L11n313

L11n312

L11n314

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Link Presentations

[edit Notes on L11n313's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X9,20,10,21 X7,16,8,17 X13,18,14,19 X19,14,20,15 X15,22,16,11 X17,10,18,5 X21,8,22,9 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 9, -3, 8}, {11, -2, -5, 6, -7, 4, -8, 5, -6, 3, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)t(3)^{4}+t(1)t(2)^{2}t(3)^{3}+t(2)^{2}t(3)^{3}-t(1)t(3)^{3}-t(1)t(2)t(3)^{3}-t(2)t(3)^{3}-2t(2)^{2}t(3)^{2}+2t(1)t(3)^{2}-t(1)t(2)t(3)^{2}+t(2)t(3)^{2}+t(2)^{2}t(3)-t(1)t(3)+t(1)t(2)t(3)+t(2)t(3)-t(3)-t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{-3}-q^{-4}+q^{-5}+q^{-6}+3q^{-8}-2q^{-9}+3q^{-10}-3q^{-11}+2q^{-12}-q^{-13}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}+4a^{12}z^{-2}+5a^{12}-a^{10}z^{4}-9a^{10}z^{2}-5a^{10}z^{-2}-13a^{10}+a^{8}z^{6}+6a^{8}z^{4}+9a^{8}z^{2}+2a^{8}z^{-2}+7a^{8}+a^{6}z^{6}+5a^{6}z^{4}+5a^{6}z^{2}+a^{6}$ (db)
Kauffman polynomial	$z^{7}a^{15}-5z^{5}a^{15}+7z^{3}a^{15}-4za^{15}+a^{15}z^{-1}+2z^{8}a^{14}-10z^{6}a^{14}+13z^{4}a^{14}-7z^{2}a^{14}-a^{14}z^{-2}+3a^{14}+z^{9}a^{13}-2z^{7}a^{13}-11z^{5}a^{13}+26z^{3}a^{13}-19za^{13}+5a^{13}z^{-1}+4z^{8}a^{12}-23z^{6}a^{12}+38z^{4}a^{12}-29z^{2}a^{12}-4a^{12}z^{-2}+15a^{12}+z^{9}a^{11}-2z^{7}a^{11}-16z^{5}a^{11}+42z^{3}a^{11}-33za^{11}+9a^{11}z^{-1}+3z^{8}a^{10}-21z^{6}a^{10}+44z^{4}a^{10}-42z^{2}a^{10}-5a^{10}z^{-2}+20a^{10}+2z^{7}a^{9}-15z^{5}a^{9}+27z^{3}a^{9}-18za^{9}+5a^{9}z^{-1}+z^{8}a^{8}-7z^{6}a^{8}+14z^{4}a^{8}-15z^{2}a^{8}-2a^{8}z^{-2}+8a^{8}+z^{7}a^{7}-5z^{5}a^{7}+4z^{3}a^{7}+z^{6}a^{6}-5z^{4}a^{6}+5z^{2}a^{6}-a^{6}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                       1 1 -7                     1 1 0 -9                 1 1     0 -11               2 1 1     2 -13             2 4 1       1 -15           3 1 1         3 -17         2 5 2           1 -19       2 2 1             1 -21     1 2 1               0 -23   1 2                   -1 -25   1                     1 -27 1                       -1

Integral Khovanov Homology (db, data source)

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

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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