L11n265

L11n264

L11n266

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Link Presentations

[edit Notes on L11n265's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,19,12,18 X15,21,16,20 X17,9,18,22 X21,17,22,16 X19,13,20,12 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, -5, 9, -4, 3, -6, 8, -7, 5, -9, 6, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-2uvw^{4}+3uvw^{3}-3uvw^{2}+2uvw-uv+uw^{4}-2uw^{3}+2uw^{2}-uw+vw^{4}-2vw^{3}+2vw^{2}-vw+w^{5}-2w^{4}+3w^{3}-3w^{2}+2w}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{11}+2q^{10}-6q^{9}+9q^{8}-11q^{7}+12q^{6}-10q^{5}+10q^{4}-4q^{3}+3q^{2}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-2z^{6}a^{-6}+3z^{4}a^{-4}-10z^{4}a^{-6}+3z^{4}a^{-8}+10z^{2}a^{-4}-21z^{2}a^{-6}+11z^{2}a^{-8}-z^{2}a^{-10}+10a^{-4}-22a^{-6}+15a^{-8}-3a^{-10}+3a^{-4}z^{-2}-8a^{-6}z^{-2}+7a^{-8}z^{-2}-2a^{-10}z^{-2}$ (db)
Kauffman polynomial	$z^{9}a^{-7}+z^{9}a^{-9}+4z^{8}a^{-6}+7z^{8}a^{-8}+3z^{8}a^{-10}+3z^{7}a^{-5}+7z^{7}a^{-7}+7z^{7}a^{-9}+3z^{7}a^{-11}-13z^{6}a^{-6}-18z^{6}a^{-8}-3z^{6}a^{-10}+2z^{6}a^{-12}-9z^{5}a^{-5}-26z^{5}a^{-7}-21z^{5}a^{-9}-3z^{5}a^{-11}+z^{5}a^{-13}+6z^{4}a^{-4}+31z^{4}a^{-6}+32z^{4}a^{-8}+4z^{4}a^{-10}-3z^{4}a^{-12}+21z^{3}a^{-5}+48z^{3}a^{-7}+28z^{3}a^{-9}-2z^{3}a^{-11}-3z^{3}a^{-13}-16z^{2}a^{-4}-41z^{2}a^{-6}-35z^{2}a^{-8}-10z^{2}a^{-10}-24za^{-5}-45za^{-7}-21za^{-9}+3za^{-11}+3za^{-13}+13a^{-4}+28a^{-6}+22a^{-8}+7a^{-10}+a^{-12}+8a^{-5}z^{-1}+15a^{-7}z^{-1}+7a^{-9}z^{-1}-a^{-11}z^{-1}-a^{-13}z^{-1}-3a^{-4}z^{-2}-8a^{-6}z^{-2}-7a^{-8}z^{-2}-2a^{-10}z^{-2}$ (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   \ 0 1 2 3 4 5 6 7 8 9 χ 23                   1 -1 21                 1   1 19               5 1   -4 17             4 1     3 15           7 5       -2 13         5 4         1 11       6 8           2 9     4 4             0 7     6               6 5 3 4                 -1 3 3                   3

Integral Khovanov Homology (db, data source)

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

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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