L10a92

L10a91

L10a93

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Link Presentations

[edit Notes on L10a92's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X20,12,9,11 X2,9,3,10 X18,14,19,13 X14,5,15,6 X4,17,5,18 X16,7,17,8 X6,15,7,16 X8,20,1,19
Gauss code	{1, -4, 2, -7, 6, -9, 8, -10}, {4, -1, 3, -2, 5, -6, 9, -8, 7, -5, 10, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{3}v^{3}-2u^{3}v^{2}+u^{3}v-2u^{2}v^{3}+5u^{2}v^{2}-5u^{2}v+u^{2}+uv^{3}-5uv^{2}+5uv-2u+v^{2}-2v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{7/2}+3q^{5/2}-5q^{3/2}+8{\sqrt {q}}-{\frac {11}{\sqrt {q}}}+{\frac {11}{q^{3/2}}}-{\frac {11}{q^{5/2}}}+{\frac {8}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-7a^{3}z^{3}-7a^{3}z-a^{3}z^{-1}+az^{7}+5az^{5}-z^{5}a^{-1}+9az^{3}-3z^{3}a^{-1}+5az-2za^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-2a^{7}z^{3}+a^{7}z+3a^{6}z^{6}-6a^{6}z^{4}+2a^{6}z^{2}+4a^{5}z^{7}-7a^{5}z^{5}+3a^{5}z^{3}-3a^{5}z+a^{5}z^{-1}+3a^{4}z^{8}-2a^{4}z^{6}-3a^{4}z^{4}+2a^{4}z^{2}-a^{4}+a^{3}z^{9}+6a^{3}z^{7}-17a^{3}z^{5}+z^{5}a^{-3}+18a^{3}z^{3}-2z^{3}a^{-3}-8a^{3}z+a^{3}z^{-1}+6a^{2}z^{8}-11a^{2}z^{6}+3z^{6}a^{-2}+8a^{2}z^{4}-7z^{4}a^{-2}+3z^{2}a^{-2}+az^{9}+6az^{7}+4z^{7}a^{-1}-19az^{5}-9z^{5}a^{-1}+22az^{3}+7z^{3}a^{-1}-7az-3za^{-1}+3z^{8}-3z^{6}-2z^{4}+3z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   2   -2 4                 3 1   2 2               5 2     -3 0             6 3       3 -2           6 6         0 -4         5 5           0 -6       3 6             3 -8     3 5               -2 -10   1 4                 3 -12   2                   -2 -14 1                     1

Integral Khovanov Homology (db, data source)

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

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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