L11a473

L11a472

L11a474

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Link Presentations

[edit Notes on L11a473's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X22,15,17,16 X18,10,19,9 X8,18,9,17 X20,13,21,14 X14,21,15,22 X16,19,5,20 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {6, -5, 9, -7, 8, -4}, {10, -1, 3, -6, 5, -2, 11, -3, 7, -8, 4, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(w-1)\left(v^{2}w^{3}-v^{2}w^{2}+v^{2}w-vw^{3}+3vw^{2}-3vw+v-w^{2}+w-1\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}-7q^{3}+12q^{2}-15q+19-17q^{-1}+16q^{-2}-11q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+6z^{6}+a^{4}z^{4}-9a^{2}z^{4}-4z^{4}a^{-2}+15z^{4}+3a^{4}z^{2}-15a^{2}z^{2}-6z^{2}a^{-2}+18z^{2}+3a^{4}-10a^{2}-3a^{-2}+10+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+8az^{9}+5z^{9}a^{-1}+4a^{4}z^{8}+10a^{2}z^{8}+8z^{8}a^{-2}+14z^{8}+3a^{5}z^{7}+a^{3}z^{7}-13az^{7}-5z^{7}a^{-1}+6z^{7}a^{-3}+a^{6}z^{6}-7a^{4}z^{6}-32a^{2}z^{6}-22z^{6}a^{-2}+3z^{6}a^{-4}-49z^{6}-8a^{5}z^{5}-15a^{3}z^{5}-7z^{5}a^{-1}-13z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+40a^{2}z^{4}+30z^{4}a^{-2}-5z^{4}a^{-4}+72z^{4}+6a^{5}z^{3}+13a^{3}z^{3}+16az^{3}+21z^{3}a^{-1}+10z^{3}a^{-3}-2z^{3}a^{-5}+3a^{6}z^{2}-33a^{2}z^{2}-18z^{2}a^{-2}-48z^{2}-a^{5}z-7a^{3}z-13az-10za^{-1}-3za^{-3}-a^{6}+3a^{4}+13a^{2}+5a^{-2}+15+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     2   2 7                   5 1   -4 5                 7 2     5 3               8 5       -3 1             11 7         4 -1           8 10           2 -3         8 9             -1 -5       5 10               5 -7     2 6                 -4 -9   1 5                   4 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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