L11n261

L11n260

L11n262

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Link Presentations

[edit Notes on L11n261's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,12,19,11 X22,20,9,19 X20,16,21,15 X16,22,17,21 X12,18,13,17 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -9, -4, 3, 7, -8, 9, -5, 6, -7, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uvw^{3}-uvw+uv-uw^{3}+2uw^{2}-2uw+u-vw^{3}+2vw^{2}-2vw+v-w^{3}+w^{2}-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{10}+q^{9}-q^{8}-q^{7}+3q^{6}-3q^{5}+5q^{4}-3q^{3}+5q^{2}-2q+1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-a^{-10}z^{-2}-a^{-10}+2z^{2}a^{-8}+3a^{-8}z^{-2}+5a^{-8}-2z^{2}a^{-6}-2a^{-6}z^{-2}-5a^{-6}-z^{6}a^{-4}-4z^{4}a^{-4}-4z^{2}a^{-4}-a^{-4}z^{-2}-2a^{-4}+z^{4}a^{-2}+3z^{2}a^{-2}+a^{-2}z^{-2}+3a^{-2}$ (db)
Kauffman polynomial	$z^{8}a^{-4}+z^{8}a^{-6}+z^{8}a^{-8}+z^{8}a^{-10}+2z^{7}a^{-3}+4z^{7}a^{-5}+3z^{7}a^{-7}+2z^{7}a^{-9}+z^{7}a^{-11}+z^{6}a^{-2}+z^{6}a^{-4}-2z^{6}a^{-6}-8z^{6}a^{-8}-6z^{6}a^{-10}-6z^{5}a^{-3}-15z^{5}a^{-5}-20z^{5}a^{-7}-17z^{5}a^{-9}-6z^{5}a^{-11}-4z^{4}a^{-2}-14z^{4}a^{-4}-3z^{4}a^{-6}+15z^{4}a^{-8}+8z^{4}a^{-10}+z^{3}a^{-3}+16z^{3}a^{-5}+42z^{3}a^{-7}+37z^{3}a^{-9}+10z^{3}a^{-11}+6z^{2}a^{-2}+12z^{2}a^{-4}+2z^{2}a^{-6}-7z^{2}a^{-8}-3z^{2}a^{-10}+4za^{-3}-10za^{-5}-34za^{-7}-27za^{-9}-7za^{-11}-4a^{-2}-3a^{-4}+4a^{-6}+5a^{-8}+a^{-10}-2a^{-3}z^{-1}+2a^{-5}z^{-1}+10a^{-7}z^{-1}+8a^{-9}z^{-1}+2a^{-11}z^{-1}+a^{-2}z^{-2}+a^{-4}z^{-2}-2a^{-6}z^{-2}-3a^{-8}z^{-2}-a^{-10}z^{-2}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 21                       1 -1 19                         0 17                   1 1   0 15               3 1       -2 13             1   1       2 11           4 5 1         0 9         2 1 1           2 7       1 4 1             2 5     4 2                 2 3   1 4                   3 1   1                     -1 -1 1                       1

Integral Khovanov Homology (db, data source)

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

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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