L11a502

L11a501

L11a503

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Link Presentations

[edit Notes on L11a502's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X20,15,21,16 X14,5,15,6 X4,13,5,14 X12,19,7,20 X18,11,19,12 X22,17,13,18 X16,21,17,22 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 7, -6}, {5, -4, 3, -9, 8, -7, 6, -3, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)^{2}t(3)^{3}+2t(1)t(2)^{2}t(3)^{3}-t(2)^{2}t(3)^{3}+t(1)t(3)^{3}+2t(1)^{2}t(2)t(3)^{3}-3t(1)t(2)t(3)^{3}+t(2)t(3)^{3}+t(1)^{2}t(3)^{2}-2t(1)t(2)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}-2t(1)^{2}t(2)t(3)^{2}+4t(1)t(2)t(3)^{2}-2t(2)t(3)^{2}-t(1)^{2}t(3)+2t(1)t(2)^{2}t(3)-t(2)^{2}t(3)+2t(1)t(3)+2t(1)^{2}t(2)t(3)-4t(1)t(2)t(3)+2t(2)t(3)+t(1)^{2}-t(1)t(2)^{2}+t(2)^{2}-2t(1)-t(1)^{2}t(2)+3t(1)t(2)-2t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-3}-2q^{-4}+6q^{-5}-9q^{-6}+14q^{-7}-15q^{-8}+17q^{-9}-14q^{-10}+11q^{-11}-7q^{-12}+3q^{-13}-q^{-14}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}-a^{12}z^{4}-a^{12}z^{2}+4a^{12}z^{-2}+4a^{12}+a^{10}z^{6}+a^{10}z^{4}-6a^{10}z^{2}-5a^{10}z^{-2}-10a^{10}+2a^{8}z^{6}+7a^{8}z^{4}+7a^{8}z^{2}+2a^{8}z^{-2}+4a^{8}+a^{6}z^{6}+4a^{6}z^{4}+5a^{6}z^{2}+2a^{6}$ (db)
Kauffman polynomial	$z^{5}a^{17}-2z^{3}a^{17}+3z^{6}a^{16}-5z^{4}a^{16}+6z^{7}a^{15}-14z^{5}a^{15}+12z^{3}a^{15}-5za^{15}+a^{15}z^{-1}+7z^{8}a^{14}-18z^{6}a^{14}+22z^{4}a^{14}-12z^{2}a^{14}-a^{14}z^{-2}+4a^{14}+4z^{9}a^{13}-2z^{7}a^{13}-13z^{5}a^{13}+30z^{3}a^{13}-21za^{13}+5a^{13}z^{-1}+z^{10}a^{12}+9z^{8}a^{12}-30z^{6}a^{12}+44z^{4}a^{12}-33z^{2}a^{12}-4a^{12}z^{-2}+17a^{12}+6z^{9}a^{11}-8z^{7}a^{11}-9z^{5}a^{11}+35z^{3}a^{11}-33za^{11}+9a^{11}z^{-1}+z^{10}a^{10}+5z^{8}a^{10}-17z^{6}a^{10}+27z^{4}a^{10}-33z^{2}a^{10}-5a^{10}z^{-2}+20a^{10}+2z^{9}a^{9}+2z^{7}a^{9}-16z^{5}a^{9}+21z^{3}a^{9}-16za^{9}+5a^{9}z^{-1}+3z^{8}a^{8}-7z^{6}a^{8}+6z^{4}a^{8}-7z^{2}a^{8}-2a^{8}z^{-2}+6a^{8}+2z^{7}a^{7}-5z^{5}a^{7}+2z^{3}a^{7}+za^{7}+z^{6}a^{6}-4z^{4}a^{6}+5z^{2}a^{6}-2a^{6}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                       1 1 -7                     2 1 -1 -9                   4     4 -11                 5 2     -3 -13               9 4       5 -15             7 6         -1 -17           10 8           2 -19         6 9             3 -21       5 8               -3 -23     2 6                 4 -25   1 5                   -4 -27   2                     2 -29 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-7$ $i=-5$ 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 r=-11} ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-8$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-7$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ 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}^{6}} $r=-6$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=-5$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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