L11n58

L11n57

L11n59

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Link Presentations

[edit Notes on L11n58's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X18,11,19,12 X22,19,5,20 X20,15,21,16 X16,21,17,22 X12,17,13,18 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 4, -8, -9, 3, 6, -7, 8, -4, 5, -6, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{5}+3t(1)t(2)^{4}-3t(2)^{4}-6t(1)t(2)^{3}+6t(2)^{3}+6t(1)t(2)^{2}-6t(2)^{2}-3t(1)t(2)+3t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {2}{q^{23/2}}}-{\frac {5}{q^{21/2}}}+{\frac {9}{q^{19/2}}}-{\frac {12}{q^{17/2}}}+{\frac {13}{q^{15/2}}}-{\frac {13}{q^{13/2}}}+{\frac {10}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	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 -2 z a^{11}-2 a^{11} z^{-1} +5 z^3 a^9+11 z a^9+5 a^9 z^{-1} -3 z^5 a^7-10 z^3 a^7-10 z a^7-3 a^7 z^{-1} -z^5 a^5-2 z^3 a^5-z a^5} (db)
Kauffman polynomial	$-3z^{4}a^{14}+4z^{2}a^{14}-a^{14}-z^{7}a^{13}-3z^{5}a^{13}+4z^{3}a^{13}-za^{13}-2z^{8}a^{12}-z^{6}a^{12}+z^{4}a^{12}+z^{2}a^{12}-z^{9}a^{11}-6z^{7}a^{11}+10z^{5}a^{11}-6z^{3}a^{11}+4za^{11}-2a^{11}z^{-1}-6z^{8}a^{10}+z^{6}a^{10}+16z^{4}a^{10}-17z^{2}a^{10}+5a^{10}-z^{9}a^{9}-11z^{7}a^{9}+27z^{5}a^{9}-26z^{3}a^{9}+17za^{9}-5a^{9}z^{-1}-4z^{8}a^{8}-z^{6}a^{8}+16z^{4}a^{8}-15z^{2}a^{8}+5a^{8}-6z^{7}a^{7}+13z^{5}a^{7}-14z^{3}a^{7}+11za^{7}-3a^{7}z^{-1}-3z^{6}a^{6}+4z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+2z^{3}a^{5}-za^{5}$ (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 -4                   1 1 -6                 3 1 -2 -8               5     5 -10             5 3     -2 -12           8 5       3 -14         6 6         0 -16       6 7           -1 -18     3 6             3 -20   2 6               -4 -22   3                 3 -24 2                   -2

Integral Khovanov Homology (db, data source)

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

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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