L10a153

L10a152

L10a154

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Link Presentations

[edit Notes on L10a153's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(w-1)\left(2uv^{2}w^{2}-uvw^{2}+2uvw-uw+u+v^{2}w^{3}-v^{2}w^{2}+2vw^{2}-vw+2w\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{-3}-2q^{-4}+5q^{-5}-6q^{-6}+9q^{-7}-9q^{-8}+9q^{-9}-7q^{-10}+5q^{-11}-2q^{-12}+q^{-13}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{12}z^{2}+a^{12}z^{-2}+3a^{12}-3a^{10}z^{4}-11a^{10}z^{2}-2a^{10}z^{-2}-10a^{10}+2a^{8}z^{6}+9a^{8}z^{4}+12a^{8}z^{2}+a^{8}z^{-2}+6a^{8}+a^{6}z^{6}+4a^{6}z^{4}+4a^{6}z^{2}+a^{6}$ (db)
Kauffman polynomial	$a^{16}z^{4}-2a^{16}z^{2}+a^{16}+2a^{15}z^{5}-2a^{15}z^{3}+3a^{14}z^{6}-3a^{14}z^{4}+a^{14}z^{2}+3a^{13}z^{7}-2a^{13}z^{5}+3a^{12}z^{8}-6a^{12}z^{6}+11a^{12}z^{4}-11a^{12}z^{2}-a^{12}z^{-2}+6a^{12}+a^{11}z^{9}+4a^{11}z^{7}-17a^{11}z^{5}+24a^{11}z^{3}-13a^{11}z+2a^{11}z^{-1}+6a^{10}z^{8}-21a^{10}z^{6}+34a^{10}z^{4}-32a^{10}z^{2}-2a^{10}z^{-2}+14a^{10}+a^{9}z^{9}+3a^{9}z^{7}-19a^{9}z^{5}+25a^{9}z^{3}-13a^{9}z+2a^{9}z^{-1}+3a^{8}z^{8}-11a^{8}z^{6}+15a^{8}z^{4}-14a^{8}z^{2}-a^{8}z^{-2}+7a^{8}+2a^{7}z^{7}-6a^{7}z^{5}+3a^{7}z^{3}+a^{6}z^{6}-4a^{6}z^{4}+4a^{6}z^{2}-a^{6}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                     1 1 -7                   2 1 -1 -9                 3     3 -11               3 2     -1 -13             6 3       3 -15           4 4         0 -17         5 5           0 -19       3 5             2 -21     2 4               -2 -23     3                 3 -25 1 2                   -1 -27 1                     1

Integral Khovanov Homology (db, data source)

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

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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