L11n410

L11n409

L11n411

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Link Presentations

[edit Notes on L11n410's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(w-1)\left(2vw^{2}-2vw+v-w^{2}+2w-2\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$1-3q^{-1}+7q^{-2}-9q^{-3}+14q^{-4}-13q^{-5}+13q^{-6}-10q^{-7}+7q^{-8}-3q^{-9}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+a^{8}z^{4}+3a^{8}z^{2}+4a^{8}z^{-2}+6a^{8}-a^{6}z^{6}-3a^{6}z^{4}-6a^{6}z^{2}-5a^{6}z^{-2}-10a^{6}-a^{4}z^{6}-2a^{4}z^{4}+a^{4}z^{2}+2a^{4}z^{-2}+4a^{4}+a^{2}z^{4}+2a^{2}z^{2}+a^{2}$ (db)
Kauffman polynomial	$6z^{3}a^{11}-5za^{11}+a^{11}z^{-1}+3z^{6}a^{10}+3z^{4}a^{10}-6z^{2}a^{10}-a^{10}z^{-2}+3a^{10}+8z^{7}a^{9}-19z^{5}a^{9}+29z^{3}a^{9}-20za^{9}+5a^{9}z^{-1}+7z^{8}a^{8}-16z^{6}a^{8}+25z^{4}a^{8}-23z^{2}a^{8}-4a^{8}z^{-2}+13a^{8}+2z^{9}a^{7}+9z^{7}a^{7}-33z^{5}a^{7}+41z^{3}a^{7}-30za^{7}+9a^{7}z^{-1}+11z^{8}a^{6}-29z^{6}a^{6}+30z^{4}a^{6}-25z^{2}a^{6}-5a^{6}z^{-2}+16a^{6}+2z^{9}a^{5}+4z^{7}a^{5}-22z^{5}a^{5}+23z^{3}a^{5}-15za^{5}+5a^{5}z^{-1}+4z^{8}a^{4}-9z^{6}a^{4}+5z^{4}a^{4}-5z^{2}a^{4}-2a^{4}z^{-2}+6a^{4}+3z^{7}a^{3}-8z^{5}a^{3}+5z^{3}a^{3}+z^{6}a^{2}-3z^{4}a^{2}+3z^{2}a^{2}-a^{2}$ (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                 2   -2 -3               5 1   4 -5             6 4     -2 -7           8 3       5 -9         5 6         1 -11       8 8           0 -13     4 7             3 -15   3 6               -3 -17   4                 4 -19 3                   -3

Integral Khovanov Homology (db, data source)

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

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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