L11n445

L11n444

L11n446

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Link Presentations

[edit Notes on L11n445's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(3)-1)(t(4)-1)\left(t(4)^{2}+t(1)t(4)-2t(1)t(2)t(4)+t(2)t(4)-2t(4)+t(1)t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}t(4)^{3/2}}}$ (db)
Jones polynomial	$-10q^{9/2}+9q^{7/2}-12q^{5/2}+9q^{3/2}-{\frac {2}{q^{3/2}}}+q^{15/2}-3q^{13/2}+6q^{11/2}-9{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}+a^{-7}z^{-1}-3z^{3}a^{-5}-a^{-5}z^{-3}-6za^{-5}-5a^{-5}z^{-1}+2z^{5}a^{-3}+7z^{3}a^{-3}+3a^{-3}z^{-3}+12za^{-3}+10a^{-3}z^{-1}-4z^{3}a^{-1}+az^{-3}-3a^{-1}z^{-3}+2az-9za^{-1}+3az^{-1}-9a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-3}-z^{9}a^{-5}-4z^{8}a^{-2}-7z^{8}a^{-4}-3z^{8}a^{-6}-4z^{7}a^{-1}-10z^{7}a^{-3}-9z^{7}a^{-5}-3z^{7}a^{-7}+8z^{6}a^{-2}+13z^{6}a^{-4}+3z^{6}a^{-6}-z^{6}a^{-8}-z^{6}+14z^{5}a^{-1}+43z^{5}a^{-3}+38z^{5}a^{-5}+9z^{5}a^{-7}+9z^{4}a^{-4}+12z^{4}a^{-6}+3z^{4}a^{-8}-3az^{3}-33z^{3}a^{-1}-66z^{3}a^{-3}-45z^{3}a^{-5}-9z^{3}a^{-7}-24z^{2}a^{-2}-30z^{2}a^{-4}-14z^{2}a^{-6}-3z^{2}a^{-8}-5z^{2}+7az+30za^{-1}+48za^{-3}+31za^{-5}+6za^{-7}+21a^{-2}+18a^{-4}+6a^{-6}+a^{-8}+9-5az^{-1}-14a^{-1}z^{-1}-18a^{-3}z^{-1}-11a^{-5}z^{-1}-2a^{-7}z^{-1}-6a^{-2}z^{-2}-3a^{-4}z^{-2}-3z^{-2}+az^{-3}+3a^{-1}z^{-3}+3a^{-3}z^{-3}+a^{-5}z^{-3}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 16                   1 -1 14                 2   2 12               4 1   -3 10             6 2     4 8           5 6       1 6         7 4         3 4     1 3 5           3 2     7 7             0 0   2 8               6 -2   1                 -1 -4 2                   2

Integral Khovanov Homology (db, data source)

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

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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