L11n385

L11n384

L11n386

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Link Presentations

[edit Notes on L11n385's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X3849 X9,16,10,17 X17,19,18,22 X11,20,12,21 X19,10,20,11 X21,5,22,18 X13,2,14,3
Gauss code	{1, 11, -5, -3}, {-9, 8, -10, 7}, {-4, -1, 2, 5, -6, 9, -8, 4, -11, -2, 3, 6, -7, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(v-1)(w-1)\left(uw^{4}-1\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{-1}-q^{-2}+3q^{-3}-q^{-4}+3q^{-5}-2q^{-6}+2q^{-7}-q^{-8}+q^{-9}-q^{-10}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-z^{2}a^{10}-a^{10}z^{-2}-2a^{10}+z^{6}a^{8}+6z^{4}a^{8}+11z^{2}a^{8}+4a^{8}z^{-2}+10a^{8}-z^{8}a^{6}-7z^{6}a^{6}-17z^{4}a^{6}-21z^{2}a^{6}-5a^{6}z^{-2}-16a^{6}+z^{6}a^{4}+6z^{4}a^{4}+11z^{2}a^{4}+2a^{4}z^{-2}+8a^{4}$ (db)
Kauffman polynomial	$a^{13}z+a^{12}z^{2}+a^{11}z^{3}-2a^{11}z+a^{11}z^{-1}+2a^{10}z^{4}-7a^{10}z^{2}-a^{10}z^{-2}+4a^{10}+2a^{9}z^{7}-11a^{9}z^{5}+19a^{9}z^{3}-17a^{9}z+5a^{9}z^{-1}+3a^{8}z^{8}-19a^{8}z^{6}+39a^{8}z^{4}-36a^{8}z^{2}-4a^{8}z^{-2}+17a^{8}+a^{7}z^{9}-3a^{7}z^{7}-8a^{7}z^{5}+30a^{7}z^{3}-29a^{7}z+9a^{7}z^{-1}+4a^{6}z^{8}-26a^{6}z^{6}+54a^{6}z^{4}-47a^{6}z^{2}-5a^{6}z^{-2}+22a^{6}+a^{5}z^{9}-5a^{5}z^{7}+3a^{5}z^{5}+12a^{5}z^{3}-15a^{5}z+5a^{5}z^{-1}+a^{4}z^{8}-7a^{4}z^{6}+17a^{4}z^{4}-19a^{4}z^{2}-2a^{4}z^{-2}+10a^{4}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ -1                   1 1 -3                     0 -5               3 1   2 -7           1 1 2     2 -9           3 1       2 -11       2 2 1         1 -13     1 4 3           0 -15   1 1 1             1 -17   2 2               0 -19 1 1                 0 -21 1                   -1

Integral Khovanov Homology (db, data source)

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

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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