L11n377

L11n376

L11n378

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Link Presentations

[edit Notes on L11n377's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X15,1,16,4 X5,10,6,11 X3849 X17,22,18,19 X11,20,12,21 X19,12,20,13 X21,18,22,5 X9,16,10,17 X2,14,3,13
Gauss code	{1, -11, -5, 3}, {-8, 7, -9, 6}, {-4, -1, 2, 5, -10, 4, -7, 8, 11, -2, -3, 10, -6, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(3)-1)\left(t(3)^{2}+2t(2)t(3)-2t(3)-t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-10}+2q^{-9}-4q^{-8}+6q^{-7}-7q^{-6}+9q^{-5}-6q^{-4}+7q^{-3}-4q^{-2}+2q^{-1}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+3a^{8}z^{2}+4a^{8}z^{-2}+6a^{8}-2a^{6}z^{4}-6a^{6}z^{2}-5a^{6}z^{-2}-10a^{6}-a^{4}z^{4}+a^{4}z^{2}+2a^{4}z^{-2}+4a^{4}+2a^{2}z^{2}+a^{2}$ (db)
Kauffman polynomial	$z^{7}a^{11}-5z^{5}a^{11}+8z^{3}a^{11}-5za^{11}+a^{11}z^{-1}+2z^{8}a^{10}-9z^{6}a^{10}+12z^{4}a^{10}-6z^{2}a^{10}-a^{10}z^{-2}+3a^{10}+z^{9}a^{9}+z^{7}a^{9}-19z^{5}a^{9}+33z^{3}a^{9}-20za^{9}+5a^{9}z^{-1}+6z^{8}a^{8}-24z^{6}a^{8}+31z^{4}a^{8}-23z^{2}a^{8}-4a^{8}z^{-2}+13a^{8}+z^{9}a^{7}+5z^{7}a^{7}-29z^{5}a^{7}+41z^{3}a^{7}-30za^{7}+9a^{7}z^{-1}+4z^{8}a^{6}-12z^{6}a^{6}+18z^{4}a^{6}-25z^{2}a^{6}-5a^{6}z^{-2}+16a^{6}+5z^{7}a^{5}-14z^{5}a^{5}+19z^{3}a^{5}-15za^{5}+5a^{5}z^{-1}+3z^{6}a^{4}-z^{4}a^{4}-5z^{2}a^{4}-2a^{4}z^{-2}+6a^{4}+z^{5}a^{3}+3z^{3}a^{3}+3z^{2}a^{2}-a^{2}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -1                   2 2 -3                 4 2 -2 -5               3     3 -7             3 4     1 -9           6 3       3 -11         3 5         2 -13       3 4           -1 -15     1 3             2 -17   1 3               -2 -19   1                 1 -21 1                   -1

Integral Khovanov Homology (db, data source)

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

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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