L11a409

L11a408

L11a410

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Link Presentations

[edit Notes on L11a409's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X8,18,9,17 X14,8,15,7 X18,10,19,9 X10,12,5,11 X22,19,11,20 X20,15,21,16 X16,21,17,22 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -3, 5, -6}, {6, -2, 11, -4, 8, -9, 3, -5, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{u v^2 w^4-2 u v^2 w^3+2 u v^2 w^2-u v w^4+4 u v w^3-5 u v w^2+2 u v w-2 u w^3+4 u w^2-3 u w+u-v^2 w^4+3 v^2 w^3-4 v^2 w^2+2 v^2 w-2 v w^3+5 v w^2-4 v w+v-2 w^2+2 w-1}{\sqrt{u} v w^2}$ (db)
Jones polynomial	$q^7-3 q^6+7 q^5-11 q^4- q^{-4} +16 q^3+3 q^{-3} -16 q^2-7 q^{-2} +18 q+11 q^{-1} -14$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^8 a^{-2} -6 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-15 z^4 a^{-2} +4 z^4 a^{-4} +9 z^4-3 a^2 z^2-19 z^2 a^{-2} +6 z^2 a^{-4} +15 z^2-3 a^2-13 a^{-2} +4 a^{-4} +12-a^2 z^{-2} -5 a^{-2} z^{-2} +2 a^{-4} z^{-2} +4 z^{-2}$ (db)
Kauffman polynomial	$z^4 a^{-8} -z^2 a^{-8} +3 z^5 a^{-7} -2 z^3 a^{-7} +6 z^6 a^{-6} -7 z^4 a^{-6} +5 z^2 a^{-6} - a^{-6} +8 z^7 a^{-5} -10 z^5 a^{-5} +5 z^3 a^{-5} +9 z^8 a^{-4} -18 z^6 a^{-4} +21 z^4 a^{-4} -17 z^2 a^{-4} -2 a^{-4} z^{-2} +8 a^{-4} +5 z^9 a^{-3} +a^3 z^7+z^7 a^{-3} -4 a^3 z^5-26 z^5 a^{-3} +6 a^3 z^3+33 z^3 a^{-3} -4 a^3 z-18 z a^{-3} +a^3 z^{-1} +5 a^{-3} z^{-1} +z^{10} a^{-2} +3 a^2 z^8+16 z^8 a^{-2} -11 a^2 z^6-58 z^6 a^{-2} +13 a^2 z^4+72 z^4 a^{-2} -7 a^2 z^2-50 z^2 a^{-2} -a^2 z^{-2} -5 a^{-2} z^{-2} +3 a^2+20 a^{-2} +3 a z^9+8 z^9 a^{-1} -4 a z^7-12 z^7 a^{-1} -15 a z^5-24 z^5 a^{-1} +30 a z^3+50 z^3 a^{-1} -19 a z-33 z a^{-1} +5 a z^{-1} +9 a^{-1} z^{-1} +z^{10}+10 z^8-45 z^6+56 z^4-34 z^2-4 z^{-2} +15$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3 1 -2 11                   4     4 9                 7 3     -4 7               9 4       5 5             8 8         0 3           10 8           2 1         6 10             4 -1       5 8               -3 -3     2 6                 4 -5   1 5                   -4 -7   2                     2 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}$ ${\mathbb Z}$

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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