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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X10,13,5,14 X20,15,21,16 X14,7,15,8 X22,17,11,18 X16,21,17,22 X8,20,9,19 X18,10,19,9 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-11 + 3q-10 - 7q-9 + 11q-8 - 15q-7 + 18q-6 - 16q-5 + 16q-4 - 10q-3 + 7q-2 - 3q-1 + 1

A2 (sl(3)) Invariant:

- q-34 - q-32 - 4q-28 - q-22 + 6q-20 + 2q-18 + 8q-16 + 6q-14 + 4q-12 + 7q-10 - q-8 + 2q-6 - q-2 + 1

HOMFLY-PT Polynomial:

2a2z2 + a2z4 + 2a4z-2 + 7a4 + 5a4z2 - a4z4 - a4z6 - 5a6z-2 - 14a6 - 14a6z2 - 8a6z4 - 2a6z6 + 4a8z-2 + 9a8 + 9a8z2 + 3a8z4 - a10z-2 - 2a10 - a10z2

Kauffman Polynomial:

2a2z2 - 3a2z4 + a2z6 + 4a3z3 - 8a3z5 + 3a3z7 - 2a4z-2 + 9a4 - 18a4z2 + 18a4z4 - 15a4z6 + 5a4z8 + 5a5z-1 - 13a5z + 13a5z3 - 5a5z5 - 6a5z7 + 4a5z9 - 5a6z-2 + 21a6 - 57a6z2 + 77a6z4 - 50a6z6 + 12a6z8 + a6z10 + 9a7z-1 - 29a7z + 36a7z3 - 6a7z5 - 14a7z7 + 8a7z9 - 4a8z-2 + 18a8 - 48a8z2 + 72a8z4 - 48a8z6 + 13a8z8 + a8z10 + 5a9z-1 - 21a9z + 33a9z3 - 18a9z5 + 4a9z9 - a10z-2 + 5a10 - 10a10z2 + 11a10z4 - 11a10z6 + 6a10z8 + a11z-1 - 4a11z + 4a11z3 - 8a11z5 + 5a11z7 + a12z2 - 5a12z4 + 3a12z6 + a13z - 2a13z3 + a13z5

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                       1 j = -1                     2   j = -3                   5 1   j = -5                 6 3     j = -7               10 4       j = -9             8 8         j = -11           10 8           j = -13         6 9             j = -15       5 9               j = -17     2 6                 j = -19   1 5                   j = -21   2                     j = -23 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 405]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 405]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[10, 13, 5, 14], X[20, 15, 21, 16], > X[14, 7, 15, 8], X[22, 17, 11, 18], X[16, 21, 17, 22], X[8, 20, 9, 19], > X[18, 10, 19, 9], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 5, -8, 9, -3}, > {11, -2, 3, -5, 4, -7, 6, -9, 8, -4, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-11 3 7 11 15 18 16 16 10 7 3 1 - q + --- - -- + -- - -- + -- - -- + -- - -- + -- - - 10 9 8 7 6 5 4 3 2 q q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -32 4 -22 6 2 8 6 4 7 -8 2 1 - q - q - --- - q + --- + --- + --- + --- + --- + --- - q + -- - 28 20 18 16 14 12 10 6 q q q q q q q q -2 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 405]][a, z]
Out[8]=	4 6 8 10 4 6 8 10 2 a 5 a 4 a a 2 2 4 2 7 a - 14 a + 9 a - 2 a + ---- - ---- + ---- - --- + 2 a z + 5 a z - 2 2 2 2 z z z z 6 2 8 2 10 2 2 4 4 4 6 4 8 4 4 6 > 14 a z + 9 a z - a z + a z - a z - 8 a z + 3 a z - a z - 6 6 > 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 405]][a, z]
Out[9]=	4 6 8 10 5 7 9 4 6 8 10 2 a 5 a 4 a a 5 a 9 a 5 a 9 a + 21 a + 18 a + 5 a - ---- - ---- - ---- - --- + ---- + ---- + ---- + 2 2 2 2 z z z z z z z 11 a 5 7 9 11 13 2 2 4 2 > --- - 13 a z - 29 a z - 21 a z - 4 a z + a z + 2 a z - 18 a z - z 6 2 8 2 10 2 12 2 3 3 5 3 7 3 > 57 a z - 48 a z - 10 a z + a z + 4 a z + 13 a z + 36 a z + 9 3 11 3 13 3 2 4 4 4 6 4 8 4 > 33 a z + 4 a z - 2 a z - 3 a z + 18 a z + 77 a z + 72 a z + 10 4 12 4 3 5 5 5 7 5 9 5 11 5 > 11 a z - 5 a z - 8 a z - 5 a z - 6 a z - 18 a z - 8 a z + 13 5 2 6 4 6 6 6 8 6 10 6 12 6 > a z + a z - 15 a z - 50 a z - 48 a z - 11 a z + 3 a z + 3 7 5 7 7 7 11 7 4 8 6 8 8 8 > 3 a z - 6 a z - 14 a z + 5 a z + 5 a z + 12 a z + 13 a z + 10 8 5 9 7 9 9 9 6 10 8 10 > 6 a z + 4 a z + 8 a z + 4 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 2 1 5 2 6 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q q q t q t q t q t q t q t q t 9 6 9 10 8 8 8 10 4 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t q t 6 t 2 t 2 > ---- + -- + --- + q t 5 3 q q t q

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