L11a480

L11a479

L11a481

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Link Presentations

[edit Notes on L11a480's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X18,7,19,8 X14,21,15,22 X20,10,21,9 X8,13,9,14 X22,15,17,16 X16,17,5,18 X12,20,13,19 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {8, -3, 9, -5, 4, -7}, {10, -1, 3, -6, 5, -2, 11, -9, 6, -4, 7, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)^{3}(t(2)t(3)+1)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{3}-4q^{2}+9q-13+19q^{-1}-20q^{-2}+21q^{-3}-17q^{-4}+13q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{4}-3a^{6}z^{2}-2a^{6}+2a^{4}z^{6}+8a^{4}z^{4}+11a^{4}z^{2}+a^{4}z^{-2}+6a^{4}-a^{2}z^{8}-5a^{2}z^{6}-10a^{2}z^{4}-11a^{2}z^{2}-2a^{2}z^{-2}-7a^{2}+z^{6}+3z^{4}+3z^{2}+z^{-2}+3$ (db)
Kauffman polynomial	$2a^{4}z^{10}+2a^{2}z^{10}+5a^{5}z^{9}+12a^{3}z^{9}+7az^{9}+6a^{6}z^{8}+10a^{4}z^{8}+12a^{2}z^{8}+8z^{8}+5a^{7}z^{7}-2a^{5}z^{7}-26a^{3}z^{7}-15az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-6a^{6}z^{6}-32a^{4}z^{6}-47a^{2}z^{6}+z^{6}a^{-2}-23z^{6}+a^{9}z^{5}-6a^{7}z^{5}-5a^{5}z^{5}+18a^{3}z^{5}+7az^{5}-9z^{5}a^{-1}-5a^{8}z^{4}+a^{6}z^{4}+39a^{4}z^{4}+56a^{2}z^{4}-2z^{4}a^{-2}+21z^{4}-2a^{9}z^{3}+a^{7}z^{3}+6a^{5}z^{3}+a^{3}z^{3}+az^{3}+3z^{3}a^{-1}+3a^{8}z^{2}-23a^{4}z^{2}-30a^{2}z^{2}-10z^{2}+a^{9}z+a^{7}z-3a^{5}z-6a^{3}z-3az-a^{8}+7a^{4}+9a^{2}+4+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   6 1   5 1                 7 3     -4 -1               12 6       6 -3             10 9         -1 -5           11 10           1 -7         8 12             4 -9       5 9               -4 -11     2 8                 6 -13   1 5                   -4 -15   2                     2 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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