L11n456

L11n455

L11n457

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Link Presentations

[edit Notes on L11n456's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X15,18,16,11 X9,21,10,20 X13,19,14,22 X21,15,22,14 X19,5,20,10 X17,8,18,9 X7,16,8,17 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {-7, 4, -6, 5}, {10, -1, -9, 8, -4, 7}, {11, -2, -5, 6, -3, 9, -8, 3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {-uvwx+uvw+uvx-uv-uw^{2}x^{2}+2uwx-uw-ux+u+vw^{2}x^{2}-vw^{2}x-vwx^{2}+2vwx-v-w^{2}x^{2}+w^{2}x+wx^{2}-wx}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-q^{5/2}-{\frac {7}{q^{5/2}}}+{\frac {3}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {4}{q^{11/2}}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+2z^{3}a^{5}+3za^{5}+2a^{5}z^{-1}+a^{5}z^{-3}-z^{5}a^{3}-4z^{3}a^{3}-9za^{3}-7a^{3}z^{-1}-3a^{3}z^{-3}+z^{5}a+7z^{3}a+11za+8az^{-1}+3az^{-3}-z^{3}a^{-1}-4za^{-1}-3a^{-1}z^{-1}-a^{-1}z^{-3}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+a^{8}z^{2}+3a^{7}z^{7}-11a^{7}z^{5}+9a^{7}z^{3}-3a^{7}z+3a^{6}z^{8}-10a^{6}z^{6}+5a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{9}+a^{5}z^{7}-16a^{5}z^{5}+22a^{5}z^{3}-a^{5}z^{-3}-12a^{5}z+5a^{5}z^{-1}+4a^{4}z^{8}-16a^{4}z^{6}+16a^{4}z^{4}+3a^{4}z^{2}+3a^{4}z^{-2}-10a^{4}+a^{3}z^{9}-3a^{3}z^{7}-2a^{3}z^{5}+17a^{3}z^{3}-3a^{3}z^{-3}-21a^{3}z+12a^{3}z^{-1}+a^{2}z^{8}-5a^{2}z^{6}+4a^{2}z^{4}+15a^{2}z^{2}+6a^{2}z^{-2}-19a^{2}+z^{7}a^{-1}-4az^{5}-7z^{5}a^{-1}+18az^{3}+14z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-23az-11za^{-1}+12az^{-1}+5a^{-1}z^{-1}-4z^{4}+12z^{2}+3z^{-2}-10$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                       1 1 2                 1 1     0 0               4         4 -2             2 4 1       1 -4           6 2           4 -6         3 6 1           2 -8       3 3 1             1 -10     2 4                 2 -12   1 2                   -1 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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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