L11a496

L11a495

L11a497

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Link Presentations

[edit Notes on L11a496's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(t(3)^{4}-t(3)^{3}+t(3)^{2}-t(3)+1\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$-q^{9}+3q^{8}-6q^{7}+8q^{6}-11q^{5}+13q^{4}-11q^{3}+11q^{2}-7q+6-2q^{-1}+q^{-2}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{8}a^{-4}-2z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-10z^{4}a^{-2}+14z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}-17z^{2}a^{-2}+18z^{2}a^{-4}-5z^{2}a^{-6}+4z^{2}-14a^{-2}+13a^{-4}-4a^{-6}+5-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+7z^{9}a^{-3}+5z^{9}a^{-5}+3z^{8}a^{-2}+11z^{8}a^{-4}+9z^{8}a^{-6}+z^{8}-9z^{7}a^{-1}-25z^{7}a^{-3}-7z^{7}a^{-5}+9z^{7}a^{-7}-34z^{6}a^{-2}-60z^{6}a^{-4}-24z^{6}a^{-6}+8z^{6}a^{-8}-6z^{6}+10z^{5}a^{-1}+10z^{5}a^{-3}-22z^{5}a^{-5}-16z^{5}a^{-7}+6z^{5}a^{-9}+70z^{4}a^{-2}+88z^{4}a^{-4}+19z^{4}a^{-6}-10z^{4}a^{-8}+3z^{4}a^{-10}+14z^{4}+4z^{3}a^{-1}+34z^{3}a^{-3}+41z^{3}a^{-5}+5z^{3}a^{-7}-5z^{3}a^{-9}+z^{3}a^{-11}-57z^{2}a^{-2}-57z^{2}a^{-4}-14z^{2}a^{-6}+2z^{2}a^{-8}-16z^{2}-12za^{-1}-31za^{-3}-25za^{-5}-4za^{-7}+2za^{-9}+23a^{-2}+22a^{-4}+7a^{-6}+9+5a^{-1}z^{-1}+9a^{-3}z^{-1}+5a^{-5}z^{-1}+a^{-7}z^{-1}-5a^{-2}z^{-2}-4a^{-4}z^{-2}-a^{-6}z^{-2}-2z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 19                       1 -1 17                     2   2 15                   4 1   -3 13                 4 2     2 11               7 4       -3 9             6 4         2 7           5 7           2 5         6 6             0 3       5 9               4 1     1 2                 -1 -1   1 5                   4 -3   1                     -1 -5 1                       1

Integral Khovanov Homology (db, data source)

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

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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