L10a14

L10a13

L10a15

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Link Presentations

[edit Notes on L10a14's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)^{3}\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+12{\sqrt {q}}-{\frac {16}{\sqrt {q}}}+{\frac {16}{q^{3/2}}}-{\frac {15}{q^{5/2}}}+{\frac {12}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+4az^{5}-z^{5}a^{-1}+a^{5}z^{3}-6a^{3}z^{3}+7az^{3}-2z^{3}a^{-1}+2a^{5}z-7a^{3}z+7az-2za^{-1}+a^{5}z^{-1}-3a^{3}z^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{3}z^{9}-2az^{9}-5a^{4}z^{8}-12a^{2}z^{8}-7z^{8}-5a^{5}z^{7}-11a^{3}z^{7}-14az^{7}-8z^{7}a^{-1}-3a^{6}z^{6}+3a^{4}z^{6}+18a^{2}z^{6}-4z^{6}a^{-2}+8z^{6}-a^{7}z^{5}+7a^{5}z^{5}+30a^{3}z^{5}+38az^{5}+15z^{5}a^{-1}-z^{5}a^{-3}+5a^{6}z^{4}+5a^{4}z^{4}-4a^{2}z^{4}+5z^{4}a^{-2}+z^{4}+2a^{7}z^{3}-4a^{5}z^{3}-28a^{3}z^{3}-33az^{3}-10z^{3}a^{-1}+z^{3}a^{-3}-3a^{6}z^{2}-7a^{4}z^{2}-5a^{2}z^{2}-z^{2}a^{-2}-2z^{2}-a^{7}z+2a^{5}z+13a^{3}z+15az+5za^{-1}+a^{6}+3a^{4}+3a^{2}+2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   3   -3 4                 6 1   5 2               6 3     -3 0             10 6       4 -2           8 8         0 -4         7 8           -1 -6       5 8             3 -8     2 7               -5 -10   1 5                 4 -12   2                   -2 -14 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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