L11a444

L11a443

L11a445

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Link Presentations

[edit Notes on L11a444's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{2}t(2)^{3}-2t(3)^{2}t(2)^{3}-t(1)t(3)t(2)^{3}+t(3)t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-2t(3)^{3}t(2)^{2}-4t(1)t(3)^{2}t(2)^{2}+5t(3)^{2}t(2)^{2}-t(1)t(2)^{2}+4t(1)t(3)t(2)^{2}-4t(3)t(2)^{2}+t(2)^{2}-t(1)t(3)^{3}t(2)+t(3)^{3}t(2)+4t(1)t(3)^{2}t(2)-4t(3)^{2}t(2)+2t(1)t(2)-5t(1)t(3)t(2)+4t(3)t(2)-t(2)-t(1)t(3)^{2}+t(3)^{2}+2t(1)t(3)-t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-8q^{6}+13q^{5}-15q^{4}+18q^{3}+q^{-3}-17q^{2}-2q^{-2}+14q+6q^{-1}-9$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}+2z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}-3z^{2}a^{-2}+2z^{2}a^{-4}-z^{2}a^{-6}-4z^{2}+2a^{2}-5a^{-2}+3a^{-4}-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-9}-z^{3}a^{-9}+4z^{6}a^{-8}-6z^{4}a^{-8}+2z^{2}a^{-8}+7z^{7}a^{-7}-11z^{5}a^{-7}+3z^{3}a^{-7}+7z^{8}a^{-6}-9z^{6}a^{-6}+z^{4}a^{-6}+2z^{2}a^{-6}-2a^{-6}+4z^{9}a^{-5}+z^{7}a^{-5}-8z^{5}a^{-5}+3z^{3}a^{-5}+za^{-5}+z^{10}a^{-4}+9z^{8}a^{-4}-18z^{6}a^{-4}+12z^{4}a^{-4}-5z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+6z^{9}a^{-3}-6z^{7}a^{-3}-5z^{5}a^{-3}+13z^{3}a^{-3}-8za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+5z^{8}a^{-2}+a^{2}z^{6}-13z^{6}a^{-2}-4a^{2}z^{4}+15z^{4}a^{-2}+5a^{2}z^{2}-15z^{2}a^{-2}-2a^{-2}z^{-2}-2a^{2}+9a^{-2}+2z^{9}a^{-1}+2az^{7}+2z^{7}a^{-1}-5az^{5}-14z^{5}a^{-1}+2az^{3}+16z^{3}a^{-1}+az-8za^{-1}+2a^{-1}z^{-1}+3z^{8}-7z^{6}+6z^{4}-5z^{2}-z^{-2}+3$ (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 6 7 χ 17                       1 -1 15                     3   3 13                   5 1   -4 11                 8 3     5 9               9 7       -2 7             9 6         3 5           8 9           1 3         6 9             -3 1       4 9               5 -1     2 5                 -3 -3     4                   4 -5 1 2                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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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