L11n415

L11n414

L11n416

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Link Presentations

[edit Notes on L11n415's Link Presentations]

Planar diagram presentation	X8192 X5,15,6,14 X10,3,11,4 X13,5,14,4 X2738 X6,9,1,10 X18,12,19,11 X12,18,7,17 X20,16,21,15 X22,20,13,19 X16,22,17,21
Gauss code	{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, 7, -8}, {-4, 2, 9, -11, 8, -7, 10, -9, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{2}vw^{2}-2u^{2}vw+u^{2}v-u^{2}w^{2}+u^{2}w+2uv^{2}w^{2}-2uv^{2}w+uv^{2}+uvw^{3}-4uvw^{2}+4uvw-uv-uw^{3}+2uw^{2}-2uw-v^{2}w^{2}+v^{2}w-vw^{3}+2vw^{2}-2vw}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+9q^{5}-11q^{4}+12q^{3}-10q^{2}+9q+3q^{-1}-4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-a^{-8}+3z^{2}a^{-6}-a^{-6}z^{-2}+2a^{-6}-2z^{4}a^{-4}-z^{2}a^{-4}+4a^{-4}z^{-2}+4a^{-4}-3z^{4}a^{-2}-8z^{2}a^{-2}-5a^{-2}z^{-2}-10a^{-2}+3z^{2}+2z^{-2}+5$ (db)
Kauffman polynomial	$z^{9}a^{-3}+z^{9}a^{-5}+4z^{8}a^{-2}+7z^{8}a^{-4}+3z^{8}a^{-6}+3z^{7}a^{-1}+7z^{7}a^{-3}+8z^{7}a^{-5}+4z^{7}a^{-7}-14z^{6}a^{-2}-17z^{6}a^{-4}+3z^{6}a^{-8}-9z^{5}a^{-1}-28z^{5}a^{-3}-26z^{5}a^{-5}-6z^{5}a^{-7}+z^{5}a^{-9}+32z^{4}a^{-2}+23z^{4}a^{-4}-9z^{4}a^{-6}-6z^{4}a^{-8}+6z^{4}+18z^{3}a^{-1}+48z^{3}a^{-3}+33z^{3}a^{-5}+z^{3}a^{-7}-2z^{3}a^{-9}-38z^{2}a^{-2}-19z^{2}a^{-4}+6z^{2}a^{-6}+3z^{2}a^{-8}-16z^{2}-19za^{-1}-35za^{-3}-19za^{-5}-2za^{-7}+za^{-9}+22a^{-2}+13a^{-4}-a^{-8}+11+5a^{-1}z^{-1}+9a^{-3}z^{-1}+5a^{-5}z^{-1}+a^{-7}z^{-1}-5a^{-2}z^{-2}-4a^{-4}z^{-2}-a^{-6}z^{-2}-2z^{-2}$ (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 χ 17                   1 -1 15                 2   2 13               4 1   -3 11             5 2     3 9           7 5       -2 7         5 4         1 5       5 7           2 3     4 5             -1 1   1 6               5 -1 2 3                 -1 -3 3                   3

Integral Khovanov Homology (db, data source)

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

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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