L11n372

L11n371

L11n373

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Link Presentations

[edit Notes on L11n372's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(3)-1)\left(-t(1)t(3)^{3}+t(1)t(2)t(3)^{3}+t(1)t(2)^{2}t(3)^{2}+t(1)t(3)^{2}-2t(1)t(2)t(3)^{2}-t(3)^{2}-t(1)t(2)^{2}t(3)+t(2)^{2}t(3)-2t(2)t(3)+t(3)-t(2)^{2}+t(2)\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{4}+3q^{3}-5q^{2}+8q-8+10q^{-1}-8q^{-2}+7q^{-3}-4q^{-4}+2q^{-5}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}-a^{4}z^{4}-3a^{4}z^{2}+a^{4}z^{-2}-a^{4}+a^{2}z^{6}+3a^{2}z^{4}-z^{4}a^{-2}+a^{2}z^{2}-2a^{2}z^{-2}-2z^{2}a^{-2}-2a^{2}+z^{6}+3z^{4}+2z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$3a^{6}z^{2}-a^{6}+a^{5}z^{5}+3a^{5}z^{3}-a^{5}z+3a^{4}z^{6}-a^{4}z^{4}-3a^{4}z^{2}-a^{4}z^{-2}+4a^{4}+6a^{3}z^{7}+z^{7}a^{-3}-16a^{3}z^{5}-4z^{5}a^{-3}+17a^{3}z^{3}+5z^{3}a^{-3}-10a^{3}z-2za^{-3}+2a^{3}z^{-1}+6a^{2}z^{8}+3z^{8}a^{-2}-20a^{2}z^{6}-13z^{6}a^{-2}+27a^{2}z^{4}+17z^{4}a^{-2}-26a^{2}z^{2}-8z^{2}a^{-2}-2a^{2}z^{-2}+12a^{2}+2a^{-2}+2az^{9}+2z^{9}a^{-1}+2az^{7}-3z^{7}a^{-1}-25az^{5}-12z^{5}a^{-1}+30az^{3}+21z^{3}a^{-1}-14az-7za^{-1}+2az^{-1}+9z^{8}-36z^{6}+45z^{4}-28z^{2}-z^{-2}+10$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 9                   1 -1 7                 2   2 5               3 1   -2 3             5 2     3 1           4 4       0 -1         6 4         2 -3       3 5           2 -5     4 5             -1 -7   1 4               3 -9 1 3                 -2 -11 2                   2

Integral Khovanov Homology (db, data source)

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

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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