L11a418

L11a417

L11a419

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Link Presentations

[edit Notes on L11a418's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-2t(1)t(2)^{2}+2t(1)t(3)t(2)^{2}-3t(3)t(2)^{2}+2t(2)^{2}+2t(1)t(3)^{2}t(2)-3t(3)^{2}t(2)+3t(1)t(2)-5t(1)t(3)t(2)+5t(3)t(2)-2t(2)-2t(1)t(3)^{2}+2t(3)^{2}+3t(1)t(3)-2t(3)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{7}+3q^{6}-5q^{5}+8q^{4}-10q^{3}+12q^{2}-11q+11-7q^{-1}+5q^{-2}-2q^{-3}+q^{-4}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}-2z^{2}a^{2}+a^{2}z^{-2}+z^{4}-2z^{2}-2z^{-2}-3+2z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}+z^{4}a^{-4}-z^{2}a^{-6}$ (db)
Kauffman polynomial	$z^{7}a^{-7}-4z^{5}a^{-7}+4z^{3}a^{-7}+3z^{8}a^{-6}-13z^{6}a^{-6}+16z^{4}a^{-6}-5z^{2}a^{-6}+3z^{9}a^{-5}-11z^{7}a^{-5}+11z^{5}a^{-5}-4z^{3}a^{-5}+z^{10}a^{-4}+2z^{8}a^{-4}-15z^{6}a^{-4}+a^{4}z^{4}+14z^{4}a^{-4}-2a^{4}z^{2}-3z^{2}a^{-4}+a^{4}+5z^{9}a^{-3}-14z^{7}a^{-3}+2a^{3}z^{5}+10z^{5}a^{-3}-2a^{3}z^{3}-4z^{3}a^{-3}+z^{10}a^{-2}+2z^{8}a^{-2}+3a^{2}z^{6}-6z^{6}a^{-2}-3a^{2}z^{4}-4z^{4}a^{-2}+3a^{2}z^{2}+9z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-3a^{2}-5a^{-2}+2z^{9}a^{-1}+3az^{7}+z^{7}a^{-1}-7z^{5}a^{-1}-5az^{3}+z^{3}a^{-1}+6az+6za^{-1}-2az^{-1}-2a^{-1}z^{-1}+3z^{8}-z^{6}-6z^{4}+12z^{2}+2z^{-2}-8$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 15                       1 -1 13                     2   2 11                   3 1   -2 9                 5 2     3 7               5 3       -2 5             7 5         2 3           6 7           1 1         5 5             0 -1       3 7               4 -3     2 4                 -2 -5     3                   3 -7 1 2                     -1 -9 1                       1

Integral Khovanov Homology (db, data source)

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

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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