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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X7,14,8,15 X18,11,19,12 X22,19,5,20 X20,15,21,16 X16,21,17,22 X12,17,13,18 X13,8,14,9 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

2q-23/2 - 5q-21/2 + 9q-19/2 - 12q-17/2 + 13q-15/2 - 13q-13/2 + 10q-11/2 - 8q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- 2q-36 - 2q-34 + q-32 - 3q-30 + q-28 + q-26 + 5q-22 + 5q-18 + q-16 + 3q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial:

- a5z - 2a5z3 - a5z5 - 3a7z-1 - 10a7z - 10a7z3 - 3a7z5 + 5a9z-1 + 11a9z + 5a9z3 - 2a11z-1 - 2a11z

Kauffman Polynomial:

- a5z + 2a5z3 - a5z5 - a6z2 + 4a6z4 - 3a6z6 - 3a7z-1 + 11a7z - 14a7z3 + 13a7z5 - 6a7z7 + 5a8 - 15a8z2 + 16a8z4 - a8z6 - 4a8z8 - 5a9z-1 + 17a9z - 26a9z3 + 27a9z5 - 11a9z7 - a9z9 + 5a10 - 17a10z2 + 16a10z4 + a10z6 - 6a10z8 - 2a11z-1 + 4a11z - 6a11z3 + 10a11z5 - 6a11z7 - a11z9 + a12z2 + a12z4 - a12z6 - 2a12z8 - a13z + 4a13z3 - 3a13z5 - a13z7 - a14 + 4a14z2 - 3a14z4

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                   1 j = -6                 3 1 j = -8               5     j = -10             5 3     j = -12           8 5       j = -14         6 6         j = -16       6 7           j = -18     3 6             j = -20   2 6               j = -22   3                 j = -24 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, 58]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 58]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[7, 14, 8, 15], X[18, 11, 19, 12], > X[22, 19, 5, 20], X[20, 15, 21, 16], X[16, 21, 17, 22], X[12, 17, 13, 18], > X[13, 8, 14, 9], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 4, -8, -9, 3, 6, -7, 8, -4, > 5, -6, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	2 5 9 12 13 13 10 8 3 -(5/2) ----- - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - q 23/2 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 2 -32 3 -28 -26 5 5 -16 3 2 -8 --- - --- + q - --- + q + q + --- + --- + q + --- - --- + q 36 34 30 22 18 12 10 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 58]][a, z]
Out[8]=	7 9 11 -3 a 5 a 2 a 5 7 9 11 5 3 ----- + ---- - ----- - a z - 10 a z + 11 a z - 2 a z - 2 a z - z z z 7 3 9 3 5 5 7 5 > 10 a z + 5 a z - a z - 3 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 58]][a, z]
Out[9]=	7 9 11 8 10 14 3 a 5 a 2 a 5 7 9 11 5 a + 5 a - a - ---- - ---- - ----- - a z + 11 a z + 17 a z + 4 a z - z z z 13 6 2 8 2 10 2 12 2 14 2 5 3 > a z - a z - 15 a z - 17 a z + a z + 4 a z + 2 a z - 7 3 9 3 11 3 13 3 6 4 8 4 > 14 a z - 26 a z - 6 a z + 4 a z + 4 a z + 16 a z + 10 4 12 4 14 4 5 5 7 5 9 5 11 5 > 16 a z + a z - 3 a z - a z + 13 a z + 27 a z + 10 a z - 13 5 6 6 8 6 10 6 12 6 7 7 9 7 > 3 a z - 3 a z - a z + a z - a z - 6 a z - 11 a z - 11 7 13 7 8 8 10 8 12 8 9 9 11 9 > 6 a z - a z - 4 a z - 6 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 2 3 2 6 3 6 6 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q t q t q t q t q t q t q t 7 6 6 8 5 5 3 5 3 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ---- 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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