L10a110

L10a109

L10a111

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Link Presentations

[edit Notes on L10a110's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X18,14,19,13 X20,16,11,15 X14,20,15,19 X6,17,7,18 X16,7,17,8 X10,6,1,5 X4,10,5,9 X2,11,3,12
Gauss code	{1, -10, 2, -9, 8, -6, 7, -2, 9, -8}, {10, -1, 3, -5, 4, -7, 6, -3, 5, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{3}t(1)^{3}-2t(2)^{2}t(1)^{3}+2t(2)t(1)^{3}-2t(1)^{3}-2t(2)^{3}t(1)^{2}+2t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}+2t(1)^{2}+2t(2)^{3}t(1)-2t(2)^{2}t(1)+2t(2)t(1)-2t(1)-2t(2)^{3}+2t(2)^{2}-2t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{15/2}+3q^{13/2}-6q^{11/2}+9q^{9/2}-9q^{7/2}+10q^{5/2}-9q^{3/2}+6{\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-5}-3z^{3}a^{-5}-3za^{-5}-a^{-5}z^{-1}+z^{7}a^{-3}+5z^{5}a^{-3}+10z^{3}a^{-3}+11za^{-3}+5a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-9z^{3}a^{-1}+4az-14za^{-1}+4az^{-1}-8a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-5z^{8}a^{-2}-4z^{8}a^{-4}-z^{8}-az^{7}-7z^{7}a^{-3}-8z^{7}a^{-5}+11z^{6}a^{-2}-9z^{6}a^{-6}+2z^{6}+6az^{5}+12z^{5}a^{-1}+26z^{5}a^{-3}+14z^{5}a^{-5}-6z^{5}a^{-7}+9z^{4}a^{-2}+22z^{4}a^{-4}+15z^{4}a^{-6}-3z^{4}a^{-8}+5z^{4}-13az^{3}-24z^{3}a^{-1}-20z^{3}a^{-3}-4z^{3}a^{-5}+4z^{3}a^{-7}-z^{3}a^{-9}-28z^{2}a^{-2}-23z^{2}a^{-4}-9z^{2}a^{-6}-14z^{2}+12az+20za^{-1}+9za^{-3}-za^{-7}+14a^{-2}+9a^{-4}+2a^{-6}+8-4az^{-1}-8a^{-1}z^{-1}-5a^{-3}z^{-1}-a^{-5}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 16                     1 1 14                   2   -2 12                 4 1   3 10               5 2     -3 8             4 4       0 6           6 5         -1 4         3 4           -1 2       4 7             3 0     1 2               -1 -2     4                 4 -4 1 1                   0 -6 1                     1

Integral Khovanov Homology (db, data source)

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

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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