L10a26

L10a25

L10a27

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Link Presentations

[edit Notes on L10a26's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{5}+2t(1)t(2)^{4}-4t(2)^{4}-6t(1)t(2)^{3}+8t(2)^{3}+8t(1)t(2)^{2}-6t(2)^{2}-4t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {10}{q^{9/2}}}+{\frac {12}{q^{7/2}}}+q^{5/2}-{\frac {15}{q^{5/2}}}-4q^{3/2}+{\frac {13}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {6}{q^{11/2}}}+8{\sqrt {q}}-{\frac {11}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+6za^{5}+4a^{5}z^{-1}-2z^{5}a^{3}-6z^{3}a^{3}-9za^{3}-4a^{3}z^{-1}-z^{5}a+2za+az^{-1}+z^{3}a^{-1}$ (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-3a^{6}z^{8}-8a^{4}z^{8}-5a^{2}z^{8}-3a^{7}z^{7}-10a^{5}z^{7}-16a^{3}z^{7}-9az^{7}-a^{8}z^{6}+3a^{6}z^{6}+8a^{4}z^{6}-4a^{2}z^{6}-8z^{6}+9a^{7}z^{5}+34a^{5}z^{5}+41a^{3}z^{5}+12az^{5}-4z^{5}a^{-1}+3a^{8}z^{4}+9a^{6}z^{4}+16a^{4}z^{4}+20a^{2}z^{4}-z^{4}a^{-2}+9z^{4}-9a^{7}z^{3}-34a^{5}z^{3}-34a^{3}z^{3}-7az^{3}+2z^{3}a^{-1}-3a^{8}z^{2}-13a^{6}z^{2}-21a^{4}z^{2}-14a^{2}z^{2}-3z^{2}+4a^{7}z+16a^{5}z+15a^{3}z+3az+a^{8}+4a^{6}+7a^{4}+4a^{2}+1-a^{7}z^{-1}-4a^{5}z^{-1}-4a^{3}z^{-1}-az^{-1}$ (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                   3   3 2                 5 1   -4 0               6 3     3 -2             8 6       -2 -4           7 5         2 -6         5 8           3 -8       5 7             -2 -10     2 6               4 -12   1 4                 -3 -14   2                   2 -16 1                     -1

Integral Khovanov Homology (db, data source)

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

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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