L11a402

L11a401

L11a403

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Link Presentations

[edit Notes on L11a402's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{u v^2 w^4-2 u v^2 w^3+2 u v^2 w^2-u v^2 w-2 u v w^4+5 u v w^3-7 u v w^2+4 u v w-u v+u w^4-3 u w^3+4 u w^2-3 u w+u-v^2 w^4+3 v^2 w^3-4 v^2 w^2+3 v^2 w-v^2+v w^4-4 v w^3+7 v w^2-5 v w+2 v+w^3-2 w^2+2 w-1}{\sqrt{u} v w^2}$ (db)
Jones polynomial	$-q^5+4 q^4-9 q^3+16 q^2-20 q+25-23 q^{-1} +21 q^{-2} -15 q^{-3} +9 q^{-4} -4 q^{-5} + q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-7 a^2 z^4-3 z^4 a^{-2} +10 z^4+2 a^4 z^2-8 a^2 z^2-3 z^2 a^{-2} +8 z^2+a^4-2 a^2+1+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2}$ (db)
Kauffman polynomial	$a^6 z^6-2 a^6 z^4+a^6 z^2+4 a^5 z^7-9 a^5 z^5+z^5 a^{-5} +6 a^5 z^3-z^3 a^{-5} -2 a^5 z+7 a^4 z^8-15 a^4 z^6+4 z^6 a^{-4} +10 a^4 z^4-5 z^4 a^{-4} -6 a^4 z^2+z^2 a^{-4} +2 a^4+6 a^3 z^9-3 a^3 z^7+8 z^7 a^{-3} -19 a^3 z^5-11 z^5 a^{-3} +21 a^3 z^3+4 z^3 a^{-3} -7 a^3 z-z a^{-3} +2 a^2 z^{10}+17 a^2 z^8+11 z^8 a^{-2} -56 a^2 z^6-20 z^6 a^{-2} +59 a^2 z^4+18 z^4 a^{-2} -28 a^2 z^2-9 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +4 a^2+14 a z^9+8 z^9 a^{-1} -20 a z^7-5 z^7 a^{-1} -9 a z^5-11 z^5 a^{-1} +25 a z^3+15 z^3 a^{-1} -7 a z-3 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +2 z^{10}+21 z^8-64 z^6+70 z^4-31 z^2+2 z^{-2} +3$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   6 1   -5 5                 10 3     7 3               11 7       -4 1             14 9         5 -1           11 13           2 -3         10 12             -2 -5       6 12               6 -7     3 9                 -6 -9   1 6                   5 -11   3                     -3 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{14}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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