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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X20,9,21,10 X8,19,9,20 X4,21,1,22 X5,14,6,15 X10,4,11,3 X11,16,12,17 X15,12,16,13 X13,22,14,5 X2,18,3,17

Gauss Code:

{{1, -11, 7, -5}, {-6, -1, 2, -4, 3, -7, -8, 9, -10, 6, -9, 8, 11, -2, 4, -3, 5, 10}}

Jones Polynomial:

2q-19/2 - 4q-17/2 + 6q-15/2 - 7q-13/2 + 8q-11/2 - 8q-9/2 + 5q-7/2 - 5q-5/2 + 2q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-30 - 2q-28 - 2q-24 - q-20 - q-18 + 3q-16 + q-14 + 5q-12 + 2q-10 + 2q-8 + 2q-6 + q-2

HOMFLY-PT Polynomial:

- 2a3z-1 - 5a3z - 4a3z3 - a3z5 + 2a5z-1 + 5a5z + 8a5z3 + 5a5z5 + a5z7 + a7z-1 + a7z - 2a7z3 - a7z5 - a9z-1 - a9z

Kauffman Polynomial:

- 2a3z-1 + 7a3z - 9a3z3 + 5a3z5 - a3z7 + 3a4 - 3a4z2 - 7a4z4 + 8a4z6 - 2a4z8 - 2a5z-1 + 10a5z - 25a5z3 + 17a5z5 - a5z7 - a5z9 + 2a6z2 - 11a6z4 + 16a6z6 - 5a6z8 + a7z-1 + 4a7z - 19a7z3 + 21a7z5 - 4a7z7 - a7z9 - 3a8 + 6a8z2 - a8z4 + 5a8z6 - 3a8z8 + a9z-1 + 2a9z - 6a9z3 + 8a9z5 - 4a9z7 - 2a10z2 + 3a10z4 - 3a10z6 + a11z - 3a11z3 - a11z5 + a12 - 3a12z2

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                   1 j = -2                 1   j = -4               4 1   j = -6             3 3     j = -8           5 2       j = -10         3 3         j = -12       4 5           j = -14     2 3             j = -16   2 4               j = -18   2                 j = -20 2

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 81]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 81]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[20, 9, 21, 10], X[8, 19, 9, 20], > X[4, 21, 1, 22], X[5, 14, 6, 15], X[10, 4, 11, 3], X[11, 16, 12, 17], > X[15, 12, 16, 13], X[13, 22, 14, 5], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 7, -5}, {-6, -1, 2, -4, 3, -7, -8, 9, -10, 6, -9, 8, 11, -2, > 4, -3, 5, 10}]
In[6]:=	Jones[L][q]
Out[6]=	2 4 6 7 8 8 5 5 2 1 ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 2 2 -20 -18 3 -14 5 2 2 2 -2 -q - --- - --- - q - q + --- + q + --- + --- + -- + -- + q 28 24 16 12 10 8 6 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 81]][a, z]
Out[8]=	3 5 7 9 -2 a 2 a a a 3 5 7 9 3 3 5 3 ----- + ---- + -- - -- - 5 a z + 5 a z + a z - a z - 4 a z + 8 a z - z z z z 7 3 3 5 5 5 7 5 5 7 > 2 a z - a z + 5 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 81]][a, z]
Out[9]=	3 5 7 9 4 8 12 2 a 2 a a a 3 5 7 3 a - 3 a + a - ---- - ---- + -- + -- + 7 a z + 10 a z + 4 a z + z z z z 9 11 4 2 6 2 8 2 10 2 12 2 > 2 a z + a z - 3 a z + 2 a z + 6 a z - 2 a z - 3 a z - 3 3 5 3 7 3 9 3 11 3 4 4 6 4 > 9 a z - 25 a z - 19 a z - 6 a z - 3 a z - 7 a z - 11 a z - 8 4 10 4 3 5 5 5 7 5 9 5 11 5 > a z + 3 a z + 5 a z + 17 a z + 21 a z + 8 a z - a z + 4 6 6 6 8 6 10 6 3 7 5 7 7 7 > 8 a z + 16 a z + 5 a z - 3 a z - a z - a z - 4 a z - 9 7 4 8 6 8 8 8 5 9 7 9 > 4 a z - 2 a z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 2 2 2 4 2 3 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 20 7 18 6 16 6 16 5 14 5 14 4 12 4 q q q t q t q t q t q t q t q t 5 3 3 5 2 3 t t 2 > ------ + ------ + ------ + ----- + ---- + ---- + -- + -- + t 12 3 10 3 10 2 8 2 8 6 4 2 q t q t q t q t q t q t q q

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