L10a138

L10a137

L10a139

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Link Presentations

[edit Notes on L10a138's Link Presentations]

Planar diagram presentation	X6172 X14,6,15,5 X8493 X2,16,3,15 X16,7,17,8 X18,10,19,9 X20,12,13,11 X12,14,5,13 X4,17,1,18 X10,20,11,19
Gauss code	{1, -4, 3, -9}, {2, -1, 5, -3, 6, -10, 7, -8}, {8, -2, 4, -5, 9, -6, 10, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(t(2)^{2}t(3)^{2}+t(2)t(3)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{9}+3q^{8}-4q^{7}+6q^{6}-7q^{5}+8q^{4}-6q^{3}+6q^{2}-3q+3-q^{-1}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-6}-4z^{4}a^{-6}-3z^{2}a^{-6}+a^{-6}z^{-2}+a^{-6}+z^{8}a^{-4}+6z^{6}a^{-4}+11z^{4}a^{-4}+5z^{2}a^{-4}-2a^{-4}z^{-2}-4a^{-4}-z^{6}a^{-2}-4z^{4}a^{-2}-2z^{2}a^{-2}+a^{-2}z^{-2}+3a^{-2}$ (db)
Kauffman polynomial	$z^{3}a^{-11}+3z^{4}a^{-10}-2z^{2}a^{-10}+4z^{5}a^{-9}-3z^{3}a^{-9}+4z^{6}a^{-8}-3z^{4}a^{-8}-2z^{2}a^{-8}+a^{-8}+4z^{7}a^{-7}-6z^{5}a^{-7}+4z^{8}a^{-6}-12z^{6}a^{-6}+12z^{4}a^{-6}-6z^{2}a^{-6}+a^{-6}z^{-2}-a^{-6}+2z^{9}a^{-5}-4z^{7}a^{-5}-3z^{5}a^{-5}+3z^{3}a^{-5}+4za^{-5}-2a^{-5}z^{-1}+7z^{8}a^{-4}-31z^{6}a^{-4}+39z^{4}a^{-4}-13z^{2}a^{-4}+2a^{-4}z^{-2}-4a^{-4}+2z^{9}a^{-3}-7z^{7}a^{-3}+3z^{5}a^{-3}+2z^{3}a^{-3}+4za^{-3}-2a^{-3}z^{-1}+3z^{8}a^{-2}-15z^{6}a^{-2}+21z^{4}a^{-2}-7z^{2}a^{-2}+a^{-2}z^{-2}-3a^{-2}+z^{7}a^{-1}-4z^{5}a^{-1}+3z^{3}a^{-1}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 χ 19                     1 -1 17                   2   2 15                 2 1   -1 13               4 2     2 11             5 4       -1 9           3 2         1 7         3 5           2 5       3 3             0 3     2 5               3 1   1 1                 0 -1   2                   2 -3 1                     -1

Integral Khovanov Homology (db, data source)

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

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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