L11n431

L11n430

L11n432

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Link Presentations

[edit Notes on L11n431's Link Presentations]

Planar diagram presentation	X8192 X20,10,21,9 X5,15,6,14 X12,14,7,13 X16,8,17,7 X22,18,13,17 X10,4,11,3 X18,11,19,12 X15,1,16,6 X4,20,5,19 X2,21,3,22
Gauss code	{1, -11, 7, -10, -3, 9}, {5, -1, 2, -7, 8, -4}, {4, 3, -9, -5, 6, -8, 10, -2, 11, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(w-1)\left(-u^{2}v^{2}w+u^{2}v^{2}+u^{2}vw-u^{2}v+uv^{2}w-uv^{2}+uvw^{2}-3uvw+uv-uw^{2}+uw-vw^{2}+vw+w^{2}-w\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{9}-4q^{8}+7q^{7}-9q^{6}+12q^{5}-11q^{4}+11q^{3}-7q^{2}+5q-1$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-8}-2z^{4}a^{-6}-3z^{2}a^{-6}+a^{-6}z^{-2}+z^{6}a^{-4}+3z^{4}a^{-4}+3z^{2}a^{-4}-2a^{-4}z^{-2}-2a^{-4}-z^{4}a^{-2}+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-10}-2z^{4}a^{-10}+4z^{7}a^{-9}-11z^{5}a^{-9}+5z^{3}a^{-9}+6z^{8}a^{-8}-19z^{6}a^{-8}+15z^{4}a^{-8}-4z^{2}a^{-8}+3z^{9}a^{-7}-3z^{7}a^{-7}-10z^{5}a^{-7}+8z^{3}a^{-7}+11z^{8}a^{-6}-36z^{6}a^{-6}+37z^{4}a^{-6}-11z^{2}a^{-6}+a^{-6}z^{-2}-2a^{-6}+3z^{9}a^{-5}-5z^{7}a^{-5}+z^{5}a^{-5}+2z^{3}a^{-5}+2za^{-5}-2a^{-5}z^{-1}+5z^{8}a^{-4}-16z^{6}a^{-4}+25z^{4}a^{-4}-10z^{2}a^{-4}+2a^{-4}z^{-2}-3a^{-4}+2z^{7}a^{-3}+2za^{-3}-2a^{-3}z^{-1}+5z^{4}a^{-2}-3z^{2}a^{-2}+a^{-2}z^{-2}-2a^{-2}+z^{3}a^{-1}$ (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   \ -1 0 1 2 3 4 5 6 7 8 χ 19                   1 1 17                 3   -3 15               4 1   3 13             5 3     -2 11           7 4       3 9         5 6         1 7       6 6           0 5     3 7             4 3   2 4               -2 1   4                 4 -1 1                   -1

Integral Khovanov Homology (db, data source)

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

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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