L11n314

L11n313

L11n315

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Link Presentations

[edit Notes on L11n314's Link Presentations]

Planar diagram presentation	X6172 X3,12,4,13 X7,17,8,16 X9,20,10,21 X11,18,12,19 X19,22,20,11 X15,9,16,8 X21,10,22,5 X17,14,18,15 X2536 X13,4,14,1
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 7, -4, 8}, {-5, 2, -11, 9, -7, 3, -9, 5, -6, 4, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-3t(1)t(3)t(2)^{2}+t(3)t(2)^{2}-t(2)^{2}-3t(1)t(3)^{2}t(2)+t(3)^{2}t(2)-t(1)t(2)+3t(1)t(3)t(2)-3t(3)t(2)+3t(2)+t(1)t(3)^{2}-t(3)^{2}-t(1)t(3)+3t(3)-2}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$1-2q^{-1}+6q^{-2}-7q^{-3}+10q^{-4}-10q^{-5}+10q^{-6}-7q^{-7}+5q^{-8}-2q^{-9}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}+a^{8}z^{4}+3a^{8}z^{2}+2a^{8}-a^{6}z^{6}-3a^{6}z^{4}-2a^{6}z^{2}+a^{6}z^{-2}-a^{4}z^{6}-3a^{4}z^{4}-3a^{4}z^{2}-2a^{4}z^{-2}-4a^{4}+a^{2}z^{4}+3a^{2}z^{2}+a^{2}z^{-2}+3a^{2}$ (db)
Kauffman polynomial	$3a^{11}z^{3}-2a^{11}z+a^{10}z^{6}+4a^{10}z^{4}-5a^{10}z^{2}+2a^{10}+3a^{9}z^{7}-4a^{9}z^{5}+8a^{9}z^{3}-4a^{9}z+3a^{8}z^{8}-5a^{8}z^{6}+11a^{8}z^{4}-12a^{8}z^{2}+4a^{8}+a^{7}z^{9}+4a^{7}z^{7}-9a^{7}z^{5}+4a^{7}z^{3}+5a^{6}z^{8}-8a^{6}z^{6}+a^{6}z^{2}+a^{6}z^{-2}-2a^{6}+a^{5}z^{9}+3a^{5}z^{7}-10a^{5}z^{5}+6a^{5}z-2a^{5}z^{-1}+2a^{4}z^{8}-a^{4}z^{6}-11a^{4}z^{4}+14a^{4}z^{2}+2a^{4}z^{-2}-7a^{4}+2a^{3}z^{7}-5a^{3}z^{5}+a^{3}z^{3}+4a^{3}z-2a^{3}z^{-1}+a^{2}z^{6}-4a^{2}z^{4}+6a^{2}z^{2}+a^{2}z^{-2}-4a^{2}$ (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 χ 1                   1 1 -1                 1   -1 -3               5 1   4 -5             3 2     -1 -7           7 4       3 -9         5 5         0 -11       5 5           0 -13     3 6             3 -15   2 4               -2 -17   3                 3 -19 2                   -2

Integral Khovanov Homology (db, data source)

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

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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