L10a28

L10a27

L10a29

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Link Presentations

[edit Notes on L10a28's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)^{2}}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+3q^{5/2}-6q^{3/2}+8{\sqrt {q}}-{\frac {12}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {11}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-7a^{3}z^{3}-8a^{3}z-3a^{3}z^{-1}+az^{7}+5az^{5}-z^{5}a^{-1}+10az^{3}-3z^{3}a^{-1}+9az-3za^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-3a^{4}z^{8}-7a^{2}z^{8}-4z^{8}-4a^{5}z^{7}-8a^{3}z^{7}-9az^{7}-5z^{7}a^{-1}-3a^{6}z^{6}+13a^{2}z^{6}-3z^{6}a^{-2}+7z^{6}-a^{7}z^{5}+6a^{5}z^{5}+24a^{3}z^{5}+31az^{5}+13z^{5}a^{-1}-z^{5}a^{-3}+6a^{6}z^{4}+9a^{4}z^{4}-7a^{2}z^{4}+6z^{4}a^{-2}-4z^{4}+2a^{7}z^{3}-a^{5}z^{3}-28a^{3}z^{3}-40az^{3}-13z^{3}a^{-1}+2z^{3}a^{-3}-3a^{6}z^{2}-8a^{4}z^{2}-5a^{2}z^{2}-z^{2}a^{-2}-z^{2}-a^{7}z+2a^{5}z+15a^{3}z+19az+7za^{-1}+a^{6}+3a^{4}+3a^{2}+2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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                   2   -2 4                 4 1   3 2               4 2     -2 0             8 4       4 -2           6 6         0 -4         5 6           -1 -6       4 6             2 -8     2 5               -3 -10   1 4                 3 -12   2                   -2 -14 1                     1

Integral Khovanov Homology (db, data source)

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

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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