L11a449

L11a448

L11a450

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Link Presentations

[edit Notes on L11a449's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{3}t(2)^{3}-t(3)^{3}t(2)^{3}-t(1)t(3)^{2}t(2)^{3}+2t(3)^{2}t(2)^{3}-t(3)t(2)^{3}-2t(1)t(3)^{3}t(2)^{2}+t(3)^{3}t(2)^{2}+4t(1)t(3)^{2}t(2)^{2}-4t(3)^{2}t(2)^{2}-2t(1)t(3)t(2)^{2}+5t(3)t(2)^{2}-t(2)^{2}+t(1)t(3)^{3}t(2)-5t(1)t(3)^{2}t(2)+2t(3)^{2}t(2)-t(1)t(2)+4t(1)t(3)t(2)-4t(3)t(2)+2t(2)+t(1)t(3)^{2}+t(1)-2t(1)t(3)+t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}-6q^{3}+10q^{2}-13q+17-15q^{-1}+14q^{-2}-10q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+6z^{6}+a^{4}z^{4}-9a^{2}z^{4}-4z^{4}a^{-2}+14z^{4}+3a^{4}z^{2}-14a^{2}z^{2}-5z^{2}a^{-2}+15z^{2}+3a^{4}-9a^{2}-2a^{-2}+8+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-8a^{5}z^{5}+z^{5}a^{-5}+5a^{5}z^{3}-2z^{3}a^{-5}-a^{5}z+4a^{4}z^{8}-8a^{4}z^{6}+3z^{6}a^{-4}+a^{4}z^{4}-6z^{4}a^{-4}-2a^{4}z^{2}+z^{2}a^{-4}-a^{4}z^{-2}+3a^{4}+3a^{3}z^{9}-a^{3}z^{7}+5z^{7}a^{-3}-13a^{3}z^{5}-11z^{5}a^{-3}+14a^{3}z^{3}+7z^{3}a^{-3}-8a^{3}z-2za^{-3}+2a^{3}z^{-1}+a^{2}z^{10}+8a^{2}z^{8}+6z^{8}a^{-2}-30a^{2}z^{6}-16z^{6}a^{-2}+39a^{2}z^{4}+21z^{4}a^{-2}-28a^{2}z^{2}-13z^{2}a^{-2}-2a^{2}z^{-2}+11a^{2}+3a^{-2}+7az^{9}+4z^{9}a^{-1}-14az^{7}-5z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+16az^{3}+16z^{3}a^{-1}-12az-7za^{-1}+2az^{-1}+z^{10}+10z^{8}-40z^{6}+62z^{4}-37z^{2}-z^{-2}+11$ (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                   4 1   -3 5                 6 2     4 3               7 4       -3 1             10 6         4 -1           8 10           2 -3         6 7             -1 -5       4 8               4 -7     3 6                 -3 -9   1 5                   4 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

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

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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