L10a52

L10a51

L10a53

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Link Presentations

[edit Notes on L10a52's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X16,11,17,12 X12,5,13,6 X4,18,5,17 X14,20,15,19 X18,14,19,13 X6,15,1,16
Gauss code	{1, -4, 2, -7, 6, -10}, {4, -1, 3, -2, 5, -6, 9, -8, 10, -5, 7, -9, 8, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+t(2)^{4}-3t(1)^{2}t(2)^{3}+6t(1)t(2)^{3}-3t(2)^{3}+4t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+4t(2)^{2}-3t(1)^{2}t(2)+6t(1)t(2)-3t(2)+t(1)^{2}-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {1}{q^{9/2}}}+8q^{7/2}-{\frac {4}{q^{7/2}}}-12q^{5/2}+{\frac {7}{q^{5/2}}}+15q^{3/2}-{\frac {12}{q^{3/2}}}+q^{11/2}-16{\sqrt {q}}+{\frac {14}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-a^{3}z^{3}+5az^{3}-6z^{3}a^{-1}+2z^{3}a^{-3}-a^{3}z+2az-3za^{-1}+za^{-3}+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-2az^{9}-2z^{9}a^{-1}-5a^{2}z^{8}-7z^{8}a^{-2}-12z^{8}-4a^{3}z^{7}-9az^{7}-15z^{7}a^{-1}-10z^{7}a^{-3}-a^{4}z^{6}+10a^{2}z^{6}+2z^{6}a^{-2}-8z^{6}a^{-4}+21z^{6}+11a^{3}z^{5}+34az^{5}+40z^{5}a^{-1}+13z^{5}a^{-3}-4z^{5}a^{-5}+2a^{4}z^{4}-2a^{2}z^{4}+11z^{4}a^{-2}+8z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}-9a^{3}z^{3}-27az^{3}-27z^{3}a^{-1}-7z^{3}a^{-3}+2z^{3}a^{-5}-a^{4}z^{2}-2a^{2}z^{2}-6z^{2}a^{-2}-3z^{2}a^{-4}-4z^{2}+a^{3}z+5az+6za^{-1}+2za^{-3}-a^{2}+a^{3}z^{-1}+az^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                     1 -1 10                   3   3 8                 5 1   -4 6               7 3     4 4             8 5       -3 2           8 7         1 0         7 9           2 -2       5 7             -2 -4     3 8               5 -6   1 4                 -3 -8   3                   3 -10 1                     -1

Integral Khovanov Homology (db, data source)

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

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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