L11n399

L11n398

L11n400

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Link Presentations

[edit Notes on L11n399's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X22,16,19,15 X7,20,8,21 X19,8,20,9 X13,18,14,5 X11,14,12,15 X17,12,18,13 X16,22,17,21 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, 9, -3}, {10, -1, -4, 5, 11, -2, -7, 8, -6, 7, 3, -9, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(2)-1)(t(3)-1)\left(t(1)t(3)^{2}-1\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$1-2q^{-1}+4q^{-2}-3q^{-3}+4q^{-4}-2q^{-5}+3q^{-6}-q^{-8}+q^{-9}-q^{-10}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+2a^{8}z^{2}+4a^{8}z^{-2}+5a^{8}-2a^{6}z^{2}-5a^{6}z^{-2}-7a^{6}-a^{4}z^{6}-4a^{4}z^{4}-3a^{4}z^{2}+2a^{4}z^{-2}+a^{4}+a^{2}z^{4}+3a^{2}z^{2}+2a^{2}$ (db)
Kauffman polynomial	$a^{11}z^{7}-6a^{11}z^{5}+10a^{11}z^{3}-6a^{11}z+a^{11}z^{-1}+a^{10}z^{8}-6a^{10}z^{6}+9a^{10}z^{4}-6a^{10}z^{2}-a^{10}z^{-2}+4a^{10}+2a^{9}z^{7}-16a^{9}z^{5}+33a^{9}z^{3}-22a^{9}z+5a^{9}z^{-1}+a^{8}z^{8}-9a^{8}z^{6}+22a^{8}z^{4}-22a^{8}z^{2}-4a^{8}z^{-2}+15a^{8}+2a^{7}z^{7}-15a^{7}z^{5}+34a^{7}z^{3}-30a^{7}z+9a^{7}z^{-1}+a^{6}z^{8}-5a^{6}z^{6}+11a^{6}z^{4}-19a^{6}z^{2}-5a^{6}z^{-2}+16a^{6}+3a^{5}z^{7}-12a^{5}z^{5}+15a^{5}z^{3}-13a^{5}z+5a^{5}z^{-1}+a^{4}z^{8}-a^{4}z^{6}-6a^{4}z^{4}+2a^{4}z^{2}-2a^{4}z^{-2}+4a^{4}+2a^{3}z^{7}-7a^{3}z^{5}+4a^{3}z^{3}+a^{3}z+a^{2}z^{6}-4a^{2}z^{4}+5a^{2}z^{2}-2a^{2}$ (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 1 2 χ 1                       1 1 -1                     1   -1 -3                   3 1   2 -5                 2 3     1 -7             1 3 1       1 -9           1 1 2         2 -11         1 5 3           1 -13       1   2             3 -15       1 2               -1 -17   1 1                   0 -19                         0 -21 1                       -1

Integral Khovanov Homology (db, data source)

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

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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