z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                     1 -1 2                   2   2 0                 2 1   -1 -2               5 2     3 -4             5 4       -1 -6           5 3         2 -8         3 5           2 -10       4 5             -1 -12     1 3               2 -14   1 4                 -3 -16   1                   1 -18 1                     -1

Integral Khovanov Homology (db, data source)

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

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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