L10a133

L10a132

L10a134

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Link Presentations

[edit Notes on L10a133's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,12,15,11 X20,16,11,15 X8,18,9,17 X16,8,17,7 X18,10,19,9 X10,20,5,19 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 6, -5, 7, -8}, {3, -2, 10, -3, 4, -6, 5, -7, 8, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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