L11n391

L11n390

L11n392

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Link Presentations

[edit Notes on L11n391's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X8493 X13,22,14,19 X9,20,10,21 X19,10,20,11 X21,14,22,15 X11,18,12,5 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-8, 7, -9, 6}, {-4, -1, 2, -5, -7, 8, -10, 4, -6, 9, 11, -2, 3, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(w-1)^{2}(vw+1)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$1-3q^{-1}+6q^{-2}-5q^{-3}+7q^{-4}-5q^{-5}+5q^{-6}-2q^{-7}+q^{-9}-q^{-10}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+2a^{8}z^{2}+4a^{8}z^{-2}+6a^{8}-4a^{6}z^{2}-5a^{6}z^{-2}-10a^{6}-a^{4}z^{6}-3a^{4}z^{4}+2a^{4}z^{-2}+4a^{4}+a^{2}z^{4}+2a^{2}z^{2}+a^{2}$ (db)
Kauffman polynomial	$a^{11}z^{7}-6a^{11}z^{5}+9a^{11}z^{3}-5a^{11}z+a^{11}z^{-1}+a^{10}z^{8}-7a^{10}z^{6}+12a^{10}z^{4}-8a^{10}z^{2}-a^{10}z^{-2}+3a^{10}+a^{9}z^{7}-12a^{9}z^{5}+29a^{9}z^{3}-20a^{9}z+5a^{9}z^{-1}+2a^{8}z^{8}-15a^{8}z^{6}+33a^{8}z^{4}-25a^{8}z^{2}-4a^{8}z^{-2}+13a^{8}+a^{7}z^{9}-2a^{7}z^{7}-11a^{7}z^{5}+35a^{7}z^{3}-30a^{7}z+9a^{7}z^{-1}+4a^{6}z^{8}-17a^{6}z^{6}+25a^{6}z^{4}-23a^{6}z^{2}-5a^{6}z^{-2}+16a^{6}+a^{5}z^{9}+a^{5}z^{7}-14a^{5}z^{5}+19a^{5}z^{3}-15a^{5}z+5a^{5}z^{-1}+3a^{4}z^{8}-8a^{4}z^{6}+a^{4}z^{4}-3a^{4}z^{2}-2a^{4}z^{-2}+6a^{4}+3a^{3}z^{7}-9a^{3}z^{5}+4a^{3}z^{3}+a^{2}z^{6}-3a^{2}z^{4}+3a^{2}z^{2}-a^{2}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                     2   -2 -3                   4 1   3 -5                 3 4     1 -7             1 5 2       2 -9           1 2 3         2 -11         1 6 5           0 -13       1 1 3             3 -15       1 3               -2 -17   1 1 1                 1 -19                         0 -21 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $i=-1$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{6}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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