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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X18,9,19,10 X8,17,9,18 X19,1,20,4 X5,14,6,15 X3,10,4,11 X11,20,12,21 X13,22,14,5 X21,12,22,13 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-23/2 - 2q-21/2 + 5q-19/2 - 7q-17/2 + 8q-15/2 - 8q-13/2 + 7q-11/2 - 6q-9/2 + 2q-7/2 - 2q-5/2

A2 (sl(3)) Invariant:

- q-34 - 3q-30 - 2q-28 - q-26 - 2q-24 + 2q-22 - q-20 + 3q-18 + 3q-16 + 3q-14 + 5q-12 + q-10 + 2q-8

HOMFLY-PT Polynomial:

- 4a5z-1 - 12a5z - 9a5z3 - 2a5z5 + 7a7z-1 + 18a7z + 15a7z3 + 6a7z5 + a7z7 - 3a9z-1 - 6a9z - 4a9z3 - a9z5

Kauffman Polynomial:

4a5z-1 - 14a5z + 12a5z3 - 3a5z5 - 7a6 + 17a6z2 - 11a6z4 + 4a6z6 - a6z8 + 7a7z-1 - 22a7z + 27a7z3 - 14a7z5 + 4a7z7 - a7z9 - 7a8 + 19a8z2 - 15a8z4 + 8a8z6 - 3a8z8 + 3a9z-1 - 8a9z + 11a9z3 - 6a9z5 + a9z7 - a9z9 - a10z2 + a10z6 - 2a10z8 - 2a11z3 + 3a11z5 - 3a11z7 - a12z2 + 3a12z4 - 3a12z6 + 2a13z3 - 2a13z5 - a14 + 2a14z2 - a14z4

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                   2 j = -6                 2 2 j = -8               4     j = -10             3 2     j = -12           5 4       j = -14         3 3         j = -16       4 5           j = -18     1 3             j = -20   1 4               j = -22   1                 j = -24 1

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, 80]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 80]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[18, 9, 19, 10], X[8, 17, 9, 18], > X[19, 1, 20, 4], X[5, 14, 6, 15], X[3, 10, 4, 11], X[11, 20, 12, 21], > X[13, 22, 14, 5], X[21, 12, 22, 13], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, -7, 5}, {-6, -1, 2, -4, 3, 7, -8, 10, -9, 6, 11, -2, 4, -3, > -5, 8, -10, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 2 5 7 8 8 7 6 2 2 q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 3 2 -26 2 2 -20 3 3 3 5 -10 2 -q - --- - --- - q - --- + --- - q + --- + --- + --- + --- + q + -- 30 28 24 22 18 16 14 12 8 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 80]][a, z]
Out[8]=	5 7 9 -4 a 7 a 3 a 5 7 9 5 3 7 3 ----- + ---- - ---- - 12 a z + 18 a z - 6 a z - 9 a z + 15 a z - z z z 9 3 5 5 7 5 9 5 7 7 > 4 a z - 2 a z + 6 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 80]][a, z]
Out[9]=	5 7 9 6 8 14 4 a 7 a 3 a 5 7 9 -7 a - 7 a - a + ---- + ---- + ---- - 14 a z - 22 a z - 8 a z + z z z 6 2 8 2 10 2 12 2 14 2 5 3 7 3 > 17 a z + 19 a z - a z - a z + 2 a z + 12 a z + 27 a z + 9 3 11 3 13 3 6 4 8 4 12 4 14 4 > 11 a z - 2 a z + 2 a z - 11 a z - 15 a z + 3 a z - a z - 5 5 7 5 9 5 11 5 13 5 6 6 8 6 > 3 a z - 14 a z - 6 a z + 3 a z - 2 a z + 4 a z + 8 a z + 10 6 12 6 7 7 9 7 11 7 6 8 8 8 > a z - 3 a z + 4 a z + a z - 3 a z - a z - 3 a z - 10 8 7 9 9 9 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 4 1 3 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q q q t q t q t q t q t q t q t 5 3 3 5 4 3 2 4 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ---- 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 6 q t q t q t q t q t q t q t q t q t

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L11n79 L11n81