L11n307

L11n306

L11n308

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Link Presentations

[edit Notes on L11n307's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-3uv^{2}w^{2}+3uv^{2}w-uv^{2}-uvw^{3}+3uvw^{2}-2uvw+uw^{3}-uw^{2}+v^{2}w^{2}-v^{2}w+2vw^{3}-3vw^{2}+vw+w^{4}-3w^{3}+3w^{2}}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{10}+3q^{9}-6q^{8}+8q^{7}-9q^{6}+11q^{5}-8q^{4}+8q^{3}-4q^{2}+2q$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-4}-3z^{4}a^{-6}+2z^{2}a^{-2}+2z^{2}a^{-4}-9z^{2}a^{-6}+4z^{2}a^{-8}+a^{-2}+5a^{-4}-12a^{-6}+7a^{-8}-a^{-10}+2a^{-4}z^{-2}-5a^{-6}z^{-2}+4a^{-8}z^{-2}-a^{-10}z^{-2}$ (db)
Kauffman polynomial	$z^{7}a^{-11}-4z^{5}a^{-11}+6z^{3}a^{-11}-4za^{-11}+a^{-11}z^{-1}+3z^{8}a^{-10}-12z^{6}a^{-10}+14z^{4}a^{-10}-6z^{2}a^{-10}-a^{-10}z^{-2}+3a^{-10}+2z^{9}a^{-9}-z^{7}a^{-9}-21z^{5}a^{-9}+36z^{3}a^{-9}-19za^{-9}+5a^{-9}z^{-1}+10z^{8}a^{-8}-39z^{6}a^{-8}+48z^{4}a^{-8}-32z^{2}a^{-8}-4a^{-8}z^{-2}+15a^{-8}+2z^{9}a^{-7}+5z^{7}a^{-7}-38z^{5}a^{-7}+52z^{3}a^{-7}-33za^{-7}+9a^{-7}z^{-1}+7z^{8}a^{-6}-24z^{6}a^{-6}+35z^{4}a^{-6}-37z^{2}a^{-6}-5a^{-6}z^{-2}+20a^{-6}+7z^{7}a^{-5}-20z^{5}a^{-5}+25z^{3}a^{-5}-18za^{-5}+5a^{-5}z^{-1}+3z^{6}a^{-4}+z^{4}a^{-4}-8z^{2}a^{-4}-2a^{-4}z^{-2}+8a^{-4}+z^{5}a^{-3}+3z^{3}a^{-3}+3z^{2}a^{-2}-a^{-2}$ (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 21                   1 -1 19                 2   2 17               4 1   -3 15             4 2     2 13           6 5       -1 11         5 3         2 9       3 6           3 7     5 5             0 5     4               4 3 2 4                 -2 1 2                   2

Integral Khovanov Homology (db, data source)

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

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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