L10a135

L10a134

L10a136

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Link Presentations

[edit Notes on L10a135's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,5,15,6 X16,12,17,11 X10,13,5,14 X20,18,11,17 X8,20,9,19 X18,8,19,7 X2,9,3,10 X4,16,1,15
Gauss code	{1, -9, 2, -10}, {3, -1, 8, -7, 9, -5}, {4, -2, 5, -3, 10, -4, 6, -8, 7, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uv^{2}w^{2}-3uv^{2}w+uv^{2}-2uvw^{2}+6uvw-3uv-2uw+2u-2v^{2}w^{2}+2v^{2}w+3vw^{2}-6vw+2v-w^{2}+3w-2}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{6}-4q^{5}+7q^{4}+q^{-4}-10q^{3}-3q^{-3}+14q^{2}+7q^{-2}-13q-10q^{-1}+14$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}-3z^{4}+2a^{2}z^{2}+z^{2}a^{-2}+z^{2}a^{-4}-5z^{2}+2a^{2}+4a^{-2}-a^{-4}-5+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-6}-2z^{4}a^{-6}+z^{2}a^{-6}+4z^{7}a^{-5}-11z^{5}a^{-5}+7z^{3}a^{-5}+za^{-5}+5z^{8}a^{-4}-12z^{6}a^{-4}+a^{4}z^{4}+6z^{4}a^{-4}-a^{4}z^{2}+z^{2}a^{-4}-2a^{-4}+2z^{9}a^{-3}+5z^{7}a^{-3}+3a^{3}z^{5}-20z^{5}a^{-3}-2a^{3}z^{3}+8z^{3}a^{-3}+3za^{-3}+10z^{8}a^{-2}+6a^{2}z^{6}-18z^{6}a^{-2}-8a^{2}z^{4}+8a^{2}z^{2}+11z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-4a^{2}-8a^{-2}+2z^{9}a^{-1}+7az^{7}+8z^{7}a^{-1}-8az^{5}-20z^{5}a^{-1}+3az^{3}+6z^{3}a^{-1}+3az+5za^{-1}-2az^{-1}-2a^{-1}z^{-1}+5z^{8}+z^{6}-17z^{4}+20z^{2}+2z^{-2}-9$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                     1 1 11                   3   -3 9                 4 1   3 7               6 3     -3 5             8 4       4 3           6 7         1 1         8 7           1 -1       4 8             4 -3     3 6               -3 -5   1 5                 4 -7   2                   -2 -9 1                     1

Integral Khovanov Homology (db, data source)

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

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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