L10a77

L10a76

L10a78

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Link Presentations

[edit Notes on L10a77's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {4u^{2}v^{3}-4u^{2}v^{2}+u^{2}v+uv^{4}-5uv^{3}+9uv^{2}-5uv+u+v^{3}-4v^{2}+4v}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{25/2}}}+{\frac {3}{q^{23/2}}}-{\frac {6}{q^{21/2}}}+{\frac {10}{q^{19/2}}}-{\frac {12}{q^{17/2}}}+{\frac {13}{q^{15/2}}}-{\frac {13}{q^{13/2}}}+{\frac {9}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{11}z^{3}+a^{11}z-a^{11}z^{-1}-a^{9}z^{5}+5a^{9}z+3a^{9}z^{-1}-3a^{7}z^{5}-10a^{7}z^{3}-9a^{7}z-2a^{7}z^{-1}-a^{5}z^{5}-2a^{5}z^{3}$ (db)
Kauffman polynomial	$a^{15}z^{5}-2a^{15}z^{3}+a^{15}z+3a^{14}z^{6}-6a^{14}z^{4}+3a^{14}z^{2}+4a^{13}z^{7}-5a^{13}z^{5}-a^{13}z^{3}+a^{13}z+4a^{12}z^{8}-5a^{12}z^{6}+3a^{12}z^{4}-3a^{12}z^{2}-a^{12}+2a^{11}z^{9}+2a^{11}z^{7}-5a^{11}z^{5}+a^{11}z^{3}+a^{11}z^{-1}+9a^{10}z^{8}-20a^{10}z^{6}+18a^{10}z^{4}-2a^{10}z^{2}-3a^{10}+2a^{9}z^{9}+4a^{9}z^{7}-16a^{9}z^{5}+19a^{9}z^{3}-9a^{9}z+3a^{9}z^{-1}+5a^{8}z^{8}-9a^{8}z^{6}+4a^{8}z^{4}+4a^{8}z^{2}-3a^{8}+6a^{7}z^{7}-16a^{7}z^{5}+17a^{7}z^{3}-9a^{7}z+2a^{7}z^{-1}+3a^{6}z^{6}-5a^{6}z^{4}+a^{5}z^{5}-2a^{5}z^{3}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                     1 1 -6                   3 1 -2 -8                 4     4 -10               5 3     -2 -12             8 4       4 -14           5 5         0 -16         7 8           -1 -18       4 6             2 -20     2 6               -4 -22   1 4                 3 -24   2                   -2 -26 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-10$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-7$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-6$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=-5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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