L10a124

L10a123

L10a125

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Link Presentations

[edit Notes on L10a124's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,11,19,12 X20,17,9,18 X16,14,17,13 X8,16,5,15 X14,8,15,7 X12,19,13,20 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 7, -6}, {10, -2, 3, -8, 5, -7, 6, -5, 4, -3, 8, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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