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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X4,19,1,20 X5,12,6,13 X8493 X9,16,10,17 X13,22,14,5 X15,10,16,11 X21,14,22,15 X11,20,12,21 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

2q-17/2 - 3q-15/2 + 6q-13/2 - 9q-11/2 + 9q-9/2 - 10q-7/2 + 7q-5/2 - 6q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-28 - 3q-26 - q-24 - 2q-22 - 2q-20 + 3q-18 + q-16 + 4q-14 + 3q-12 + 2q-10 + 4q-8 - q-6 + 2q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - 2a3z-1 - 2a3z + a3z3 + a3z5 + 2a5z-1 + 3a5z + 2a5z3 + a5z5 + a7z-1 - a7z3 - a9z-1

Kauffman Polynomial:

- az + 2az3 - az5 - a2z2 + 6a2z4 - 3a2z6 - 2a3z-1 + 5a3z - 4a3z3 + 8a3z5 - 4a3z7 + 2a4 - 3a4z2 + a4z4 + 4a4z6 - 3a4z8 - 2a5z-1 + 11a5z - 22a5z3 + 16a5z5 - 4a5z7 - a5z9 - 4a6 + 11a6z2 - 20a6z4 + 14a6z6 - 5a6z8 + a7z-1 + 7a7z - 17a7z3 + 8a7z5 - a7z7 - a7z9 - 9a8 + 21a8z2 - 18a8z4 + 7a8z6 - 2a8z8 + a9z-1 + 2a9z - a9z3 + a9z5 - a9z7 - 4a10 + 8a10z2 - 3a10z4

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                   1 j = 0                 2   j = -2               4 1   j = -4             5 4     j = -6           5 2       j = -8         4 5         j = -10       5 5           j = -12     1 4             j = -14   2 5               j = -16   1                 j = -18 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, 31]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 31]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[4, 19, 1, 20], X[5, 12, 6, 13], > X[8, 4, 9, 3], X[9, 16, 10, 17], X[13, 22, 14, 5], X[15, 10, 16, 11], > X[21, 14, 22, 15], X[11, 20, 12, 21], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 8, -10, 4, -7, 9, -8, 6, 11, -2, > 3, 10, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 6 9 9 10 7 6 3 ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - Sqrt[q] 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
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 3 -24 2 2 3 -16 4 3 2 4 -6 -1 - q - --- - q - --- - --- + --- + q + --- + --- + --- + -- - q + 26 22 20 18 14 12 10 8 q q q q q q q q 2 2 > -- + q 4 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 31]][a, z]
Out[8]=	3 5 7 9 -2 a 2 a a a 3 5 3 3 3 5 3 ----- + ---- + -- - -- - a z - 2 a z + 3 a z - a z + a z + 2 a z - z z z z 7 3 3 5 5 5 > a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 31]][a, z]
Out[9]=	3 5 7 9 4 6 8 10 2 a 2 a a a 3 5 2 a - 4 a - 9 a - 4 a - ---- - ---- + -- + -- - a z + 5 a z + 11 a z + z z z z 7 9 2 2 4 2 6 2 8 2 10 2 > 7 a z + 2 a z - a z - 3 a z + 11 a z + 21 a z + 8 a z + 3 3 3 5 3 7 3 9 3 2 4 4 4 > 2 a z - 4 a z - 22 a z - 17 a z - a z + 6 a z + a z - 6 4 8 4 10 4 5 3 5 5 5 7 5 > 20 a z - 18 a z - 3 a z - a z + 8 a z + 16 a z + 8 a z + 9 5 2 6 4 6 6 6 8 6 3 7 5 7 > a z - 3 a z + 4 a z + 14 a z + 7 a z - 4 a z - 4 a z - 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > a z - a z - 3 a z - 5 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 4 2 1 2 5 1 4 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 5 4 5 5 2 5 t 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 2 t + -- + q t 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q

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