L10a54

L10a53

L10a55

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Link Presentations

[edit Notes on L10a54's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X18,11,19,12 X12,6,13,5 X4,18,5,17 X14,7,15,8 X16,14,17,13 X20,15,7,16 X6,19,1,20
Gauss code	{1, -2, 3, -6, 5, -10}, {7, -1, 2, -3, 4, -5, 8, -7, 9, -8, 6, -4, 10, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-4t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-t(2)^{3}+4t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+4t(2)^{2}-t(1)^{2}t(2)+5t(1)t(2)-4t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {12}{q^{5/2}}}-{\frac {13}{q^{7/2}}}+{\frac {12}{q^{9/2}}}-{\frac {10}{q^{11/2}}}+{\frac {6}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-2a^{7}z+2a^{5}z^{5}+6a^{5}z^{3}+4a^{5}z-a^{3}z^{7}-4a^{3}z^{5}-5a^{3}z^{3}-2a^{3}z+a^{3}z^{-1}+az^{5}+2az^{3}-az-az^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{10}+z^{2}a^{10}-3z^{5}a^{9}+3z^{3}a^{9}-za^{9}-5z^{6}a^{8}+5z^{4}a^{8}-2z^{2}a^{8}-6z^{7}a^{7}+7z^{5}a^{7}-4z^{3}a^{7}+za^{7}-5z^{8}a^{6}+5z^{6}a^{6}-z^{4}a^{6}-2z^{9}a^{5}-6z^{7}a^{5}+21z^{5}a^{5}-18z^{3}a^{5}+5za^{5}-10z^{8}a^{4}+25z^{6}a^{4}-19z^{4}a^{4}+6z^{2}a^{4}-2z^{9}a^{3}-4z^{7}a^{3}+23z^{5}a^{3}-19z^{3}a^{3}+2za^{3}+a^{3}z^{-1}-5z^{8}a^{2}+14z^{6}a^{2}-10z^{4}a^{2}+3z^{2}a^{2}-a^{2}-4z^{7}a+12z^{5}a-8z^{3}a-za+az^{-1}-z^{6}+2z^{4}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                     1 -1 2                   3   3 0                 3 1   -2 -2               7 3     4 -4             6 4       -2 -6           7 6         1 -8         6 7           1 -10       4 6             -2 -12     2 6               4 -14   1 4                 -3 -16   2                   2 -18 1                     -1

Integral Khovanov Homology (db, data source)

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

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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