L11n341

L11n340

L11n342

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Link Presentations

[edit Notes on L11n341's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,12,19,11 X14,8,15,7 X8,14,9,13 X19,22,20,13 X15,20,16,21 X21,16,22,17 X12,18,5,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -9}, {5, -4, -7, 8, 9, -3, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(2)-1)\left(t(1)t(3)^{3}-t(1)t(2)t(3)^{3}+2t(2)t(3)^{3}+2t(1)t(2)t(3)^{2}-2t(2)t(3)^{2}-2t(1)t(2)t(3)+2t(2)t(3)+t(2)^{2}+2t(1)t(2)-t(2)\right)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-2q^{-6}+5q^{-5}-8q^{-4}+q^{3}+11q^{-3}-2q^{2}-10q^{-2}+6q+11q^{-1}-8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{-2}-a^{6}-z^{4}a^{4}+z^{2}a^{4}+4a^{4}z^{-2}+6a^{4}+z^{6}a^{2}+2z^{4}a^{2}-2z^{2}a^{2}-5a^{2}z^{-2}-7a^{2}-2z^{4}-4z^{2}+2z^{-2}+z^{2}a^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+4a^{4}z^{8}+7a^{2}z^{8}+3z^{8}+4a^{5}z^{7}+8a^{3}z^{7}+6az^{7}+2z^{7}a^{-1}+a^{6}z^{6}-9a^{4}z^{6}-19a^{2}z^{6}+z^{6}a^{-2}-8z^{6}-9a^{5}z^{5}-29a^{3}z^{5}-25az^{5}-5z^{5}a^{-1}+4a^{6}z^{4}+19a^{4}z^{4}+27a^{2}z^{4}-4z^{4}a^{-2}+8z^{4}+3a^{7}z^{3}+20a^{5}z^{3}+43a^{3}z^{3}+28az^{3}+2z^{3}a^{-1}-4a^{6}z^{2}-24a^{4}z^{2}-34a^{2}z^{2}+5z^{2}a^{-2}-9z^{2}-3a^{7}z-17a^{5}z-33a^{3}z-18az+za^{-1}+3a^{6}+16a^{4}+21a^{2}-2a^{-2}+7+a^{7}z^{-1}+5a^{5}z^{-1}+9a^{3}z^{-1}+5az^{-1}-a^{6}z^{-2}-4a^{4}z^{-2}-5a^{2}z^{-2}-2z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                 2 1 -1 3               4     4 1             4 2     -2 -1           7 4       3 -3         5 6         1 -5       6 5           1 -7     2 5             3 -9   3 6               -3 -11   3                 3 -13 2                   -2

Integral Khovanov Homology (db, data source)

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

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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