L11n375

L11n374

L11n376

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Link Presentations

[edit Notes on L11n375's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 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 X13,2,14,3
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 $u$, $v$, $w$, ...)	${\frac {(t(3)-1)\left(t(1)t(3)^{4}-t(1)t(3)^{3}+2t(1)t(2)t(3)^{3}-t(2)t(3)+2t(3)+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{-3}-2q^{-4}+4q^{-5}-3q^{-6}+6q^{-7}-5q^{-8}+5q^{-9}-3q^{-10}+2q^{-11}-q^{-12}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+a^{12}z^{2}+4a^{12}z^{-2}+6a^{12}-3a^{10}z^{4}-14a^{10}z^{2}-5a^{10}z^{-2}-15a^{10}+2a^{8}z^{6}+10a^{8}z^{4}+14a^{8}z^{2}+2a^{8}z^{-2}+8a^{8}+a^{6}z^{6}+4a^{6}z^{4}+3a^{6}z^{2}+a^{6}$ (db)
Kauffman polynomial	$z^{3}a^{15}-2za^{15}+a^{15}z^{-1}+2z^{4}a^{14}-4z^{2}a^{14}-a^{14}z^{-2}+2a^{14}+z^{7}a^{13}-5z^{5}a^{13}+14z^{3}a^{13}-15za^{13}+5a^{13}z^{-1}+2z^{8}a^{12}-11z^{6}a^{12}+25z^{4}a^{12}-25z^{2}a^{12}-4a^{12}z^{-2}+14a^{12}+z^{9}a^{11}-z^{7}a^{11}-14z^{5}a^{11}+40z^{3}a^{11}-33za^{11}+9a^{11}z^{-1}+5z^{8}a^{10}-26z^{6}a^{10}+47z^{4}a^{10}-40z^{2}a^{10}-5a^{10}z^{-2}+21a^{10}+z^{9}a^{9}-16z^{5}a^{9}+30z^{3}a^{9}-20za^{9}+5a^{9}z^{-1}+3z^{8}a^{8}-14z^{6}a^{8}+20z^{4}a^{8}-16z^{2}a^{8}-2a^{8}z^{-2}+9a^{8}+2z^{7}a^{7}-7z^{5}a^{7}+3z^{3}a^{7}+z^{6}a^{6}-4z^{4}a^{6}+3z^{2}a^{6}-a^{6}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                   1 1 -7                 2 1 -1 -9               2     2 -11           1 2 2     1 -13         1 6 2       3 -15         2 3         1 -17       4 4           0 -19   1 1 2             2 -21   2 3               -1 -23   1                 1 -25 1                   -1

Integral Khovanov Homology (db, data source)

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

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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