L10a70

L10a69

L10a71

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Link Presentations

[edit Notes on L10a70's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X14,6,15,5 X16,12,17,11 X10,4,11,3 X12,7,13,8 X20,15,7,16 X6,14,1,13 X4,19,5,20 X2,18,3,17
Gauss code	{1, -10, 5, -9, 3, -8}, {6, -1, 2, -5, 4, -6, 8, -3, 7, -4, 10, -2, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+t(2)^{4}-3t(1)^{2}t(2)^{3}+7t(1)t(2)^{3}-4t(2)^{3}+5t(1)^{2}t(2)^{2}-9t(1)t(2)^{2}+5t(2)^{2}-4t(1)^{2}t(2)+7t(1)t(2)-3t(2)+t(1)^{2}-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {4}{q^{9/2}}}-5q^{7/2}+{\frac {9}{q^{7/2}}}+9q^{5/2}-{\frac {14}{q^{5/2}}}-14q^{3/2}+{\frac {17}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+17{\sqrt {q}}-{\frac {19}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-2a^{3}z^{3}+z^{3}a^{-3}-2a^{3}z-a^{-3}z^{-1}+az^{7}+4az^{5}-2z^{5}a^{-1}+7az^{3}-4z^{3}a^{-1}+4az-za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}+4a^{5}z^{5}-a^{5}z^{3}+9a^{4}z^{6}+z^{6}a^{-4}-8a^{4}z^{4}-z^{4}a^{-4}+3a^{4}z^{2}+13a^{3}z^{7}+5z^{7}a^{-3}-19a^{3}z^{5}-11z^{5}a^{-3}+12a^{3}z^{3}+5z^{3}a^{-3}-3a^{3}z+2za^{-3}-a^{-3}z^{-1}+11a^{2}z^{8}+8z^{8}a^{-2}-13a^{2}z^{6}-19z^{6}a^{-2}+a^{2}z^{4}+11z^{4}a^{-2}+a^{2}z^{2}-z^{2}a^{-2}+a^{-2}+4az^{9}+4z^{9}a^{-1}+13az^{7}+5z^{7}a^{-1}-42az^{5}-30z^{5}a^{-1}+27az^{3}+19z^{3}a^{-1}-5az-a^{-1}z^{-1}+19z^{8}-42z^{6}+22z^{4}-3z^{2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                     1 -1 8                   4   4 6                 5 1   -4 4               9 4     5 2             9 6       -3 0           10 8         2 -2         8 10           2 -4       6 9             -3 -6     3 8               5 -8   1 6                 -5 -10   3                   3 -12 1                     -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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